Research ArticleLie Group Solution for Free Convective Flow of a Nanofluid Pasta Chemically Reacting Horizontal Plate in a Porous Media
M M Rashidi12 E Momoniat3 M Ferdows4 and A Basiriparsa15
1 Mechanical Engineering Department Engineering Faculty of Bu-Ali Sina University Hamedan 65178-38695 Iran2Mechanical Engineering Department University of Michigan-Shanghai Jiao Tong University Joint InstituteShanghai Jiao Tong University Shanghai 201101 China
3 Centre for Differential Equations Continuum Mechanics and Applications School of Computational and Applied MathematicsUniversity of the Witwatersrand Johannesburg Private Bag 3 Wits 2050 South Africa
4Department of Mathematics Dhaka University Dhaka 1000 Bangladesh5 Young Researchers and Elites Club Hamadan Branch Islamic Azad University Hamadan 65178-38695 Iran
Correspondence should be addressed to M M Rashidi mm rashidiyahoocom
Received 13 August 2013 Accepted 4 November 2013 Published 11 February 2014
Academic Editor Mohamed Abd El Aziz
Copyright copy 2014 M M Rashidi et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The optimal homotopy analysis method (OHAM) is employed to investigate the steady laminar incompressible free convectiveflow of a nanofluid past a chemically reacting upward facing horizontal plate in a porous medium taking into account heatgenerationabsorption and the thermal slip boundary condition Using similarity transformations developed by Lie group analysisthe continuity momentum energy and nanoparticle volume fraction equations are transformed into a set of coupled similarityequations The OHAM solutions are obtained and verified by numerical results using a Runge-Kutta-Fehlberg fourth-fifth ordermethod The effect of the emerging flow controlling parameters on the dimensionless velocity temperature and nanoparticlevolume fraction have been presented graphically and discussed Good agreement is found between analytical and numericalresults of the present paper with published results This close agreement supports our analysis and the accuracy of the numericalcomputations This paper also includes a representative set of numerical results for reduced Nusselt and Sherwood numbers in atable for various values of the parameters It is concluded that the reduced Nusselt number increases with the Lewis number andreaction parameter whist it decreases with the order of the chemical reaction thermal slip and generation parameters
1 Introduction
Research in micro- and nanofluids has become a populararea of research in engineering At micro- and nanoscaleconventional ideas of classical fluid mechanics do not applyand traditional approaches to fluidmechanics problems needto be changed to correctly reflect the importance of theinteraction between a fluid and a solid boundary Conven-tional heat transfer fluids like oil water and ethylene glycolmixtures are poor heat transfer fluids because of their poorthermal conductivity Many attempts have been made byvarious investigators during the recent years to enhancethe thermal conductivity of these fluids by suspendingnanomicroparticles in liquids [1 2] Researchers haveobserved that the thermal conductivity of a nanofluid is
much higher than that of the base fluid even for low solidvolume fraction of nanoparticles in the mixture [3ndash5] Theeffect of temperature on thermal conductivity in a model hasbeen considered by Kumar et al [6] Patel et al [7] haveimproved the model given in [6] by incorporating the effectof microconvection due to particle movement
Nano- and microfluidics is a new area with significantpotential for novel engineering applications especially for thedevelopment of new biomedical devices and procedures [8]Napoli et al [9] reviewed applications of nanofluidic phe-nomena to various nanofabricated devices in particular onesdesigned for biomolecule transport and manipulation Therehas been significant interest in nanofluidsThis interest is dueto its diverse applications ranging from laser-assisted drugdelivery to electronic chip cooling Nanofluids are made of
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 239082 21 pageshttpdxdoiorg1011552014239082
2 Mathematical Problems in Engineering
ultrafine nanoparticles (lt100 nm) suspended in a base fluidwhich can be water or an organic solvent Nanofluids possesssuperior thermophysical properties like high thermal con-ductivity minimal clogging in flow passages long term sta-bility and homogeneity Industrial applications of nanofluidinclude electronics automotive and nuclear applicationsNanobiotechnology is also a fast developing field of researchwith application in many domains such as medicine phar-macy cosmetics and agroindustry Many of these industrialprocesses involve nanofluid flow and nanoparticle volumefraction past various geometries In these applications thediffusing species can be generatedabsorbed due to chemicalreaction with the ambient fluid This can greatly affect theflow and hence the properties and quality of the final product[10 11]
Different industrial applications of internal heat gen-eration include polymer production and the manufactureof ceramics or glassware phase change processes thermalcombustion processes and the development of a metal wastefrom spent nuclear fuel [12] A review of convective transportin nanofluids was conducted by Buongiorno [13] Kuznetsovand Nield [14] presented a similarity solution of naturalconvective boundary-layer flow of a nanofluid past a verticalplate They have shown that the reduced Nusselt number isa decreasing function of the buoyancy-ratio number 119873119903 aBrownianmotion number119873119887 and a thermophoresis number119873119905 Godson et al [15] presented the recent experimental andtheoretical studies on convective heat transfer in nanofluidsand their thermophysical properties and applications andclarified the challenges and opportunities for future research
Convective flow in porous media has received the atten-tion of researchers over the last several decades due to itsmany applications in mechanical chemical and civil engi-neering Examples include fibrous insulation food processingand storage thermal insulation of buildings geophysical sys-tems electrochemistry metallurgy the design of pebble bednuclear reactors underground disposal of nuclear or nonnu-clear waste and cooling system of electronic devices Excel-lent reviews of the fundamental theoretical and experimentalworks can be found in the books by Nield and Bejan [16]Vadasz [17] Vafai [18] The Cheng-Minkowycz problem [19]was investigated by Nield and Kuznetsov [20] for nanofluidwhere the model involves the effect of Brownian motionand thermophoresis The classical problem of free convectiveflow in a porous medium near a horizontal flat plate wasfirst investigated by Cheng and Chang [21] Following himmany researchers such as Chang and Cheng [22] Shiunlinand Gebhart [23] Merkin and Zhang [24] and Chaudhary etal [25] have extended the problem in various aspects Gorlaand Chamkha [26] presented a similarity analysis of freeconvective flow of nanofluid past a horizontal upward facingplate in a porous medium numerically Khan and Pop [27]extended this problem for nanofluid Very recently Aziz et al[28] extended the same problem for a water-based nanofluidcontaining gyrotactic microorganisms
Lie group analysis has been used by many investigatorsto analyze various convective phenomena under variousflow configurations arising in fluidmechanics aerodynamicsplasma physics meteorology chemical engineering and
other engineering branches [29] This method has beenapplied by many investigators to study various transportproblems For example the symmetrical properties of theturbulent boundary-layer flows and other turbulent flowsare investigated by using the Lie group techniques byAvramenko et al [30] Kuznetsov et al [31] investigateda falling bioconvection plume in a deep chamber filledwith a fluid saturated porous medium theoretically Jalil etal [32] studied mixed convective flow with mass transferusing Lie group analysis The effect of thermal radiation andconvective surface boundary condition on the boundary-layer flow was investigated by Hamad et al [33] Aziz et al[34] studied MHD flow over an inclined radiating plate withthe temperature dependent thermal conductivity variablereactive index and heat generation using scaling group oftransformations Reviews for the fundamental theory andapplications of group theory to differential equations can befound in the texts by Hansen [35] Ames [36] Seshadri andNa [37] and Shang [38]
Most scientific problems and phenomena such as theboundary-layer problem occur nonlinearly For these non-linear problems we have difficulty in finding their exactanalytical solutions Analytical solutions to these nonlinearequations are of fundamental importance Where no analyti-cal solutions can be found researchers have resorted to otherapproaches One such approach is a perturbation method[39] that is strongly dependent upon the so-called ldquosmallparametersrdquo The perturbation method cannot provide uswith a simple way to adjust and control the convergenceregion and rate of convergence of a given approximate series
Another known method is the differential transformmethod that has been used in recent years [40ndash44] In 1992Liao introduced the basic ideas of the homotopy in topologyto propose a general analytic method for nonlinear problemsnamely homotopy analysismethod (HAM) [45] that does notneed any small parameterThis method has been successfullyapplied to solve many types of nonlinear problems by others[46ndash48] As this approach is based on the homotopy oftopology the validity of the HAM is independent of whetheror not there exists a small parameter in the consideredequation Therefore the HAM can overcome the foregoingrestrictions and limitations of perturbation methods [49]This method also provides us with great freedom to selectproper base functions to approximate solutions of nonlinearproblems Using one interesting property of homotopy onecan transform any nonlinear problem into an infinite numberof linear problems
In this paper the steady flow of an Ostwald-de Waelepower-law fluid induced by a steadily rotating infinite disk toa non-Darcian fluid-saturated porous medium is consideredThe coupled governing equations are transformed into ordi-nary differential equations in the boundary layerTheOHAMis applied to solve the ODEs The validity of our solutionsis verified by the numerical results (by using a fourth-orderRunge-Kutta and shooting method)
The aim of the present study is to investigate the effect ofhigher order chemical reaction internal heat generation andthe thermal slip boundary condition on the boundary-layerflow of a nanofluid past an upward facing horizontal plate
Lie group analysis is used to develop the similarity transfor-mations and the corresponding similarity representations ofthe governing equations The coupled governing equationsare transformed into ordinary differential equations in theboundary layer The OHAM is applied to solve the ODEsThe obtained solutions are verified by the numerical results(obtained by using a Runge-Kutta-Fehlberg fourth-fifth orderand shooting method) The effect of relevant parameters ondimensionless fluid velocity temperature and nanoparticlevolume fraction are investigated and shown graphically anddiscussed A table containing data for the reduced Nusseltnumber and reduced Sherwood number is also provided toshow the effects of various parameters on them To the best ofour knowledge the effects of thermal slip boundary conditionwith internal heat generation and chemical reaction on theboundary-layer flow of a nanofluid past a horizontal plate inporous media have not been reported in the literature yet
The paper is divided up as follows In Section 2 themath-ematical formulation is presented In Section 3 we used theLie group method to reduce the system of partial differentialequations to a system of ordinary differential equations InSection 4 the basic idea of theHAM is presented In Section 5we derived the OHAM solution of the coupled system ofnonlinear ordinary differential equations In Section 6 wecompared our results with numerical solutions obtainedusing a Runge-Kutta-Fehlberg method In Section 7 weintroduced the physical quantities to be considered andcompared in this paper Section 8 contains the results anddiscussion The conclusions are summarized in Section 9
2 Formulation of the Problem
We consider a two-dimensional laminar free convectiveboundary-layer flow of a nanofluid past an upward facingchemically reacting horizontal plate in a porous media(Figure 1) We assume that a homogeneous isothermal irre-versible chemical reaction of order 119899 takes place betweenthe plate and nanofluid There is internal heat genera-tionabsorption within the fluid inside the boundary layer at
the volumetric rate 119902 Variation of density of the fluid is takeninto account using the Oberbeck-Boussinesq approximationThe conservation of mass momentum energy and nanopar-ticles describing the flow can be written in dimensional form(see [27])
nabla sdot = 0 (1)
120588119891
120576
120597
120597119905
= minus nabla119875 minus
120583
119896
+ [119862120588119875 + (1 minus 119862) 120588119891 (1 minus 120573 (119879 minus 119879infin))] 119892
minus 119896 (119909) (119862 minus 119862infin)119899
(4)
Theflow is assumed to be slow to ignore an advective termand a Forchheimer quadratic drag term in the momentumequation
We consider a steady flow where the Oberbeck-Boussinesq approximation is used In addition we assumethat the nanoparticle concentration is dilute With a suitablechoice for the reference pressure the momentum equationcan be linearized and (2) written as (see [50])
minus nabla119875 minus
120583
119896
+ [(120588119875 minus 120588119891infin) (119862 minus 119862infin)
+ (1 minus 119862infin) 120588119891infin120573 (119879 minus 119879infin)] 119892 = 0
(5)
We also consider the effect of temperature-dependentvolumetric heat generationabsorption in the flow region thatis given by Vajravelu and Hadjinicolaou [51] as
119902 =
Ra2311987601198714311990943
(119879 minus 119879infin) 119879 gt 119879infin(6)
where1198760 is the heat generationabsorption constant Also weconsider the case where the reaction rate varies as
119896 (119909) =
Ra2311989601198714311990943
(7)
where 1198960 is the constant reaction rate
4 Mathematical Problems in Engineering
With these assumptions along with standard boundary-layer approximation the governing equations can be writtenin dimensional form as
120597119906
120597119909
+
120597V120597119910
= 0 (8)
120597119875
120597119909
= minus
120583
119896
119906 (9)
120597119875
120597119910
= minus
120583
119896
V + [(1 minus 119862infin) 120588119891infin119892120573 (119879 minus 119879infin)
minus (120588119875 minus 120588119891infin) 119892 (119862 minus 119862infin)]
(10)
119906
120597119879
120597119909
+ V120597119879
120597119910
= 120572119898
1205972119879
1205971199102
+ 120591 [119863119861
120597119862
120597119910
120597119879
120597119910
+ (
119863119879
119879infin
)(
120597119879
120597119910
)
2
]
+
1198760
(120588119862)119891
Ra23
11987143
11990943
(119879 minus 119879infin)
(11)
119906
120597119862
120597119909
+ V120597119862
120597119910
= 119863119861
1205972119862
1205971199102+ (
119863119879
119879infin
)
1205972119879
1205971199102
minus
Ra2311989601198714311990943
(119862 minus 119862infin)119899
(12)
where 120572119898 = 119896119898(120588119888119875)119891 is the thermal diffusivity of the fluidand 120591 = 120576(120588119862)119901(120588119862)119891 is a parameter
where 1198631(119909) is the thermal slip factor with dimension(length)minus1 The following new nondimensional variables areintroduced to make (8)ndash(13) dimensionless
119909 =
119909
119871radicRa 119910 =
119910
119871
119906 =
119906119871
120572119898radicRa
V =V119871120572119898
120579 =
119879 minus 119879infin
Δ119879
120601 =
119862 minus 119862infin
Δ119862
Δ119879 = 119879119908 minus 119879infin Δ119862 = 119862119908 minus 119862infin
(14)
where Ra = 119892119896120573(1 minus 119862infin)Δ119879119871(120572119898]) is the Rayleigh numberbased on the characteristic length 119871 A stream function 120595
defined by
119906 =
120597120595
120597119910
V = minus120597 120595
120597119909
(15)
is introduced into (8)ndash(13) to reduce the number of depen-dent variables and the number of equations Note that (8) issatisfied identically Hence we have
The parameters in (16)ndash(19) are introduced in Nomencla-ture and defined by
119873119905 = 120591119863119879
Δ119879
120572119898
119879infin 119873119887 = 120591119863119861
Δ119862
120572119898
119873119903 =
(120588119875 minus 120588119891infin) Δ119862
120588119891infin
120573 (1 minus 119862infin) Δ119879
119876 =
11987601198712
120572119898(120588119862)119891
119870 =
11989601198712(Δ119862)119899minus1
120572119898
Le =120572119898
119863119861
(20)
3 Lie Group Analysis
We consider the following scaling group of transformationswhich is a special form of Lie group analysis [52]
Γ 119909lowast= 1199091198901205761205721 119910
lowast= 1199101198901205761205722
120595lowast= 1205951198901205761205723 120579
lowast= 1205791198901205761205724
120601lowast= 1206011198901205761205725 119863
lowast1 = 1198631119890
1205761205726
(21)
Here 120576 is the parameter of the group Γ and 120572119894 (119894 =
1 2 3 4 5 6) are arbitrary real numbers whose connec-tion will be determined by our analysis The transfor-mations in (21) can be considered as a point transfor-mation transforming the coordinates (119909 119910 120595 120579 120601 1198631) to
Mathematical Problems in Engineering 5
(119909lowast 119910lowast 120595lowast 120579lowast 120601lowast 119863lowast1 ) We now investigate the relationship
times Δ 119895 (119909 119910 120579 120601 1205973120595
1205971199103)
(119895 = 1 2 3)
(22)
Since this is the requirement that the differential formsΔ 1 Δ 2 and Δ 3 be reformed under the transformation groupin (19) by using (21) (16)ndash(18) are transformed to (see [3538])
The system will remain invariant (structure of the equa-tions same) under the group transformation Γ if we have thefollowing relationship among the exponents
1205723 minus 21205722 = 1205724 minus 1205721 = 1205725 minus 1205721
1205723 + 1205724 minus 1205721 minus 1205722 = 1205724 minus 21205722 = 1205724 + 1205725 minus 21205722
= 21205724 minus 21205722 = 1205724 minus4
3
1205721
1205723 + 1205725 minus 1205721 minus 1205722 = 1205725 minus 21205722 = 1205724 minus 21205722
= constant = 119891 (120578) (say) that is 120595 = 11990913119891 (120578)
(30b)
6 Mathematical Problems in Engineering
where 119891 is an arbitrary function of 120578 From the equations211988911990931205722119909 = 1198891205790 and 211988911990931205722119909 = 1198891206010 we obtain byintegration
Here 120578 is the similarity variable and 119891 120579 120601 are dependentvariables Note that the similarity transformations in (31) areconsistent with the well-known similarity transformationsreported in the paper of Cheng and Chang [21] for 120582 = 0
in their paper Thus the dimensionless velocity components119906 V can be expressed as
119906 =
1198911015840
11990913 V = minus
1
311990923
(119891 minus 21205781198911015840) (32)
where primes indicate differentiation with respect to 120578
32 Similarity Equations On substituting the transforma-tions in (31) into the governing (16)ndash(18) we obtain thefollowing system of ordinary differential equations
where 119887 = (1198631)0119871 is the thermal slip parameter
4 Basic Idea of the HAM
Let us consider the following differential equation
N [119906 (120591)] = 0 (37)
where N is a nonlinear operator 120591 denotes independentvariable and 119906(120591) is an unknown function respectively Forsimplicity we ignore all boundary or initial conditions whichcan be treated in the similar way By means of generalizing
the traditional homotopy method Liao [45] constructs theso-called zero-order deformation equation
(1 minus 119901)L [120593 (120591 119901) minus 1199060 (120591)] = 119901ℏ119867 (120591)N [120593 (120591 119901)] (38)
where 119901 isin [0 1] is the embedding parameter ℏ = 0 is anonzero auxiliary parameter 119867(120591) = 0 is an auxiliary func-tionL is an auxiliary linear operator 1199060(120591) is an initial guessof 119906(120591) and 120593(120591 119901) is an unknown function respectively Itis important that one has great freedom to choose auxiliarythings in the HAM Obviously when 119901 = 0 and 119901 = 1 itholds 120593(120591 0) = 1199060(120591) 120593(120591 1) = 119906(120591) respectively Thus as119901 increases from 0 to 1 the solution 120593(120591 119901) varies from theinitial guess 1199060(120591) to the solution 119906(120591) Expanding 120593(120591 119901) inTaylor series with respect to 119901 we have
If the auxiliary linear operator the initial guess theauxiliary parameter ℏ and the auxiliary function are soproperly chosen the series (39) converges at 119901 = 1 then wehave
119906 (120591) = 1199060 (120591) +
+infin
sum
119898=1
119906119898 (120591) (41)
which must be one of solutions of original nonlinear equa-tion as proved by Liao [45] As ℏ = minus1 and 119867(120591) = 1 (38)becomes
(1 minus 119901)L [120593 (120591 119901) minus 1199060 (120591)] + 119901N [120593 (120591 119901)] = 0 (42)
which is used mostly in the homotopy perturbation methodwhere the solution can be obtained directly without usingTaylor series
According to definition (39) the governing equationcan be deduced from the zero-order deformation equation(37) By defining the vector 119899 = 1199060(120591) 1199061(120591) 119906119899(120591)
and differentiating equation (37) 119898 times with respect tothe embedding parameter 119901 and then setting 119901 = 0 andfinally dividing them by119898 we have the so-called119898th-orderdeformation equation
L [119906119898 (120591) minus 120594119898119906119898minus1 (120591)] = ℏ119867 (120591) 119877119898 (119898minus1) (43)
It should be emphasized that 119906119898(120591) for 119898 ge 1 isgoverned by the linear equation (39) with the linear boundaryconditions that come from original problem which canbe easily solved by symbolic computation software such asMAPLE and MATHEMATICA
Figure 3 The ℏ-curve of 1205791015840(0) given by the various HAM-orderapproximate solution
5 HAM Solution
In this section we applied the HAM to obtain approx-imate analytical solutions of the effect of higher orderchemical reaction internal heat generation and the thermalslip boundary condition on the boundary-layer flow of ananofluid past an upward facing horizontal plate (33)ndash(35)
Figure 4 The ℏ-curve of 1206011015840(0) given by the various HAM-orderapproximate solution
We start with the initial approximation 1198650(120578) = 0 Θ0(120578) =(1(119887 + 1))Exp[minus120578]Φ0(120578) = Exp[minus120578] and the linear operator
L [119865 (120578 119901)] =
1205972119865 (120578 119901)
1205971205783
L [Θ (120578 119901)] =
1205972Θ(120578 119901)
1205971205782
+
120597Θ (120578 119901)
120597120578
L [Φ (120578 119901)] =
1205972Φ(120578 119901)
1205971205782
+
120597Φ (120578 119901)
120597120578
(45)
Furthermore from (33)ndash(35) we define the nonlinearoperators
N [119865 (120578 119901)] =
1205972119865 (120578 119901)
1205971205782
minus
2
3
120578(
120597Θ (120578 119901)
120597120578
minus 119873119903
120597Φ (120578 119901)
120597120578
)
N [Θ (120578 119901)] =
1205972Θ(120578 119901)
1205971205782
+
1
3
119865 (120578 119901)
120597Θ (120578 119901)
120597120578
+ 119873119887
120597Θ (120578 119901)
120597120578
120597Φ (120578 119901)
120597120578
+ 119873119905(
120597Θ (120578 119901)
120597120578
)
2
+ 119876Θ (120578 119901)
N [Φ (120578 119901)] =
1205972Φ(120578 119901)
1205971205782
+
Le3
119865 (120578 119901)
120597Φ (120578 119901)
120597120578
+
119873119905
119873119887
1205972Θ(120578 119901)
1205971205782
minus 119870(Φ (120578 119901))119899
(46)
8 Mathematical Problems in Engineering
Using the above definition with assumption 119867119865(120591) = 1119867Θ(120591) = Exp[minus120578] and 119867Φ(120591) = Exp[minus120578] we construct the119911119890119903119900th-order deformation equation
(1 minus 119901)L [120593 (119909 119901) minus 1199060 (119909)] = 119867 (120591) 119901ℏN [120593 (119909 119901)]
Differentiating the 119911119890119903119900th-order deformation equation(47)119898 times with respect to 119901 and finally dividing by119898 wehave the119898th-order deformation equation
L [119865119898 (119909) minus 120594119898119865119898minus1 (119909)] = ℏ119877119898 (119898minus1)
L [Θ119898 (119909) minus 120594119898Θ119898minus1 (119909)] = Exp [minus120578] ℏ119877119898 (Θ119898minus1)
L [Φ119898 (119909) minus 120594119898Φ119898minus1 (119909)] = Exp [minus120578] ℏ119877119898 (Φ119898minus1)
By using the symbolic software MATHEMATICA tosolve the system of linear equations (49) with the boundaryconditions (46) it can be obtained finally
1198651 (120578) = minus
4
3 (1 + 119887)
ℏ +
4119890minus120578
3 (1 + 119887)
ℏ +
4119873119903
3 (1 + 119887)
ℏ
+
4119887119873119903
3 (1 + 119887)
ℏ minus
4119873119903119890minus120578
3 (1 + 119887)
ℏ minus
4119887119873119903119890minus120578
3 (1 + 119887)
ℏ
+
2119890minus120578120578
3 (1 + 119887)
ℏ minus
2119873119903119890minus120578120578
3 (1 + 119887)
ℏ minus
2119887119873119903119890minus120578120578
3 (1 + 119887)
ℏ
Θ1 (120578) =119890minus2120578
2 (1 + 119887)
ℏ minus
119890minus120578
2(1 + 119887)2ℏ minus
119887119890minus120578
2(1 + 119887)2ℏ
minus
119873119887119890minus3120578
6(1 + 119887)2ℏ +
119887119873119887119890minus3120578
6(1 + 119887)2ℏ minus
119873119887119890minus120578
6(1 + 119887)2ℏ
minus
119887119873119887119890minus120578
6(1 + 119887)2ℏ +
119873119905119890minus3120578
6(1 + 119887)2ℏ minus
119873119905119890minus120578
6(1 + 119887)2ℏ
+
119876119890minus2120578
2 (1 + 119887)
ℏ minus
119876119890minus120578
2(1 + 119887)2ℏ minus
119887119876119890minus120578
2(1 + 119887)2ℏ
Φ1 (120578) =119890minus2120578
2 (1 + 119887)
ℏ +
119887119890minus2120578
2 (1 + 119887)
ℏ minus
119890minus120578
2 (1 + 119887)
ℏ
minus
119887119890minus120578
2 (1 + 119887)
ℏ minus
119870119890minus2120578
2 (1 + 119887)
ℏ minus
119887119870eminus2120578
2 (1 + 119887)
ℏ
+
119870119890minus120578
2 (1 + 119887)
ℏ +
119887119870119890minus120578
2 (1 + 119887)
ℏ +
119873119905
119873119887
119890minus2120578
2 (1 + 119887)
ℏ
minus
119873119905
119873119887
119890minus120578
2 (1 + 119887)
ℏ
(56)
The higher orders solutions of 119865(120578) Θ(120578) and Φ(120578) weretoo long to be mentioned here therefore they are showngraphically
51 Convergence of the HAM Solution As mentioned by Liao[45] HAM provides us with great freedom in choosing thesolution of a nonlinear problem by different base functionsThis has a great effect on the convergence region because theconvergence region and rate of a series are chiefly determinedby the base functions used to express the solution We usedseveral terms in evaluating the approximate solution 119865app asympsum119899119894=0 119865119894Θapp asymp sum
119899119894=0Θ119894Φapp asymp sum
119899119894=0Φ119894 note that the solution
series contains the auxiliary parameter ℏ which provides uswith a simple way to adjust and control the convergence ofthe solution series Generally by means of the so-called ℏ-curve that is a curve of a versus ℏ As pointed by Liao [45]the valid region of ℏ is a horizontal line segment Figures2 3 and 4 show the ℏ-curve with the various order of theHAM for 11989110158401015840(0) 1205791015840(0) 1206011015840(0) respectively when 119873119887 = 01
Figure 6 The residual error of 120579(120578) for (34) using 12th order ofapproximations
119873119905 = 01 119873119903 = 01 119876 = 01 Le = 10 119870 = 10 119899 = 10119887 = 00 It can be seen that when the order of series is 12 thesegment of the horizontal line is more than the other ordersFor example it can be found that for 11986510158401015840(0) the acceptablerange ofℏ is betweenminus05 andminus20 for 12th order of theHAMbut for 8th order of the HAM the acceptable range of ℏ isbetween minus10 and minus20 or minus10 and minus15 for 4th order of theHAM so horizontal line segment of 12th order of the HAM ismore than others Therefore it is straightforward to choosean appropriate range for ℏ which ensures the convergence
10 Mathematical Problems in Engineering
minus00002
120578
= minus10
= minus111 (optimal value)= minus12
Nt = 01Nr = 01
Q = 01Le = 10K = 10n = 10b = 00
0006
0004
0002
0
0 2 4 6 8 10
Nb = 01
Res (120601
(120578))
ℏ
ℏ
ℏ
Figure 7 The residual error of 120601(120578) for (35) using 12th order ofapproximations
of the solution series To choose optimal value of auxiliaryparameter the averaged residual errors (see [53] for moredetails) are defined as
119864119865119898 =1
119870
119870
sum
119894=0
[N119865(119898
sum
119894=0
119865119894 (119894Δ119909))]
2
119864Θ119898 =1
119870
119870
sum
119894=0
[NΘ(119898
sum
119894=0
Θ119894 (119894Δ119909))]
2
119864Φ119898 =1
119870
119870
sum
119894=0
[NΦ(119898
sum
119894=0
Φ119894 (119894Δ119909))]
2
(57)
where Δ119909 = 10119870 and 119870 = 20 For given order ofapproximation 119898 and the optimal values ℏ are given bythe minimum of 119864119898 corresponding to nonlinear algebraicequations
119889119864119865119898
119889ℏ
= 0
119889119864Θ119898
119889ℏ
= 0
119889119864Φ119898
119889ℏ
= 0 (58)
It is noted that the optimal value of ℏ is replaced intothe equations Table 1 shows optimal values obtained for theauxiliary parameter ℏ for various quantities of119873119905 119876 and 119899when119873119887 = 01119873119903 = 01 Le = 10 119870 = 10 119887 = 00 To see
the accuracy of the solutions the residual errors are definedfor the system as (for119898 order approximation)
Res119865119898
=
1198892119865119898
1198891205782minus
2
3
120578(
119889Θ119898
119889120578
minus 119873119903
119889Φ119898
119889120578
) (59)
ResΘ119898
=
1198892Θ119898
1198891205782+
1
3
119865119898
119889Θ119898
d120578+ 119873119887
119889Θ119898
119889120578
119889Φ119898
119889120578
+ 119873119905(
119889Θ119898
119889120578
)
2
+ 119876Θ119898
(60)
ResΦ119898
=
1198892Φ119898
1198891205782+
Le3
119865119898
119889Φ119898
119889120578
+
119873119905
119873119887
1198892Θ119898
1198891205782
minus 119870(
119889Φ119898
119889120578
)
119899
(61)
Figures 5 6 and 7 show the residual errors for 12th-orderdeformation solutions when 119873119887 = 01 119873119905 = 01 119873119903 = 01119876 = 01 Le = 10 119870 = 10 119899 = 10 119887 = 00 For example ℏ =minus145 has the minimum range of residual curve in Figure 5and so on
Graphical representation of results is very useful todemonstrate the efficiency and accuracy of the HAM for theabove problem
6 Comparisons and Verification
It is worth citing that for isothermal plate (119887 = 0) and in theabsence of internal heat generationabsorption (119876 = 0) andchemical reaction (119870 = 0) our problem reduces to Gorla andChamkha [26] and Khan and Pop [27] To verify the accuracyof our results the present results are compared in Table 2with Gorla and Chamkha [26] and are found to be in goodagreement
7 Physical Quantities
Theparameters of physical interest of the present problem arethe local skin friction factor 119862119891119909 the local Nusselt numberNu119909 and the local Sherwood number Sh119909 respectivelyPhysically 119862119891119909 indicates wall shear stress Nu119909 indicates therate of wall heat transfer whilst Sh119909 indicates the rate ofwall nanoparticle volume fractionThe following relations areused to find these quantities
119862119891119909 =2120583
1205881198802119903
(
120597119906
120597119910
)
119910=0
Nu119909 =minus119909
119879119891 minus 119879infin
(
120597119879
120597119910
)
119910=0
Sh119909 =minus119909
119862119908 minus 119862infin
(
120597119862
120597119910
)
119910=0
(62)
Mathematical Problems in Engineering 11
Table 2 Comparison of present solution with Gorla and Chamkha[26] for different values of buoyancy and nanofluid parameters
Present results (HAM) Gorla and Chamkha [26]minus1205791015840(0) minus120601
1015840(0) minus120579
1015840(0) minus120601
1015840(0)
Nr 119873119887 = 03119873119905 = 01 Le = 10 119887 = 119870 = 119876 = 0
01 032578 148242 326119864 minus 01 148416402 032385 146704 325119864 minus 01 146816103 032188 145125 322119864 minus 01 145266404 031985 143503 321119864 minus 01 143639205 031777 141833 319119864 minus 01 1419499Nt 119873119887 = 03119873119903 = 05 Le = 10 119887 = 119870 = 119876 = 0
01 031777 141833 319119864 minus 01 141949902 030486 141491 305119864 minus 01 141653603 02927 141561 293119864 minus 01 141686604 028125 141991 282119864 minus 01 142158205 027046 142737 271119864 minus 01 1429226Nb 119873119905 = 01119873119903 = 05 Le = 10 119887 = 119870 = 119876 = 0
01 03672 132611 368119864 minus 01 132745402 034271 139216 343119864 minus 01 139361503 031777 141833 319119864 minus 01 141949904 029399 143428 294119864 minus 01 143546405 027161 144598 272119864 minus 01 144772
Figure 10 Comparison of dimensionless concentration profile 120601(120578)obtained by various orders of the HAM
Here Ra119909 = 119892119870120573(1minus119862infin)Δ119879119909(120572119898]) is the local Rayleighnumber Pr = ]120572119898 is the Prandtl number and 119880119903 = (1 minus
119862infin)119892119870120573Δ119879120572119898 is the reference velocity
8 Results and Discussion
Graphical representation of results is very useful to discussthe physical features presented by the solutions This sectiondescribes the influence of some interesting parameters on
12 Mathematical Problems in Engineering
0 2 4 6 8 10
02
04
06
08
1
Nr = 01Numerical
Numerical
NumericalNr = 03Nr = 05
f998400 (120578)
120578
Nt = 01Nb = 01Q = 01Le = 10K = 10n = 10b = 00
Figure 11 Effect of buoyancy ratio119873119903 on the dimensionless velocityprofile
0 2 4 6 8 10
02
04
06
08
1
Nt = 01Nb = 01Q = 01Le = 10K = 10n = 10b = 00
Nr = 01Numerical
Numerical
NumericalNr = 03Nr = 05
120578
120579(120578
)
Figure 12 Effect of buoyancy ratio 119873119903 on the dimensionlesstemperature profile
the velocity and temperature fields Equations (33) to (35)with boundary conditions in (36) were solved analytically byHAM and numerically using Runge-Kutta-Fehlberg fourth-fifth order proposed by Aziz [54]
Figures 8 9 and 10 respectively for 1198911015840(120578) 120579(120578) and 120601(120578)show the comparisons between various order approximationof the optimal HAM and the numerical solutions for the case
0 2 4 6 8 10
02
04
06
08
1
Nr = 01Numerical
Numerical
NumericalNr = 03Nr = 05
Nt = 01Nb = 01Q = 01Le = 10K = 10n = 10b = 00
120578
120601(120578
)Figure 13 Effect of buoyancy ratio 119873119903 on the dimensionlessconcentration profile
0 2 4 6 8 10
02
04
06
08
1
12
NumericalNumerical
Numerical
f998400 (120578
)
120578
Nb = 01
Nb = 03
Nb = 05
Nt = 01Nr = 01Q = 01Le = 10K = 10n = 10b = 00
Figure 14 Effect of Brownian motion 119873119887 on the dimensionlessvelocity profile
119873119887 = 01 119873119905 = 01 119873119903 = 01 119876 = 01 Le = 10 119870 = 10119899 = 10 119887 = 00 For better presentation these results arepresented in Tables 3 4 and 5 Also Tables 6 7 and 8 presentthe comparisons between various order approximation of theoptimal HAM and the numerical solutions for the case119873119887 =03 119873119905 = 005 119873119903 = 03 119876 = 01 Le = 20 119870 = 05
Mathematical Problems in Engineering 13
0 2 4 6 8 10
02
04
06
08
1
0
120578
NumericalNumerical
Numerical
Nb = 01
Nb = 03
Nb = 05
Nt = 01Nr = 01Q = 01Le = 10K = 10n = 10b = 00
120579(120578
)
Figure 15 Effect of Brownian motion 119873119887 on the dimensionlesstemperature profile
02
04
06
08
1
00
2 4 6 8 10120578
NumericalNumerical
Numerical
Nb = 01
Nb = 03
Nb = 05
Nt = 01Nr = 01Q = 01Le = 10K = 10n = 10b = 00
120601(120578
)
Figure 16 Effect of Brownian motion 119873119887 on the dimensionlessconcentration profile
119899 = 20 119887 = 01 It is observed that the results of 12th-order approximation of the optimal HAM are very close tothe numerical solutions which confirm the validity of thesemethods
In the following figures effects of various physicalparameters on the dimensionless velocity temperature and
Table 3 Comparison of values of 1198911015840(120578) obtained by various ordersof the HAM with numerical solution when 119873119887 = 01119873119905 =
temperature profiles will be investigated These results havebeen obtained by 12th order of the HAM and have beenvalidated by numerical results Figures 11ndash13 respectivelyrepresent that the comparison of solutions of 1198911015840(120578) 120579(120578)and 120601(120578) for fix values 119873119887 = 01 119873119905 = 01 119876 = 01Le = 10 119870 = 10 119899 = 10 119887 = 00 and different values ofbuoyancy ratio119873119903 In Figure 11 it is clear that velocity of thefluid tremendously decreases in the near of horizontal platewith an increase in the buoyancy ratio It is observed that thetemperature increases slightly but concentration of the fluiddoes not vary sensibly (Figures 12 and 13) The effects of theBrownian motion 119873119887 are depicted in Figures 14 15 and 16when119873119903 = 01119873119905 = 01119876 = 01 Le = 10119870 = 10 119899 = 10
14 Mathematical Problems in Engineering
Table 6 Comparison of values of 1198911015840(120578) obtained by various ordersof the HAM with numerical solution when 119873119887 = 03119873119905 =
119887 = 00 If 119873119887 increases 1198911015840(120578) and 120579(120578) increase lightly and120601(120578) decreases The dimensionless velocity dimensionlesstemperature and dimensionless concentration profiles fordifferent values of thermophoresis 119873119905 with constant values119873119903 = 01 119873119887 = 01 119876 = 01 Le = 10 119870 = 10119899 = 10 119887 = 00 are presented in Figures 17 18 and19 It is observed that the velocity and temperature of thefluid are not impressible from119873119905 but concentration increasesslightly In Figures 20 21 and 22 respectively comparison ofsolutions of 1198911015840(120578) 120579(120578) and 120601(120578) for 119873119903 = 01 119873119887 = 01119873119905 = 01 Le = 10 119870 = 10 119899 = 10 119887 = 00 anddifferent values of heat generationabsorption parameter 119876are shown It is observed that in presence of heat source1198911015840(120578) 120579(120578) both increase while heat sink causes that 1198911015840(120578)
120579(120578) both decrease extremely These effects are more visible
Table 9 Values of reduced Nusselt number and Sherwood numberobtained by HAM for different values of the parameters Le 119870 119899 119887and 119876 when119873119887 = 119873119903 = 119873119905 = 01
Figure 17 Effect of thermophoresis 119873119905 on the dimensionlessvelocity profile
for temperature distribution For 120601(120578) the effect of changing119876 is seen to be almost insignificant This is in agreementwith the physical factThe dimensionless profiles for differentvalues of Lewis number Le with constant parameters 119873119903 =01 119873119887 = 01 119873119905 = 01 119876 = 01 119870 = 10 119899 = 10119887 = 00 are shown in Figures 23 24 and 25 It is clear that forthese constant parameters variation of Lewis number has anyconsiderable effect on the velocity and temperature profilesbut concentration profile decreases with the increase of Lewisnumber What is similar to these results can be detected inFigures 26 27 and 28 for chemical reaction parameter 119870Figures 29 30 and 31 respectively present the comparisonof solutions of 1198911015840(120578) 120579(120578) and 120601(120578) for 119873119903 = 05 119873119887 = 05119873119905 = 01 119876 = 001 Le = 10 119870 = 10 119887 = 00 and differentvalues of order of chemical reaction 119899 It is observed that
Mathematical Problems in Engineering 15
0 2 4 6 80
1
120578
Nt = 00
Nt = 005
Nt = 10NumericalNumerical
Numerical
08
06
04
02
10
Q = 01Le = 10K = 10n = 10b = 00
Nb = 01
120579(120578
)
Nr = 01
Figure 18 Effect of thermophoresis 119873119905 on the dimensionlesstemperature profile
0 2 4 6 80
1
120601(120578
)
120578
Nt = 00
Nt = 005
Nt = 10NumericalNumerical
Numerical
08
06
04
02
10
Q = 01Le = 10K = 10n = 10b = 00
Nb = 01Nr = 01
Figure 19 Effect of thermophoresis 119873119905 on the dimensionlessconcentration profile
for higher order of chemical reaction velocity decreases andtemperature and concentration profiles increase extremelyUltimately Figures 32 33 and 34 depict the effect of thermalslip parameter 119887 on velocity temperature and concentrationfunctions when 119873119903 = 01 119873119887 = 01 119873119905 = 01 119876 = 01Le = 20 119870 = 15 119899 = 10 It is seen that the velocityand temperature of the fluid decrease with the increase of the
0 2 4 6 80
1
12
10
02
04
06
08
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numericalf998400 (120578
)
120578
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 20 Effect of generationabsorption heat parameter119876 on thedimensionless velocity profile
0 2 4 6 8 100
02
04
06
08
1
120579(120578
)
120578
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numerical
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 21 Effect of generationabsorption heat parameter 119876 on thedimensionless temperature profile
thermal slip parameter in the near of horizontal plate whilethe concentration of the fluid does not vary patently
Also for investigation of the parameters of physicalinterest Table 9 is presented In this table numerical valuesof reduced Nusselt number and Sherwood number obtainedby HAM for different values of the parameters 119890 119870 119899 119887 and119876 can be compared when119873119887 = 119873119903 = 119873119905 = 01
16 Mathematical Problems in Engineering
10
02
04
06
08
0 2 4 6 80
1
120578
120601(120578
)
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numerical
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 22 Effect of generationabsorption heat parameter119876 on thedimensionless concentration profile
0 2 4 6 80
1
12
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 23 Effect of Lewis number Le on the dimensionless velocityprofile
9 Conclusions
In this paper we studied the steady laminar incompressiblefree convective flow of a nanofluid past a chemically reactingupward facing horizontal plate in porous medium takinginto account heat generation and the thermal slip boundarycondition
0 2 4 6 80
1
120579(120578
)
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 24 Effect of Lewis number Le on the dimensionlesstemperature profile
0 2 4 6 80
1
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 25 Effect of Lewis number Le on the dimensionlessconcentration profile
The governing partial differential equations have beentransformed by similarity transformations into a system ofordinary differential equations which are solved by OHAMand numerical method (fourth-order Runge-Kutta schemewith the shooting method) Dimensionless velocity temper-ature and concentration functions are presented for various
Mathematical Problems in Engineering 17
0 2 4 6 80
1
12
K = 00
K = 05
K = 10NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 26 Effect of chemical reaction parameter 119870 on the dimen-sionless velocity profile
0 2 4 6 80
1
K = 00
K = 05
K = 10NumericalNumerical
Numerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 27 Effect of chemical reaction parameter 119870 on the dimen-sionless temperature profile
values of parameters for the problem for example buoy-ancy ratio 119873119903 Brownian motion 119873119887 thermophoresis 119873119905heat generationabsorption parameter 119876 Lewis number Lechemical reaction parameter119870 order of chemical reaction 119899and thermal slip parameter 119887 From the present investigationthe following may be concluded
(i) The velocity 1198911015840(120578) increases with increasing 119873119887 anddecreases with increasing119873119903 119899 and 119887
0 2 4 6 80
1
K = 00
K = 05
K = 10NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 28 Effect of chemical reaction parameter 119870 on the dimen-sionless concentration profile
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 29 Effect of order of chemical reaction 119899 on the dimension-less velocity profile
(ii) The temperature 120579(120578) increases with 119873119903 119873119887 119899 anddecreases with 119887
(iii) Parameters of119873119905 Le and119870 have no effect on velocityand temperature profiles
(iv) The concentration distribution 120601(120578) increases withincreasing N119905 119899 and decreases with 119873119887 Le and 119870but119873119903 119876 and 119887 are not affected by it
18 Mathematical Problems in Engineering
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 30 Effect of order of chemical reaction 119899 on the dimension-less temperature profile
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 31 Effect of order of chemical reaction 119899 on the dimension-less concentration profile
(v) In presence of heat source (119876 gt 0) 1198911015840(120578) 120579(120578) bothincrease while heat sink (119876 lt 0) causes that 1198911015840(120578)120579(120578) both decrease extremely These effects are morevisible for temperature distribution
In addition numerical results for the reducedNusselt andSherwood numbers are tabulated It is concluded that thereduced Nusselt number increases with the Lewis numberand reaction parameter whist it decreases with the order of
0 2 4 6 8
1
12
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 32 Effect of order of thermal slip parameter 119887 on thedimensionless velocity profile
0 2 4 6 80
1
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 33 Effect of order of thermal slip parameter 119887 on thedimensionless temperature profile
chemical reaction thermal slip and generation parametersIt is further concluded that the reduced Sherwood numberenhances with the Lewis number generation and reactionparameters whist it suppresses with thermal slip and orderof chemical reaction
The convergence of the solution series and the power ofHAM in controlling and adjusting the convergence regionand rate of solution series were discussed The proper range
Mathematical Problems in Engineering 19
0 2 4 6 80
1
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 34 Effect of order of thermal slip parameter 119887 on thedimensionless concentration profile
of auxiliary parameter ℏ to ensure the convergence of thesolution series was obtained through the so called ℏ-curveThe HAM provides us with a convenient way to control theconvergence of approximation series which is a fundamentalqualitative difference in analysis between the HAM andother methods The numerical results of the above problemsdisplay a fast convergence with minimal calculations Thisshows that the HAM is a very efficient method Finally theagreement between analytical and numerical results of thepresent study with previous published results is excellent
Nomenclature
119887 Thermal slip parameter119862 Dimensional concentration119862119891 Skin friction factor1198631 Thermal slip factor119863119861 Brownian diffusion coefficient119863119879 Thermophoretic diffusion coefficient119891 Dimensionless velocity functions119892 Gravitation acceleration119870 Chemical reaction parameter119896 Permeability of the porous media1198960 The constant reaction rate119896119898 Effective thermal conductivity of the porous
medium119896(119909) Variable reaction rateL Linear operator of the HAM119871 Length of horizontal plateLe Lewis numberN Nonlinear operator of the HAM
119899 Order of chemical reaction119873119887 Brownian motion119873119903 Buoyancy ratio119873119905 ThermophoresisNu Nusselt number119875 PressurePr Prandtl number119876 Heat generationabsorption parameter1198760 Heat generationabsorption constant119902 Internal heat generation rate
Ra Rayleigh numberSh Sherwood number119879 Temperature119905 Time119906 Velocity in 119909-direction119880119903 Reference velocity Velocity vectorV Velocity in 119910-direction119909 Distance along the surface119910 Distance normal to the surface
Greek Letters
120572119898 Thermal diffusivity120573 Volumetric expansion coefficient of nanofluid120576 Porosity of the porous media120601 Dimensionless concentration120578 Similarity variable120583 Dynamic viscosity120579 Dimensionless temperature120588119891 Density of the base fluid120588119901 Density of the nanoparticles(120588119862)119891 Effective heat capacity of the fluid(120588119862)119875 Effective heat capacity of the nanoparticle material120595 Stream function
Subscript Superscript
infin Conditions far away from the surface1015840 Differentiation with respect to 120578
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
M M Rashidi wishes to thank the Centre for DifferentialEquations Continuum Mechanics and Applications Schoolof Computational and Applied Mathematics University ofthe Witwatersrand for the hospitality during his visit EMomoniat thanks theNational Research Foundation of SouthAfrica for their support
20 Mathematical Problems in Engineering
References
[1] E Abu-Nada F O Hakan and I Pop ldquoBuoyancy inducedflow in a nanofluid filled enclosure partially exposed to forcedconvectionrdquo Superlattices and Microstructures vol 51 no 3 pp381ndash395 2012
[2] S Kakac and A Pramuanjaroenkij ldquoReview of convective heattransfer enhancement with nanofluidsrdquo International Journal ofHeat and Mass Transfer vol 52 no 13-14 pp 3187ndash3196 2009
[3] J A Eastman S U S Choi S Li W Yu and L J ThompsonldquoAnomalously increased effective thermal conductivities ofethylene glycol-based nanofluids containing copper nanoparti-clesrdquo Applied Physics Letters vol 78 no 6 pp 718ndash720 2001
[4] Y Xuan and Q Li ldquoHeat transfer enhancement of nanofluidsrdquoInternational Journal of Heat and Fluid Flow vol 21 no 1 pp58ndash64 2000
[5] S K Das N Putra P Thiesen and W Roetzel ldquoTemperaturedependence of thermal conductivity enhancement for nanoflu-idsrdquo Journal of Heat Transfer vol 125 no 4 pp 567ndash574 2003
[6] D H Kumar H E Patel V R R Kumar T SundararajanT Pradeep and S K Das ldquoModel for heat conduction innanofluidsrdquo Physical Review Letters vol 93 no 14 Article ID144301 2004
[7] H E Patel T Sundararajan T Pradeep A Dasgupta NDasgupta and S K Das ldquoA micro-convection model forthermal conductivity of nanofluidsrdquo Pramana vol 65 no 5 pp863ndash869 2005
[8] A V Kuznetsov ldquoNon-oscillatory and oscillatory nanofluidbio-thermal convection in a horizontal layer of finite depthrdquoEuropean Journal of Mechanics vol 30 no 2 pp 156ndash165 2011
[9] M Napoli J C T Eijkel and S Pennathur ldquoNanofluidictechnology for biomolecule applications a critical reviewrdquo Labon a Chip vol 10 no 8 pp 957ndash985 2010
[10] J C Crepeau and R Clarksean ldquoSimilarity solutions of naturalconvection with internal heat generationrdquo Journal of HeatTransfer vol 119 no 1 pp 183ndash185 1997
[11] A Z Sahin ldquoTransient heat conduction in semi-infinite solidwith spatially decaying exponential heat generationrdquo Interna-tional Communications in Heat andMass Transfer vol 19 no 3pp 349ndash358 1992
[12] O D Makind and A Aziz ldquoMixed convection from a con-vectively heated vertical plate to a fluid with internal heatgenerationrdquo Journal of Heat Transfer vol 133 Article ID 1225016 pages 2011
[13] J Buongiorno ldquoConvective transport in nanofluidsrdquo Journal ofHeat Transfer vol 128 no 3 pp 240ndash250 2006
[14] AV Kuznetsov andDANield ldquoNatural convective boundary-layer flow of a nanofluid past a vertical platerdquo InternationalJournal of Thermal Sciences vol 49 no 2 pp 243ndash247 2010
[15] L B Godson D Raja L D Mohan and S WongwisescldquoEnhancement of heat transfer using nanofluids-an overviewrdquoRenewable and Sustainable Energy Reviews vol 14 no 2 pp629ndash641 2010
[16] D A Nield and A BejanConvection in PorousMedia SpringerNew York NY USA 3rd edition 2006
[17] P Vadasz Emerging Topics in Heat and Mass Transfer in PorousMedia Springer New York NY USA 2008
[18] K Vafai Porous Media Applications in Biological Systems andBiotechnology CRC Press New York NY USA 2010
[19] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application to
heat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977
[20] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for natural convective boundary-layer flow in a porousmedium saturated by a nanofluidrdquo International Journal of Heatand Mass Transfer vol 52 no 25-26 pp 5792ndash5795 2009
[21] P Cheng and I D Chang ldquoBuoyancy induced flows in asaturated porous medium adjacent to impermeable horizontalsurfacesrdquo International Journal of Heat and Mass Transfer vol19 no 11 pp 1267ndash1272 1976
[22] I D Chang and P Cheng ldquoMatched asymptotic expansionsfor free convection about an impermeable horizontal surfacein a porous mediumrdquo International Journal of Heat and MassTransfer vol 26 no 2 pp 163ndash174 1983
[23] D S Shiunlin and B Gebhart ldquoBuoyancy-induced flow adja-cent to a horizontal surface submerged in porous mediumsaturated with cold waterrdquo International Journal of Heat andMass Transfer vol 29 no 4 pp 611ndash623 1986
[24] J H Merkin and G Zhang ldquoOn the similarity solutions for freeconvection in a saturated porousmedium adjacent to imperme-able horizontal surfacesrdquoWarme- und Stoffubertragung vol 25no 3 pp 179ndash184 1990
[25] M A Chaudhary J H Merkin and I Pop ldquoNatural convectionfrom a horizontal permeable surface in a porous mediummdashnumerical and asymptotic solutionsrdquo Transport in PorousMedia vol 22 no 3 pp 327ndash344 1996
[26] R S R Gorla and A Chamkha ldquoNatural convective boundarylayer flow over a nonisothermal vertical plate embedded ina porous medium saturated with a nanofluidrdquo Nanoscale andMicroscale Thermophysical Engineering vol 15 no 2 pp 81ndash942011
[27] W A Khan and I Pop ldquoFree convection boundary layer flowpast a horizontal flat plate embedded in a porous medium filledwith a nanofluidrdquo Journal of Heat Transfer vol 133 no 9 ArticleID 094501 2011
[28] A Aziz W A Khan and I Pop ldquoFree convection boundarylayer flow past a horizontal flat plate embedded in porousmedium filled by nanofluid containing gyrotactic microorgan-ismsrdquo International Journal of Thermal Sciences vol 56 pp 48ndash57 2012
[29] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer New York NYUSA 2009
[30] A A Avramenko S G Kobzar I V Shevchuk A V Kuznetsovand L T Iwanisov ldquoSymmetry of turbulent boundary-layerflows investigation of different eddy viscosity modelsrdquo ActaMechanica vol 151 no 1-2 pp 1ndash14 2001
[31] A V Kuznetsov A A Avramenko and P Geng ldquoAnalyticalinvestigation of a falling plume caused by bioconvection ofoxytactic bacteria in a fluid saturated porous mediumrdquo Inter-national Journal of Engineering Science vol 42 no 5-6 pp 557ndash569 2004
[32] M Jalil S Asghar and M Mushtaq ldquoLie group analysis ofmixed convection flow with mass transfer over a stretchingsurface with suction or injectionrdquo Mathematical Problems inEngineering vol 2010 Article ID 264901 14 pages 2010
[33] M A A Hamad M J Uddin and A I M Ismail ldquoRadiationeffects on heat and mass transfer in MHD stagnation-pointflow over a permeable flat plate with thermal convectivesurface boundary condition temperature dependent viscosityand thermal conductivityrdquoNuclear Engineering and Design vol242 pp 194ndash200 2012
Mathematical Problems in Engineering 21
[34] A Aziz M J Uddin M A A Hamad and A I MdIsmail ldquoMHD flow over an inclined radiating plate with thetemperature-dependent thermal conductivity variable reactiveindex and heat generationrdquoHeat Transfer vol 41 no 3 pp 241ndash259 2012
[35] A G Hansen Similarity Analysis of Boundary Layer Problemsin Engineering Prentice Hall Englewood Cliffs NJ USA 1964
[36] W FAmesNonlinear Partial Differential Equations in Engineer-ing Academic Press New York NY USA 1972
[37] R Seshadri and T Y Na Group Invariance in EngineeringBoundary Value Problems Springer New York NY USA 1985
[38] D Shang Theory of Heat Transfer with Forced Convection FilmFlows vol 3 of Heat and Mass Transfer 2010
[40] E Erfani M M Rashidi and A B Parsa ldquoThe modifieddifferential transform method for solving off-centered stag-nation flow toward a rotating discrdquo International Journal ofComputational Methods vol 7 no 4 pp 655ndash670 2010
[41] M M Rashidi N Laraqi and A Basiri Parsa ldquoAnalyticalmodeling of heat convection in magnetized micropolar fluid byusing modified differential transform methodrdquo Heat Transfervol 40 no 3 pp 187ndash204 2011
[42] M M Rashidi and M Keimanesh ldquoUsing differential trans-form method and pade approximant for solving mhd flow ina laminar liquid film from a horizontal stretching surfacerdquoMathematical Problems in Engineering vol 2010 Article ID491319 14 pages 2010
[43] M M Rashidi and G Domairry ldquoNew analytical solution ofthe three-dimensional NavierStokes equationsrdquoModern PhysicsLetters B vol 23 no 26 pp 3147ndash3155 2009
[44] M M Rashidi M Keimanesh O A Beg and T K HungldquoMagnetohydrodynamic biorheological transport phenomenain a porous medium a simulation of magnetic blood flowcontrol and filtrationrdquo International Journal for NumericalMethods in Biomedical Engineering vol 27 no 6 pp 805ndash8212011
[45] S J Liao The proposed homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992
[46] M M Rashidi G Domairry and S Dinarvand ldquoApproximatesolutions for the Burger and regularized long wave equationsby means of the homotopy analysis methodrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 14 no 3 pp708ndash717 2009
[47] Z Ziabakhsh and G Domairry ldquoAnalytic solution of naturalconvection flow of a non-Newtonian fluid between two verticalflat plates using homotopy analysis methodrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 14 no 5pp 1868ndash1880 2009
[48] M M Rashidi S A Mohimanian Pour T Hayat and SObaidat ldquoAnalytic approximate solutions for steady flow over arotating disk in porousmediumwith heat transfer by homotopyanalysis methodrdquo Computers and Fluids vol 54 no 1 pp 1ndash92012
[49] M Sajid T Hayat and S Asghar ldquoComparison between theHAM and HPM solutions of thin film flows of non-Newtonianfluids on a moving beltrdquo Nonlinear Dynamics vol 50 no 1-2pp 27ndash35 2007
[50] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for the double-diffusive natural convective boundary
layer flow in a porous medium saturated by a nanofluidrdquoInternational Journal of Heat and Mass Transfer vol 54 no 1-3pp 374ndash378 2011
[51] K Vajravelu and A Hadjinicolaou ldquoConvective heat transferin an electrically conducting fluid at a stretching surfacewith uniform free streamrdquo International Journal of EngineeringScience vol 35 pp 1237ndash1244 1997
[52] S Mukhopadhyay and G C Layek ldquoEffects of variable fluidviscosity on flow past a heated stretching sheet embedded ina porous medium in presence of heat sourcesinkrdquo Meccanicavol 47 no 4 pp 863ndash876 2012
[53] S J Liao ldquoAn optimal homotopy-analysis approach for stronglynonlinear differential equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 8 pp 2003ndash20162010
[54] A Aziz ldquoA similarity solution for laminar thermal boundarylayer over a flat plate with a convective surface boundary con-ditionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1064ndash1068 2009
ultrafine nanoparticles (lt100 nm) suspended in a base fluidwhich can be water or an organic solvent Nanofluids possesssuperior thermophysical properties like high thermal con-ductivity minimal clogging in flow passages long term sta-bility and homogeneity Industrial applications of nanofluidinclude electronics automotive and nuclear applicationsNanobiotechnology is also a fast developing field of researchwith application in many domains such as medicine phar-macy cosmetics and agroindustry Many of these industrialprocesses involve nanofluid flow and nanoparticle volumefraction past various geometries In these applications thediffusing species can be generatedabsorbed due to chemicalreaction with the ambient fluid This can greatly affect theflow and hence the properties and quality of the final product[10 11]
Different industrial applications of internal heat gen-eration include polymer production and the manufactureof ceramics or glassware phase change processes thermalcombustion processes and the development of a metal wastefrom spent nuclear fuel [12] A review of convective transportin nanofluids was conducted by Buongiorno [13] Kuznetsovand Nield [14] presented a similarity solution of naturalconvective boundary-layer flow of a nanofluid past a verticalplate They have shown that the reduced Nusselt number isa decreasing function of the buoyancy-ratio number 119873119903 aBrownianmotion number119873119887 and a thermophoresis number119873119905 Godson et al [15] presented the recent experimental andtheoretical studies on convective heat transfer in nanofluidsand their thermophysical properties and applications andclarified the challenges and opportunities for future research
Convective flow in porous media has received the atten-tion of researchers over the last several decades due to itsmany applications in mechanical chemical and civil engi-neering Examples include fibrous insulation food processingand storage thermal insulation of buildings geophysical sys-tems electrochemistry metallurgy the design of pebble bednuclear reactors underground disposal of nuclear or nonnu-clear waste and cooling system of electronic devices Excel-lent reviews of the fundamental theoretical and experimentalworks can be found in the books by Nield and Bejan [16]Vadasz [17] Vafai [18] The Cheng-Minkowycz problem [19]was investigated by Nield and Kuznetsov [20] for nanofluidwhere the model involves the effect of Brownian motionand thermophoresis The classical problem of free convectiveflow in a porous medium near a horizontal flat plate wasfirst investigated by Cheng and Chang [21] Following himmany researchers such as Chang and Cheng [22] Shiunlinand Gebhart [23] Merkin and Zhang [24] and Chaudhary etal [25] have extended the problem in various aspects Gorlaand Chamkha [26] presented a similarity analysis of freeconvective flow of nanofluid past a horizontal upward facingplate in a porous medium numerically Khan and Pop [27]extended this problem for nanofluid Very recently Aziz et al[28] extended the same problem for a water-based nanofluidcontaining gyrotactic microorganisms
Lie group analysis has been used by many investigatorsto analyze various convective phenomena under variousflow configurations arising in fluidmechanics aerodynamicsplasma physics meteorology chemical engineering and
other engineering branches [29] This method has beenapplied by many investigators to study various transportproblems For example the symmetrical properties of theturbulent boundary-layer flows and other turbulent flowsare investigated by using the Lie group techniques byAvramenko et al [30] Kuznetsov et al [31] investigateda falling bioconvection plume in a deep chamber filledwith a fluid saturated porous medium theoretically Jalil etal [32] studied mixed convective flow with mass transferusing Lie group analysis The effect of thermal radiation andconvective surface boundary condition on the boundary-layer flow was investigated by Hamad et al [33] Aziz et al[34] studied MHD flow over an inclined radiating plate withthe temperature dependent thermal conductivity variablereactive index and heat generation using scaling group oftransformations Reviews for the fundamental theory andapplications of group theory to differential equations can befound in the texts by Hansen [35] Ames [36] Seshadri andNa [37] and Shang [38]
Most scientific problems and phenomena such as theboundary-layer problem occur nonlinearly For these non-linear problems we have difficulty in finding their exactanalytical solutions Analytical solutions to these nonlinearequations are of fundamental importance Where no analyti-cal solutions can be found researchers have resorted to otherapproaches One such approach is a perturbation method[39] that is strongly dependent upon the so-called ldquosmallparametersrdquo The perturbation method cannot provide uswith a simple way to adjust and control the convergenceregion and rate of convergence of a given approximate series
Another known method is the differential transformmethod that has been used in recent years [40ndash44] In 1992Liao introduced the basic ideas of the homotopy in topologyto propose a general analytic method for nonlinear problemsnamely homotopy analysismethod (HAM) [45] that does notneed any small parameterThis method has been successfullyapplied to solve many types of nonlinear problems by others[46ndash48] As this approach is based on the homotopy oftopology the validity of the HAM is independent of whetheror not there exists a small parameter in the consideredequation Therefore the HAM can overcome the foregoingrestrictions and limitations of perturbation methods [49]This method also provides us with great freedom to selectproper base functions to approximate solutions of nonlinearproblems Using one interesting property of homotopy onecan transform any nonlinear problem into an infinite numberof linear problems
In this paper the steady flow of an Ostwald-de Waelepower-law fluid induced by a steadily rotating infinite disk toa non-Darcian fluid-saturated porous medium is consideredThe coupled governing equations are transformed into ordi-nary differential equations in the boundary layerTheOHAMis applied to solve the ODEs The validity of our solutionsis verified by the numerical results (by using a fourth-orderRunge-Kutta and shooting method)
The aim of the present study is to investigate the effect ofhigher order chemical reaction internal heat generation andthe thermal slip boundary condition on the boundary-layerflow of a nanofluid past an upward facing horizontal plate
Lie group analysis is used to develop the similarity transfor-mations and the corresponding similarity representations ofthe governing equations The coupled governing equationsare transformed into ordinary differential equations in theboundary layer The OHAM is applied to solve the ODEsThe obtained solutions are verified by the numerical results(obtained by using a Runge-Kutta-Fehlberg fourth-fifth orderand shooting method) The effect of relevant parameters ondimensionless fluid velocity temperature and nanoparticlevolume fraction are investigated and shown graphically anddiscussed A table containing data for the reduced Nusseltnumber and reduced Sherwood number is also provided toshow the effects of various parameters on them To the best ofour knowledge the effects of thermal slip boundary conditionwith internal heat generation and chemical reaction on theboundary-layer flow of a nanofluid past a horizontal plate inporous media have not been reported in the literature yet
The paper is divided up as follows In Section 2 themath-ematical formulation is presented In Section 3 we used theLie group method to reduce the system of partial differentialequations to a system of ordinary differential equations InSection 4 the basic idea of theHAM is presented In Section 5we derived the OHAM solution of the coupled system ofnonlinear ordinary differential equations In Section 6 wecompared our results with numerical solutions obtainedusing a Runge-Kutta-Fehlberg method In Section 7 weintroduced the physical quantities to be considered andcompared in this paper Section 8 contains the results anddiscussion The conclusions are summarized in Section 9
2 Formulation of the Problem
We consider a two-dimensional laminar free convectiveboundary-layer flow of a nanofluid past an upward facingchemically reacting horizontal plate in a porous media(Figure 1) We assume that a homogeneous isothermal irre-versible chemical reaction of order 119899 takes place betweenthe plate and nanofluid There is internal heat genera-tionabsorption within the fluid inside the boundary layer at
the volumetric rate 119902 Variation of density of the fluid is takeninto account using the Oberbeck-Boussinesq approximationThe conservation of mass momentum energy and nanopar-ticles describing the flow can be written in dimensional form(see [27])
nabla sdot = 0 (1)
120588119891
120576
120597
120597119905
= minus nabla119875 minus
120583
119896
+ [119862120588119875 + (1 minus 119862) 120588119891 (1 minus 120573 (119879 minus 119879infin))] 119892
minus 119896 (119909) (119862 minus 119862infin)119899
(4)
Theflow is assumed to be slow to ignore an advective termand a Forchheimer quadratic drag term in the momentumequation
We consider a steady flow where the Oberbeck-Boussinesq approximation is used In addition we assumethat the nanoparticle concentration is dilute With a suitablechoice for the reference pressure the momentum equationcan be linearized and (2) written as (see [50])
minus nabla119875 minus
120583
119896
+ [(120588119875 minus 120588119891infin) (119862 minus 119862infin)
+ (1 minus 119862infin) 120588119891infin120573 (119879 minus 119879infin)] 119892 = 0
(5)
We also consider the effect of temperature-dependentvolumetric heat generationabsorption in the flow region thatis given by Vajravelu and Hadjinicolaou [51] as
119902 =
Ra2311987601198714311990943
(119879 minus 119879infin) 119879 gt 119879infin(6)
where1198760 is the heat generationabsorption constant Also weconsider the case where the reaction rate varies as
119896 (119909) =
Ra2311989601198714311990943
(7)
where 1198960 is the constant reaction rate
4 Mathematical Problems in Engineering
With these assumptions along with standard boundary-layer approximation the governing equations can be writtenin dimensional form as
120597119906
120597119909
+
120597V120597119910
= 0 (8)
120597119875
120597119909
= minus
120583
119896
119906 (9)
120597119875
120597119910
= minus
120583
119896
V + [(1 minus 119862infin) 120588119891infin119892120573 (119879 minus 119879infin)
minus (120588119875 minus 120588119891infin) 119892 (119862 minus 119862infin)]
(10)
119906
120597119879
120597119909
+ V120597119879
120597119910
= 120572119898
1205972119879
1205971199102
+ 120591 [119863119861
120597119862
120597119910
120597119879
120597119910
+ (
119863119879
119879infin
)(
120597119879
120597119910
)
2
]
+
1198760
(120588119862)119891
Ra23
11987143
11990943
(119879 minus 119879infin)
(11)
119906
120597119862
120597119909
+ V120597119862
120597119910
= 119863119861
1205972119862
1205971199102+ (
119863119879
119879infin
)
1205972119879
1205971199102
minus
Ra2311989601198714311990943
(119862 minus 119862infin)119899
(12)
where 120572119898 = 119896119898(120588119888119875)119891 is the thermal diffusivity of the fluidand 120591 = 120576(120588119862)119901(120588119862)119891 is a parameter
where 1198631(119909) is the thermal slip factor with dimension(length)minus1 The following new nondimensional variables areintroduced to make (8)ndash(13) dimensionless
119909 =
119909
119871radicRa 119910 =
119910
119871
119906 =
119906119871
120572119898radicRa
V =V119871120572119898
120579 =
119879 minus 119879infin
Δ119879
120601 =
119862 minus 119862infin
Δ119862
Δ119879 = 119879119908 minus 119879infin Δ119862 = 119862119908 minus 119862infin
(14)
where Ra = 119892119896120573(1 minus 119862infin)Δ119879119871(120572119898]) is the Rayleigh numberbased on the characteristic length 119871 A stream function 120595
defined by
119906 =
120597120595
120597119910
V = minus120597 120595
120597119909
(15)
is introduced into (8)ndash(13) to reduce the number of depen-dent variables and the number of equations Note that (8) issatisfied identically Hence we have
The parameters in (16)ndash(19) are introduced in Nomencla-ture and defined by
119873119905 = 120591119863119879
Δ119879
120572119898
119879infin 119873119887 = 120591119863119861
Δ119862
120572119898
119873119903 =
(120588119875 minus 120588119891infin) Δ119862
120588119891infin
120573 (1 minus 119862infin) Δ119879
119876 =
11987601198712
120572119898(120588119862)119891
119870 =
11989601198712(Δ119862)119899minus1
120572119898
Le =120572119898
119863119861
(20)
3 Lie Group Analysis
We consider the following scaling group of transformationswhich is a special form of Lie group analysis [52]
Γ 119909lowast= 1199091198901205761205721 119910
lowast= 1199101198901205761205722
120595lowast= 1205951198901205761205723 120579
lowast= 1205791198901205761205724
120601lowast= 1206011198901205761205725 119863
lowast1 = 1198631119890
1205761205726
(21)
Here 120576 is the parameter of the group Γ and 120572119894 (119894 =
1 2 3 4 5 6) are arbitrary real numbers whose connec-tion will be determined by our analysis The transfor-mations in (21) can be considered as a point transfor-mation transforming the coordinates (119909 119910 120595 120579 120601 1198631) to
Mathematical Problems in Engineering 5
(119909lowast 119910lowast 120595lowast 120579lowast 120601lowast 119863lowast1 ) We now investigate the relationship
times Δ 119895 (119909 119910 120579 120601 1205973120595
1205971199103)
(119895 = 1 2 3)
(22)
Since this is the requirement that the differential formsΔ 1 Δ 2 and Δ 3 be reformed under the transformation groupin (19) by using (21) (16)ndash(18) are transformed to (see [3538])
The system will remain invariant (structure of the equa-tions same) under the group transformation Γ if we have thefollowing relationship among the exponents
1205723 minus 21205722 = 1205724 minus 1205721 = 1205725 minus 1205721
1205723 + 1205724 minus 1205721 minus 1205722 = 1205724 minus 21205722 = 1205724 + 1205725 minus 21205722
= 21205724 minus 21205722 = 1205724 minus4
3
1205721
1205723 + 1205725 minus 1205721 minus 1205722 = 1205725 minus 21205722 = 1205724 minus 21205722
= constant = 119891 (120578) (say) that is 120595 = 11990913119891 (120578)
(30b)
6 Mathematical Problems in Engineering
where 119891 is an arbitrary function of 120578 From the equations211988911990931205722119909 = 1198891205790 and 211988911990931205722119909 = 1198891206010 we obtain byintegration
Here 120578 is the similarity variable and 119891 120579 120601 are dependentvariables Note that the similarity transformations in (31) areconsistent with the well-known similarity transformationsreported in the paper of Cheng and Chang [21] for 120582 = 0
in their paper Thus the dimensionless velocity components119906 V can be expressed as
119906 =
1198911015840
11990913 V = minus
1
311990923
(119891 minus 21205781198911015840) (32)
where primes indicate differentiation with respect to 120578
32 Similarity Equations On substituting the transforma-tions in (31) into the governing (16)ndash(18) we obtain thefollowing system of ordinary differential equations
where 119887 = (1198631)0119871 is the thermal slip parameter
4 Basic Idea of the HAM
Let us consider the following differential equation
N [119906 (120591)] = 0 (37)
where N is a nonlinear operator 120591 denotes independentvariable and 119906(120591) is an unknown function respectively Forsimplicity we ignore all boundary or initial conditions whichcan be treated in the similar way By means of generalizing
the traditional homotopy method Liao [45] constructs theso-called zero-order deformation equation
(1 minus 119901)L [120593 (120591 119901) minus 1199060 (120591)] = 119901ℏ119867 (120591)N [120593 (120591 119901)] (38)
where 119901 isin [0 1] is the embedding parameter ℏ = 0 is anonzero auxiliary parameter 119867(120591) = 0 is an auxiliary func-tionL is an auxiliary linear operator 1199060(120591) is an initial guessof 119906(120591) and 120593(120591 119901) is an unknown function respectively Itis important that one has great freedom to choose auxiliarythings in the HAM Obviously when 119901 = 0 and 119901 = 1 itholds 120593(120591 0) = 1199060(120591) 120593(120591 1) = 119906(120591) respectively Thus as119901 increases from 0 to 1 the solution 120593(120591 119901) varies from theinitial guess 1199060(120591) to the solution 119906(120591) Expanding 120593(120591 119901) inTaylor series with respect to 119901 we have
If the auxiliary linear operator the initial guess theauxiliary parameter ℏ and the auxiliary function are soproperly chosen the series (39) converges at 119901 = 1 then wehave
119906 (120591) = 1199060 (120591) +
+infin
sum
119898=1
119906119898 (120591) (41)
which must be one of solutions of original nonlinear equa-tion as proved by Liao [45] As ℏ = minus1 and 119867(120591) = 1 (38)becomes
(1 minus 119901)L [120593 (120591 119901) minus 1199060 (120591)] + 119901N [120593 (120591 119901)] = 0 (42)
which is used mostly in the homotopy perturbation methodwhere the solution can be obtained directly without usingTaylor series
According to definition (39) the governing equationcan be deduced from the zero-order deformation equation(37) By defining the vector 119899 = 1199060(120591) 1199061(120591) 119906119899(120591)
and differentiating equation (37) 119898 times with respect tothe embedding parameter 119901 and then setting 119901 = 0 andfinally dividing them by119898 we have the so-called119898th-orderdeformation equation
L [119906119898 (120591) minus 120594119898119906119898minus1 (120591)] = ℏ119867 (120591) 119877119898 (119898minus1) (43)
It should be emphasized that 119906119898(120591) for 119898 ge 1 isgoverned by the linear equation (39) with the linear boundaryconditions that come from original problem which canbe easily solved by symbolic computation software such asMAPLE and MATHEMATICA
Figure 3 The ℏ-curve of 1205791015840(0) given by the various HAM-orderapproximate solution
5 HAM Solution
In this section we applied the HAM to obtain approx-imate analytical solutions of the effect of higher orderchemical reaction internal heat generation and the thermalslip boundary condition on the boundary-layer flow of ananofluid past an upward facing horizontal plate (33)ndash(35)
Figure 4 The ℏ-curve of 1206011015840(0) given by the various HAM-orderapproximate solution
We start with the initial approximation 1198650(120578) = 0 Θ0(120578) =(1(119887 + 1))Exp[minus120578]Φ0(120578) = Exp[minus120578] and the linear operator
L [119865 (120578 119901)] =
1205972119865 (120578 119901)
1205971205783
L [Θ (120578 119901)] =
1205972Θ(120578 119901)
1205971205782
+
120597Θ (120578 119901)
120597120578
L [Φ (120578 119901)] =
1205972Φ(120578 119901)
1205971205782
+
120597Φ (120578 119901)
120597120578
(45)
Furthermore from (33)ndash(35) we define the nonlinearoperators
N [119865 (120578 119901)] =
1205972119865 (120578 119901)
1205971205782
minus
2
3
120578(
120597Θ (120578 119901)
120597120578
minus 119873119903
120597Φ (120578 119901)
120597120578
)
N [Θ (120578 119901)] =
1205972Θ(120578 119901)
1205971205782
+
1
3
119865 (120578 119901)
120597Θ (120578 119901)
120597120578
+ 119873119887
120597Θ (120578 119901)
120597120578
120597Φ (120578 119901)
120597120578
+ 119873119905(
120597Θ (120578 119901)
120597120578
)
2
+ 119876Θ (120578 119901)
N [Φ (120578 119901)] =
1205972Φ(120578 119901)
1205971205782
+
Le3
119865 (120578 119901)
120597Φ (120578 119901)
120597120578
+
119873119905
119873119887
1205972Θ(120578 119901)
1205971205782
minus 119870(Φ (120578 119901))119899
(46)
8 Mathematical Problems in Engineering
Using the above definition with assumption 119867119865(120591) = 1119867Θ(120591) = Exp[minus120578] and 119867Φ(120591) = Exp[minus120578] we construct the119911119890119903119900th-order deformation equation
(1 minus 119901)L [120593 (119909 119901) minus 1199060 (119909)] = 119867 (120591) 119901ℏN [120593 (119909 119901)]
Differentiating the 119911119890119903119900th-order deformation equation(47)119898 times with respect to 119901 and finally dividing by119898 wehave the119898th-order deformation equation
L [119865119898 (119909) minus 120594119898119865119898minus1 (119909)] = ℏ119877119898 (119898minus1)
L [Θ119898 (119909) minus 120594119898Θ119898minus1 (119909)] = Exp [minus120578] ℏ119877119898 (Θ119898minus1)
L [Φ119898 (119909) minus 120594119898Φ119898minus1 (119909)] = Exp [minus120578] ℏ119877119898 (Φ119898minus1)
By using the symbolic software MATHEMATICA tosolve the system of linear equations (49) with the boundaryconditions (46) it can be obtained finally
1198651 (120578) = minus
4
3 (1 + 119887)
ℏ +
4119890minus120578
3 (1 + 119887)
ℏ +
4119873119903
3 (1 + 119887)
ℏ
+
4119887119873119903
3 (1 + 119887)
ℏ minus
4119873119903119890minus120578
3 (1 + 119887)
ℏ minus
4119887119873119903119890minus120578
3 (1 + 119887)
ℏ
+
2119890minus120578120578
3 (1 + 119887)
ℏ minus
2119873119903119890minus120578120578
3 (1 + 119887)
ℏ minus
2119887119873119903119890minus120578120578
3 (1 + 119887)
ℏ
Θ1 (120578) =119890minus2120578
2 (1 + 119887)
ℏ minus
119890minus120578
2(1 + 119887)2ℏ minus
119887119890minus120578
2(1 + 119887)2ℏ
minus
119873119887119890minus3120578
6(1 + 119887)2ℏ +
119887119873119887119890minus3120578
6(1 + 119887)2ℏ minus
119873119887119890minus120578
6(1 + 119887)2ℏ
minus
119887119873119887119890minus120578
6(1 + 119887)2ℏ +
119873119905119890minus3120578
6(1 + 119887)2ℏ minus
119873119905119890minus120578
6(1 + 119887)2ℏ
+
119876119890minus2120578
2 (1 + 119887)
ℏ minus
119876119890minus120578
2(1 + 119887)2ℏ minus
119887119876119890minus120578
2(1 + 119887)2ℏ
Φ1 (120578) =119890minus2120578
2 (1 + 119887)
ℏ +
119887119890minus2120578
2 (1 + 119887)
ℏ minus
119890minus120578
2 (1 + 119887)
ℏ
minus
119887119890minus120578
2 (1 + 119887)
ℏ minus
119870119890minus2120578
2 (1 + 119887)
ℏ minus
119887119870eminus2120578
2 (1 + 119887)
ℏ
+
119870119890minus120578
2 (1 + 119887)
ℏ +
119887119870119890minus120578
2 (1 + 119887)
ℏ +
119873119905
119873119887
119890minus2120578
2 (1 + 119887)
ℏ
minus
119873119905
119873119887
119890minus120578
2 (1 + 119887)
ℏ
(56)
The higher orders solutions of 119865(120578) Θ(120578) and Φ(120578) weretoo long to be mentioned here therefore they are showngraphically
51 Convergence of the HAM Solution As mentioned by Liao[45] HAM provides us with great freedom in choosing thesolution of a nonlinear problem by different base functionsThis has a great effect on the convergence region because theconvergence region and rate of a series are chiefly determinedby the base functions used to express the solution We usedseveral terms in evaluating the approximate solution 119865app asympsum119899119894=0 119865119894Θapp asymp sum
119899119894=0Θ119894Φapp asymp sum
119899119894=0Φ119894 note that the solution
series contains the auxiliary parameter ℏ which provides uswith a simple way to adjust and control the convergence ofthe solution series Generally by means of the so-called ℏ-curve that is a curve of a versus ℏ As pointed by Liao [45]the valid region of ℏ is a horizontal line segment Figures2 3 and 4 show the ℏ-curve with the various order of theHAM for 11989110158401015840(0) 1205791015840(0) 1206011015840(0) respectively when 119873119887 = 01
Figure 6 The residual error of 120579(120578) for (34) using 12th order ofapproximations
119873119905 = 01 119873119903 = 01 119876 = 01 Le = 10 119870 = 10 119899 = 10119887 = 00 It can be seen that when the order of series is 12 thesegment of the horizontal line is more than the other ordersFor example it can be found that for 11986510158401015840(0) the acceptablerange ofℏ is betweenminus05 andminus20 for 12th order of theHAMbut for 8th order of the HAM the acceptable range of ℏ isbetween minus10 and minus20 or minus10 and minus15 for 4th order of theHAM so horizontal line segment of 12th order of the HAM ismore than others Therefore it is straightforward to choosean appropriate range for ℏ which ensures the convergence
10 Mathematical Problems in Engineering
minus00002
120578
= minus10
= minus111 (optimal value)= minus12
Nt = 01Nr = 01
Q = 01Le = 10K = 10n = 10b = 00
0006
0004
0002
0
0 2 4 6 8 10
Nb = 01
Res (120601
(120578))
ℏ
ℏ
ℏ
Figure 7 The residual error of 120601(120578) for (35) using 12th order ofapproximations
of the solution series To choose optimal value of auxiliaryparameter the averaged residual errors (see [53] for moredetails) are defined as
119864119865119898 =1
119870
119870
sum
119894=0
[N119865(119898
sum
119894=0
119865119894 (119894Δ119909))]
2
119864Θ119898 =1
119870
119870
sum
119894=0
[NΘ(119898
sum
119894=0
Θ119894 (119894Δ119909))]
2
119864Φ119898 =1
119870
119870
sum
119894=0
[NΦ(119898
sum
119894=0
Φ119894 (119894Δ119909))]
2
(57)
where Δ119909 = 10119870 and 119870 = 20 For given order ofapproximation 119898 and the optimal values ℏ are given bythe minimum of 119864119898 corresponding to nonlinear algebraicequations
119889119864119865119898
119889ℏ
= 0
119889119864Θ119898
119889ℏ
= 0
119889119864Φ119898
119889ℏ
= 0 (58)
It is noted that the optimal value of ℏ is replaced intothe equations Table 1 shows optimal values obtained for theauxiliary parameter ℏ for various quantities of119873119905 119876 and 119899when119873119887 = 01119873119903 = 01 Le = 10 119870 = 10 119887 = 00 To see
the accuracy of the solutions the residual errors are definedfor the system as (for119898 order approximation)
Res119865119898
=
1198892119865119898
1198891205782minus
2
3
120578(
119889Θ119898
119889120578
minus 119873119903
119889Φ119898
119889120578
) (59)
ResΘ119898
=
1198892Θ119898
1198891205782+
1
3
119865119898
119889Θ119898
d120578+ 119873119887
119889Θ119898
119889120578
119889Φ119898
119889120578
+ 119873119905(
119889Θ119898
119889120578
)
2
+ 119876Θ119898
(60)
ResΦ119898
=
1198892Φ119898
1198891205782+
Le3
119865119898
119889Φ119898
119889120578
+
119873119905
119873119887
1198892Θ119898
1198891205782
minus 119870(
119889Φ119898
119889120578
)
119899
(61)
Figures 5 6 and 7 show the residual errors for 12th-orderdeformation solutions when 119873119887 = 01 119873119905 = 01 119873119903 = 01119876 = 01 Le = 10 119870 = 10 119899 = 10 119887 = 00 For example ℏ =minus145 has the minimum range of residual curve in Figure 5and so on
Graphical representation of results is very useful todemonstrate the efficiency and accuracy of the HAM for theabove problem
6 Comparisons and Verification
It is worth citing that for isothermal plate (119887 = 0) and in theabsence of internal heat generationabsorption (119876 = 0) andchemical reaction (119870 = 0) our problem reduces to Gorla andChamkha [26] and Khan and Pop [27] To verify the accuracyof our results the present results are compared in Table 2with Gorla and Chamkha [26] and are found to be in goodagreement
7 Physical Quantities
Theparameters of physical interest of the present problem arethe local skin friction factor 119862119891119909 the local Nusselt numberNu119909 and the local Sherwood number Sh119909 respectivelyPhysically 119862119891119909 indicates wall shear stress Nu119909 indicates therate of wall heat transfer whilst Sh119909 indicates the rate ofwall nanoparticle volume fractionThe following relations areused to find these quantities
119862119891119909 =2120583
1205881198802119903
(
120597119906
120597119910
)
119910=0
Nu119909 =minus119909
119879119891 minus 119879infin
(
120597119879
120597119910
)
119910=0
Sh119909 =minus119909
119862119908 minus 119862infin
(
120597119862
120597119910
)
119910=0
(62)
Mathematical Problems in Engineering 11
Table 2 Comparison of present solution with Gorla and Chamkha[26] for different values of buoyancy and nanofluid parameters
Present results (HAM) Gorla and Chamkha [26]minus1205791015840(0) minus120601
1015840(0) minus120579
1015840(0) minus120601
1015840(0)
Nr 119873119887 = 03119873119905 = 01 Le = 10 119887 = 119870 = 119876 = 0
01 032578 148242 326119864 minus 01 148416402 032385 146704 325119864 minus 01 146816103 032188 145125 322119864 minus 01 145266404 031985 143503 321119864 minus 01 143639205 031777 141833 319119864 minus 01 1419499Nt 119873119887 = 03119873119903 = 05 Le = 10 119887 = 119870 = 119876 = 0
01 031777 141833 319119864 minus 01 141949902 030486 141491 305119864 minus 01 141653603 02927 141561 293119864 minus 01 141686604 028125 141991 282119864 minus 01 142158205 027046 142737 271119864 minus 01 1429226Nb 119873119905 = 01119873119903 = 05 Le = 10 119887 = 119870 = 119876 = 0
01 03672 132611 368119864 minus 01 132745402 034271 139216 343119864 minus 01 139361503 031777 141833 319119864 minus 01 141949904 029399 143428 294119864 minus 01 143546405 027161 144598 272119864 minus 01 144772
Figure 10 Comparison of dimensionless concentration profile 120601(120578)obtained by various orders of the HAM
Here Ra119909 = 119892119870120573(1minus119862infin)Δ119879119909(120572119898]) is the local Rayleighnumber Pr = ]120572119898 is the Prandtl number and 119880119903 = (1 minus
119862infin)119892119870120573Δ119879120572119898 is the reference velocity
8 Results and Discussion
Graphical representation of results is very useful to discussthe physical features presented by the solutions This sectiondescribes the influence of some interesting parameters on
12 Mathematical Problems in Engineering
0 2 4 6 8 10
02
04
06
08
1
Nr = 01Numerical
Numerical
NumericalNr = 03Nr = 05
f998400 (120578)
120578
Nt = 01Nb = 01Q = 01Le = 10K = 10n = 10b = 00
Figure 11 Effect of buoyancy ratio119873119903 on the dimensionless velocityprofile
0 2 4 6 8 10
02
04
06
08
1
Nt = 01Nb = 01Q = 01Le = 10K = 10n = 10b = 00
Nr = 01Numerical
Numerical
NumericalNr = 03Nr = 05
120578
120579(120578
)
Figure 12 Effect of buoyancy ratio 119873119903 on the dimensionlesstemperature profile
the velocity and temperature fields Equations (33) to (35)with boundary conditions in (36) were solved analytically byHAM and numerically using Runge-Kutta-Fehlberg fourth-fifth order proposed by Aziz [54]
Figures 8 9 and 10 respectively for 1198911015840(120578) 120579(120578) and 120601(120578)show the comparisons between various order approximationof the optimal HAM and the numerical solutions for the case
0 2 4 6 8 10
02
04
06
08
1
Nr = 01Numerical
Numerical
NumericalNr = 03Nr = 05
Nt = 01Nb = 01Q = 01Le = 10K = 10n = 10b = 00
120578
120601(120578
)Figure 13 Effect of buoyancy ratio 119873119903 on the dimensionlessconcentration profile
0 2 4 6 8 10
02
04
06
08
1
12
NumericalNumerical
Numerical
f998400 (120578
)
120578
Nb = 01
Nb = 03
Nb = 05
Nt = 01Nr = 01Q = 01Le = 10K = 10n = 10b = 00
Figure 14 Effect of Brownian motion 119873119887 on the dimensionlessvelocity profile
119873119887 = 01 119873119905 = 01 119873119903 = 01 119876 = 01 Le = 10 119870 = 10119899 = 10 119887 = 00 For better presentation these results arepresented in Tables 3 4 and 5 Also Tables 6 7 and 8 presentthe comparisons between various order approximation of theoptimal HAM and the numerical solutions for the case119873119887 =03 119873119905 = 005 119873119903 = 03 119876 = 01 Le = 20 119870 = 05
Mathematical Problems in Engineering 13
0 2 4 6 8 10
02
04
06
08
1
0
120578
NumericalNumerical
Numerical
Nb = 01
Nb = 03
Nb = 05
Nt = 01Nr = 01Q = 01Le = 10K = 10n = 10b = 00
120579(120578
)
Figure 15 Effect of Brownian motion 119873119887 on the dimensionlesstemperature profile
02
04
06
08
1
00
2 4 6 8 10120578
NumericalNumerical
Numerical
Nb = 01
Nb = 03
Nb = 05
Nt = 01Nr = 01Q = 01Le = 10K = 10n = 10b = 00
120601(120578
)
Figure 16 Effect of Brownian motion 119873119887 on the dimensionlessconcentration profile
119899 = 20 119887 = 01 It is observed that the results of 12th-order approximation of the optimal HAM are very close tothe numerical solutions which confirm the validity of thesemethods
In the following figures effects of various physicalparameters on the dimensionless velocity temperature and
Table 3 Comparison of values of 1198911015840(120578) obtained by various ordersof the HAM with numerical solution when 119873119887 = 01119873119905 =
temperature profiles will be investigated These results havebeen obtained by 12th order of the HAM and have beenvalidated by numerical results Figures 11ndash13 respectivelyrepresent that the comparison of solutions of 1198911015840(120578) 120579(120578)and 120601(120578) for fix values 119873119887 = 01 119873119905 = 01 119876 = 01Le = 10 119870 = 10 119899 = 10 119887 = 00 and different values ofbuoyancy ratio119873119903 In Figure 11 it is clear that velocity of thefluid tremendously decreases in the near of horizontal platewith an increase in the buoyancy ratio It is observed that thetemperature increases slightly but concentration of the fluiddoes not vary sensibly (Figures 12 and 13) The effects of theBrownian motion 119873119887 are depicted in Figures 14 15 and 16when119873119903 = 01119873119905 = 01119876 = 01 Le = 10119870 = 10 119899 = 10
14 Mathematical Problems in Engineering
Table 6 Comparison of values of 1198911015840(120578) obtained by various ordersof the HAM with numerical solution when 119873119887 = 03119873119905 =
119887 = 00 If 119873119887 increases 1198911015840(120578) and 120579(120578) increase lightly and120601(120578) decreases The dimensionless velocity dimensionlesstemperature and dimensionless concentration profiles fordifferent values of thermophoresis 119873119905 with constant values119873119903 = 01 119873119887 = 01 119876 = 01 Le = 10 119870 = 10119899 = 10 119887 = 00 are presented in Figures 17 18 and19 It is observed that the velocity and temperature of thefluid are not impressible from119873119905 but concentration increasesslightly In Figures 20 21 and 22 respectively comparison ofsolutions of 1198911015840(120578) 120579(120578) and 120601(120578) for 119873119903 = 01 119873119887 = 01119873119905 = 01 Le = 10 119870 = 10 119899 = 10 119887 = 00 anddifferent values of heat generationabsorption parameter 119876are shown It is observed that in presence of heat source1198911015840(120578) 120579(120578) both increase while heat sink causes that 1198911015840(120578)
120579(120578) both decrease extremely These effects are more visible
Table 9 Values of reduced Nusselt number and Sherwood numberobtained by HAM for different values of the parameters Le 119870 119899 119887and 119876 when119873119887 = 119873119903 = 119873119905 = 01
Figure 17 Effect of thermophoresis 119873119905 on the dimensionlessvelocity profile
for temperature distribution For 120601(120578) the effect of changing119876 is seen to be almost insignificant This is in agreementwith the physical factThe dimensionless profiles for differentvalues of Lewis number Le with constant parameters 119873119903 =01 119873119887 = 01 119873119905 = 01 119876 = 01 119870 = 10 119899 = 10119887 = 00 are shown in Figures 23 24 and 25 It is clear that forthese constant parameters variation of Lewis number has anyconsiderable effect on the velocity and temperature profilesbut concentration profile decreases with the increase of Lewisnumber What is similar to these results can be detected inFigures 26 27 and 28 for chemical reaction parameter 119870Figures 29 30 and 31 respectively present the comparisonof solutions of 1198911015840(120578) 120579(120578) and 120601(120578) for 119873119903 = 05 119873119887 = 05119873119905 = 01 119876 = 001 Le = 10 119870 = 10 119887 = 00 and differentvalues of order of chemical reaction 119899 It is observed that
Mathematical Problems in Engineering 15
0 2 4 6 80
1
120578
Nt = 00
Nt = 005
Nt = 10NumericalNumerical
Numerical
08
06
04
02
10
Q = 01Le = 10K = 10n = 10b = 00
Nb = 01
120579(120578
)
Nr = 01
Figure 18 Effect of thermophoresis 119873119905 on the dimensionlesstemperature profile
0 2 4 6 80
1
120601(120578
)
120578
Nt = 00
Nt = 005
Nt = 10NumericalNumerical
Numerical
08
06
04
02
10
Q = 01Le = 10K = 10n = 10b = 00
Nb = 01Nr = 01
Figure 19 Effect of thermophoresis 119873119905 on the dimensionlessconcentration profile
for higher order of chemical reaction velocity decreases andtemperature and concentration profiles increase extremelyUltimately Figures 32 33 and 34 depict the effect of thermalslip parameter 119887 on velocity temperature and concentrationfunctions when 119873119903 = 01 119873119887 = 01 119873119905 = 01 119876 = 01Le = 20 119870 = 15 119899 = 10 It is seen that the velocityand temperature of the fluid decrease with the increase of the
0 2 4 6 80
1
12
10
02
04
06
08
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numericalf998400 (120578
)
120578
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 20 Effect of generationabsorption heat parameter119876 on thedimensionless velocity profile
0 2 4 6 8 100
02
04
06
08
1
120579(120578
)
120578
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numerical
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 21 Effect of generationabsorption heat parameter 119876 on thedimensionless temperature profile
thermal slip parameter in the near of horizontal plate whilethe concentration of the fluid does not vary patently
Also for investigation of the parameters of physicalinterest Table 9 is presented In this table numerical valuesof reduced Nusselt number and Sherwood number obtainedby HAM for different values of the parameters 119890 119870 119899 119887 and119876 can be compared when119873119887 = 119873119903 = 119873119905 = 01
16 Mathematical Problems in Engineering
10
02
04
06
08
0 2 4 6 80
1
120578
120601(120578
)
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numerical
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 22 Effect of generationabsorption heat parameter119876 on thedimensionless concentration profile
0 2 4 6 80
1
12
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 23 Effect of Lewis number Le on the dimensionless velocityprofile
9 Conclusions
In this paper we studied the steady laminar incompressiblefree convective flow of a nanofluid past a chemically reactingupward facing horizontal plate in porous medium takinginto account heat generation and the thermal slip boundarycondition
0 2 4 6 80
1
120579(120578
)
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 24 Effect of Lewis number Le on the dimensionlesstemperature profile
0 2 4 6 80
1
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 25 Effect of Lewis number Le on the dimensionlessconcentration profile
The governing partial differential equations have beentransformed by similarity transformations into a system ofordinary differential equations which are solved by OHAMand numerical method (fourth-order Runge-Kutta schemewith the shooting method) Dimensionless velocity temper-ature and concentration functions are presented for various
Mathematical Problems in Engineering 17
0 2 4 6 80
1
12
K = 00
K = 05
K = 10NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 26 Effect of chemical reaction parameter 119870 on the dimen-sionless velocity profile
0 2 4 6 80
1
K = 00
K = 05
K = 10NumericalNumerical
Numerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 27 Effect of chemical reaction parameter 119870 on the dimen-sionless temperature profile
values of parameters for the problem for example buoy-ancy ratio 119873119903 Brownian motion 119873119887 thermophoresis 119873119905heat generationabsorption parameter 119876 Lewis number Lechemical reaction parameter119870 order of chemical reaction 119899and thermal slip parameter 119887 From the present investigationthe following may be concluded
(i) The velocity 1198911015840(120578) increases with increasing 119873119887 anddecreases with increasing119873119903 119899 and 119887
0 2 4 6 80
1
K = 00
K = 05
K = 10NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 28 Effect of chemical reaction parameter 119870 on the dimen-sionless concentration profile
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 29 Effect of order of chemical reaction 119899 on the dimension-less velocity profile
(ii) The temperature 120579(120578) increases with 119873119903 119873119887 119899 anddecreases with 119887
(iii) Parameters of119873119905 Le and119870 have no effect on velocityand temperature profiles
(iv) The concentration distribution 120601(120578) increases withincreasing N119905 119899 and decreases with 119873119887 Le and 119870but119873119903 119876 and 119887 are not affected by it
18 Mathematical Problems in Engineering
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 30 Effect of order of chemical reaction 119899 on the dimension-less temperature profile
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 31 Effect of order of chemical reaction 119899 on the dimension-less concentration profile
(v) In presence of heat source (119876 gt 0) 1198911015840(120578) 120579(120578) bothincrease while heat sink (119876 lt 0) causes that 1198911015840(120578)120579(120578) both decrease extremely These effects are morevisible for temperature distribution
In addition numerical results for the reducedNusselt andSherwood numbers are tabulated It is concluded that thereduced Nusselt number increases with the Lewis numberand reaction parameter whist it decreases with the order of
0 2 4 6 8
1
12
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 32 Effect of order of thermal slip parameter 119887 on thedimensionless velocity profile
0 2 4 6 80
1
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 33 Effect of order of thermal slip parameter 119887 on thedimensionless temperature profile
chemical reaction thermal slip and generation parametersIt is further concluded that the reduced Sherwood numberenhances with the Lewis number generation and reactionparameters whist it suppresses with thermal slip and orderof chemical reaction
The convergence of the solution series and the power ofHAM in controlling and adjusting the convergence regionand rate of solution series were discussed The proper range
Mathematical Problems in Engineering 19
0 2 4 6 80
1
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 34 Effect of order of thermal slip parameter 119887 on thedimensionless concentration profile
of auxiliary parameter ℏ to ensure the convergence of thesolution series was obtained through the so called ℏ-curveThe HAM provides us with a convenient way to control theconvergence of approximation series which is a fundamentalqualitative difference in analysis between the HAM andother methods The numerical results of the above problemsdisplay a fast convergence with minimal calculations Thisshows that the HAM is a very efficient method Finally theagreement between analytical and numerical results of thepresent study with previous published results is excellent
Nomenclature
119887 Thermal slip parameter119862 Dimensional concentration119862119891 Skin friction factor1198631 Thermal slip factor119863119861 Brownian diffusion coefficient119863119879 Thermophoretic diffusion coefficient119891 Dimensionless velocity functions119892 Gravitation acceleration119870 Chemical reaction parameter119896 Permeability of the porous media1198960 The constant reaction rate119896119898 Effective thermal conductivity of the porous
medium119896(119909) Variable reaction rateL Linear operator of the HAM119871 Length of horizontal plateLe Lewis numberN Nonlinear operator of the HAM
119899 Order of chemical reaction119873119887 Brownian motion119873119903 Buoyancy ratio119873119905 ThermophoresisNu Nusselt number119875 PressurePr Prandtl number119876 Heat generationabsorption parameter1198760 Heat generationabsorption constant119902 Internal heat generation rate
Ra Rayleigh numberSh Sherwood number119879 Temperature119905 Time119906 Velocity in 119909-direction119880119903 Reference velocity Velocity vectorV Velocity in 119910-direction119909 Distance along the surface119910 Distance normal to the surface
Greek Letters
120572119898 Thermal diffusivity120573 Volumetric expansion coefficient of nanofluid120576 Porosity of the porous media120601 Dimensionless concentration120578 Similarity variable120583 Dynamic viscosity120579 Dimensionless temperature120588119891 Density of the base fluid120588119901 Density of the nanoparticles(120588119862)119891 Effective heat capacity of the fluid(120588119862)119875 Effective heat capacity of the nanoparticle material120595 Stream function
Subscript Superscript
infin Conditions far away from the surface1015840 Differentiation with respect to 120578
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
M M Rashidi wishes to thank the Centre for DifferentialEquations Continuum Mechanics and Applications Schoolof Computational and Applied Mathematics University ofthe Witwatersrand for the hospitality during his visit EMomoniat thanks theNational Research Foundation of SouthAfrica for their support
20 Mathematical Problems in Engineering
References
[1] E Abu-Nada F O Hakan and I Pop ldquoBuoyancy inducedflow in a nanofluid filled enclosure partially exposed to forcedconvectionrdquo Superlattices and Microstructures vol 51 no 3 pp381ndash395 2012
[2] S Kakac and A Pramuanjaroenkij ldquoReview of convective heattransfer enhancement with nanofluidsrdquo International Journal ofHeat and Mass Transfer vol 52 no 13-14 pp 3187ndash3196 2009
[3] J A Eastman S U S Choi S Li W Yu and L J ThompsonldquoAnomalously increased effective thermal conductivities ofethylene glycol-based nanofluids containing copper nanoparti-clesrdquo Applied Physics Letters vol 78 no 6 pp 718ndash720 2001
[4] Y Xuan and Q Li ldquoHeat transfer enhancement of nanofluidsrdquoInternational Journal of Heat and Fluid Flow vol 21 no 1 pp58ndash64 2000
[5] S K Das N Putra P Thiesen and W Roetzel ldquoTemperaturedependence of thermal conductivity enhancement for nanoflu-idsrdquo Journal of Heat Transfer vol 125 no 4 pp 567ndash574 2003
[6] D H Kumar H E Patel V R R Kumar T SundararajanT Pradeep and S K Das ldquoModel for heat conduction innanofluidsrdquo Physical Review Letters vol 93 no 14 Article ID144301 2004
[7] H E Patel T Sundararajan T Pradeep A Dasgupta NDasgupta and S K Das ldquoA micro-convection model forthermal conductivity of nanofluidsrdquo Pramana vol 65 no 5 pp863ndash869 2005
[8] A V Kuznetsov ldquoNon-oscillatory and oscillatory nanofluidbio-thermal convection in a horizontal layer of finite depthrdquoEuropean Journal of Mechanics vol 30 no 2 pp 156ndash165 2011
[9] M Napoli J C T Eijkel and S Pennathur ldquoNanofluidictechnology for biomolecule applications a critical reviewrdquo Labon a Chip vol 10 no 8 pp 957ndash985 2010
[10] J C Crepeau and R Clarksean ldquoSimilarity solutions of naturalconvection with internal heat generationrdquo Journal of HeatTransfer vol 119 no 1 pp 183ndash185 1997
[11] A Z Sahin ldquoTransient heat conduction in semi-infinite solidwith spatially decaying exponential heat generationrdquo Interna-tional Communications in Heat andMass Transfer vol 19 no 3pp 349ndash358 1992
[12] O D Makind and A Aziz ldquoMixed convection from a con-vectively heated vertical plate to a fluid with internal heatgenerationrdquo Journal of Heat Transfer vol 133 Article ID 1225016 pages 2011
[13] J Buongiorno ldquoConvective transport in nanofluidsrdquo Journal ofHeat Transfer vol 128 no 3 pp 240ndash250 2006
[14] AV Kuznetsov andDANield ldquoNatural convective boundary-layer flow of a nanofluid past a vertical platerdquo InternationalJournal of Thermal Sciences vol 49 no 2 pp 243ndash247 2010
[15] L B Godson D Raja L D Mohan and S WongwisescldquoEnhancement of heat transfer using nanofluids-an overviewrdquoRenewable and Sustainable Energy Reviews vol 14 no 2 pp629ndash641 2010
[16] D A Nield and A BejanConvection in PorousMedia SpringerNew York NY USA 3rd edition 2006
[17] P Vadasz Emerging Topics in Heat and Mass Transfer in PorousMedia Springer New York NY USA 2008
[18] K Vafai Porous Media Applications in Biological Systems andBiotechnology CRC Press New York NY USA 2010
[19] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application to
heat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977
[20] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for natural convective boundary-layer flow in a porousmedium saturated by a nanofluidrdquo International Journal of Heatand Mass Transfer vol 52 no 25-26 pp 5792ndash5795 2009
[21] P Cheng and I D Chang ldquoBuoyancy induced flows in asaturated porous medium adjacent to impermeable horizontalsurfacesrdquo International Journal of Heat and Mass Transfer vol19 no 11 pp 1267ndash1272 1976
[22] I D Chang and P Cheng ldquoMatched asymptotic expansionsfor free convection about an impermeable horizontal surfacein a porous mediumrdquo International Journal of Heat and MassTransfer vol 26 no 2 pp 163ndash174 1983
[23] D S Shiunlin and B Gebhart ldquoBuoyancy-induced flow adja-cent to a horizontal surface submerged in porous mediumsaturated with cold waterrdquo International Journal of Heat andMass Transfer vol 29 no 4 pp 611ndash623 1986
[24] J H Merkin and G Zhang ldquoOn the similarity solutions for freeconvection in a saturated porousmedium adjacent to imperme-able horizontal surfacesrdquoWarme- und Stoffubertragung vol 25no 3 pp 179ndash184 1990
[25] M A Chaudhary J H Merkin and I Pop ldquoNatural convectionfrom a horizontal permeable surface in a porous mediummdashnumerical and asymptotic solutionsrdquo Transport in PorousMedia vol 22 no 3 pp 327ndash344 1996
[26] R S R Gorla and A Chamkha ldquoNatural convective boundarylayer flow over a nonisothermal vertical plate embedded ina porous medium saturated with a nanofluidrdquo Nanoscale andMicroscale Thermophysical Engineering vol 15 no 2 pp 81ndash942011
[27] W A Khan and I Pop ldquoFree convection boundary layer flowpast a horizontal flat plate embedded in a porous medium filledwith a nanofluidrdquo Journal of Heat Transfer vol 133 no 9 ArticleID 094501 2011
[28] A Aziz W A Khan and I Pop ldquoFree convection boundarylayer flow past a horizontal flat plate embedded in porousmedium filled by nanofluid containing gyrotactic microorgan-ismsrdquo International Journal of Thermal Sciences vol 56 pp 48ndash57 2012
[29] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer New York NYUSA 2009
[30] A A Avramenko S G Kobzar I V Shevchuk A V Kuznetsovand L T Iwanisov ldquoSymmetry of turbulent boundary-layerflows investigation of different eddy viscosity modelsrdquo ActaMechanica vol 151 no 1-2 pp 1ndash14 2001
[31] A V Kuznetsov A A Avramenko and P Geng ldquoAnalyticalinvestigation of a falling plume caused by bioconvection ofoxytactic bacteria in a fluid saturated porous mediumrdquo Inter-national Journal of Engineering Science vol 42 no 5-6 pp 557ndash569 2004
[32] M Jalil S Asghar and M Mushtaq ldquoLie group analysis ofmixed convection flow with mass transfer over a stretchingsurface with suction or injectionrdquo Mathematical Problems inEngineering vol 2010 Article ID 264901 14 pages 2010
[33] M A A Hamad M J Uddin and A I M Ismail ldquoRadiationeffects on heat and mass transfer in MHD stagnation-pointflow over a permeable flat plate with thermal convectivesurface boundary condition temperature dependent viscosityand thermal conductivityrdquoNuclear Engineering and Design vol242 pp 194ndash200 2012
Mathematical Problems in Engineering 21
[34] A Aziz M J Uddin M A A Hamad and A I MdIsmail ldquoMHD flow over an inclined radiating plate with thetemperature-dependent thermal conductivity variable reactiveindex and heat generationrdquoHeat Transfer vol 41 no 3 pp 241ndash259 2012
[35] A G Hansen Similarity Analysis of Boundary Layer Problemsin Engineering Prentice Hall Englewood Cliffs NJ USA 1964
[36] W FAmesNonlinear Partial Differential Equations in Engineer-ing Academic Press New York NY USA 1972
[37] R Seshadri and T Y Na Group Invariance in EngineeringBoundary Value Problems Springer New York NY USA 1985
[38] D Shang Theory of Heat Transfer with Forced Convection FilmFlows vol 3 of Heat and Mass Transfer 2010
[40] E Erfani M M Rashidi and A B Parsa ldquoThe modifieddifferential transform method for solving off-centered stag-nation flow toward a rotating discrdquo International Journal ofComputational Methods vol 7 no 4 pp 655ndash670 2010
[41] M M Rashidi N Laraqi and A Basiri Parsa ldquoAnalyticalmodeling of heat convection in magnetized micropolar fluid byusing modified differential transform methodrdquo Heat Transfervol 40 no 3 pp 187ndash204 2011
[42] M M Rashidi and M Keimanesh ldquoUsing differential trans-form method and pade approximant for solving mhd flow ina laminar liquid film from a horizontal stretching surfacerdquoMathematical Problems in Engineering vol 2010 Article ID491319 14 pages 2010
[43] M M Rashidi and G Domairry ldquoNew analytical solution ofthe three-dimensional NavierStokes equationsrdquoModern PhysicsLetters B vol 23 no 26 pp 3147ndash3155 2009
[44] M M Rashidi M Keimanesh O A Beg and T K HungldquoMagnetohydrodynamic biorheological transport phenomenain a porous medium a simulation of magnetic blood flowcontrol and filtrationrdquo International Journal for NumericalMethods in Biomedical Engineering vol 27 no 6 pp 805ndash8212011
[45] S J Liao The proposed homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992
[46] M M Rashidi G Domairry and S Dinarvand ldquoApproximatesolutions for the Burger and regularized long wave equationsby means of the homotopy analysis methodrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 14 no 3 pp708ndash717 2009
[47] Z Ziabakhsh and G Domairry ldquoAnalytic solution of naturalconvection flow of a non-Newtonian fluid between two verticalflat plates using homotopy analysis methodrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 14 no 5pp 1868ndash1880 2009
[48] M M Rashidi S A Mohimanian Pour T Hayat and SObaidat ldquoAnalytic approximate solutions for steady flow over arotating disk in porousmediumwith heat transfer by homotopyanalysis methodrdquo Computers and Fluids vol 54 no 1 pp 1ndash92012
[49] M Sajid T Hayat and S Asghar ldquoComparison between theHAM and HPM solutions of thin film flows of non-Newtonianfluids on a moving beltrdquo Nonlinear Dynamics vol 50 no 1-2pp 27ndash35 2007
[50] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for the double-diffusive natural convective boundary
layer flow in a porous medium saturated by a nanofluidrdquoInternational Journal of Heat and Mass Transfer vol 54 no 1-3pp 374ndash378 2011
[51] K Vajravelu and A Hadjinicolaou ldquoConvective heat transferin an electrically conducting fluid at a stretching surfacewith uniform free streamrdquo International Journal of EngineeringScience vol 35 pp 1237ndash1244 1997
[52] S Mukhopadhyay and G C Layek ldquoEffects of variable fluidviscosity on flow past a heated stretching sheet embedded ina porous medium in presence of heat sourcesinkrdquo Meccanicavol 47 no 4 pp 863ndash876 2012
[53] S J Liao ldquoAn optimal homotopy-analysis approach for stronglynonlinear differential equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 8 pp 2003ndash20162010
[54] A Aziz ldquoA similarity solution for laminar thermal boundarylayer over a flat plate with a convective surface boundary con-ditionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1064ndash1068 2009
Lie group analysis is used to develop the similarity transfor-mations and the corresponding similarity representations ofthe governing equations The coupled governing equationsare transformed into ordinary differential equations in theboundary layer The OHAM is applied to solve the ODEsThe obtained solutions are verified by the numerical results(obtained by using a Runge-Kutta-Fehlberg fourth-fifth orderand shooting method) The effect of relevant parameters ondimensionless fluid velocity temperature and nanoparticlevolume fraction are investigated and shown graphically anddiscussed A table containing data for the reduced Nusseltnumber and reduced Sherwood number is also provided toshow the effects of various parameters on them To the best ofour knowledge the effects of thermal slip boundary conditionwith internal heat generation and chemical reaction on theboundary-layer flow of a nanofluid past a horizontal plate inporous media have not been reported in the literature yet
The paper is divided up as follows In Section 2 themath-ematical formulation is presented In Section 3 we used theLie group method to reduce the system of partial differentialequations to a system of ordinary differential equations InSection 4 the basic idea of theHAM is presented In Section 5we derived the OHAM solution of the coupled system ofnonlinear ordinary differential equations In Section 6 wecompared our results with numerical solutions obtainedusing a Runge-Kutta-Fehlberg method In Section 7 weintroduced the physical quantities to be considered andcompared in this paper Section 8 contains the results anddiscussion The conclusions are summarized in Section 9
2 Formulation of the Problem
We consider a two-dimensional laminar free convectiveboundary-layer flow of a nanofluid past an upward facingchemically reacting horizontal plate in a porous media(Figure 1) We assume that a homogeneous isothermal irre-versible chemical reaction of order 119899 takes place betweenthe plate and nanofluid There is internal heat genera-tionabsorption within the fluid inside the boundary layer at
the volumetric rate 119902 Variation of density of the fluid is takeninto account using the Oberbeck-Boussinesq approximationThe conservation of mass momentum energy and nanopar-ticles describing the flow can be written in dimensional form(see [27])
nabla sdot = 0 (1)
120588119891
120576
120597
120597119905
= minus nabla119875 minus
120583
119896
+ [119862120588119875 + (1 minus 119862) 120588119891 (1 minus 120573 (119879 minus 119879infin))] 119892
minus 119896 (119909) (119862 minus 119862infin)119899
(4)
Theflow is assumed to be slow to ignore an advective termand a Forchheimer quadratic drag term in the momentumequation
We consider a steady flow where the Oberbeck-Boussinesq approximation is used In addition we assumethat the nanoparticle concentration is dilute With a suitablechoice for the reference pressure the momentum equationcan be linearized and (2) written as (see [50])
minus nabla119875 minus
120583
119896
+ [(120588119875 minus 120588119891infin) (119862 minus 119862infin)
+ (1 minus 119862infin) 120588119891infin120573 (119879 minus 119879infin)] 119892 = 0
(5)
We also consider the effect of temperature-dependentvolumetric heat generationabsorption in the flow region thatis given by Vajravelu and Hadjinicolaou [51] as
119902 =
Ra2311987601198714311990943
(119879 minus 119879infin) 119879 gt 119879infin(6)
where1198760 is the heat generationabsorption constant Also weconsider the case where the reaction rate varies as
119896 (119909) =
Ra2311989601198714311990943
(7)
where 1198960 is the constant reaction rate
4 Mathematical Problems in Engineering
With these assumptions along with standard boundary-layer approximation the governing equations can be writtenin dimensional form as
120597119906
120597119909
+
120597V120597119910
= 0 (8)
120597119875
120597119909
= minus
120583
119896
119906 (9)
120597119875
120597119910
= minus
120583
119896
V + [(1 minus 119862infin) 120588119891infin119892120573 (119879 minus 119879infin)
minus (120588119875 minus 120588119891infin) 119892 (119862 minus 119862infin)]
(10)
119906
120597119879
120597119909
+ V120597119879
120597119910
= 120572119898
1205972119879
1205971199102
+ 120591 [119863119861
120597119862
120597119910
120597119879
120597119910
+ (
119863119879
119879infin
)(
120597119879
120597119910
)
2
]
+
1198760
(120588119862)119891
Ra23
11987143
11990943
(119879 minus 119879infin)
(11)
119906
120597119862
120597119909
+ V120597119862
120597119910
= 119863119861
1205972119862
1205971199102+ (
119863119879
119879infin
)
1205972119879
1205971199102
minus
Ra2311989601198714311990943
(119862 minus 119862infin)119899
(12)
where 120572119898 = 119896119898(120588119888119875)119891 is the thermal diffusivity of the fluidand 120591 = 120576(120588119862)119901(120588119862)119891 is a parameter
where 1198631(119909) is the thermal slip factor with dimension(length)minus1 The following new nondimensional variables areintroduced to make (8)ndash(13) dimensionless
119909 =
119909
119871radicRa 119910 =
119910
119871
119906 =
119906119871
120572119898radicRa
V =V119871120572119898
120579 =
119879 minus 119879infin
Δ119879
120601 =
119862 minus 119862infin
Δ119862
Δ119879 = 119879119908 minus 119879infin Δ119862 = 119862119908 minus 119862infin
(14)
where Ra = 119892119896120573(1 minus 119862infin)Δ119879119871(120572119898]) is the Rayleigh numberbased on the characteristic length 119871 A stream function 120595
defined by
119906 =
120597120595
120597119910
V = minus120597 120595
120597119909
(15)
is introduced into (8)ndash(13) to reduce the number of depen-dent variables and the number of equations Note that (8) issatisfied identically Hence we have
The parameters in (16)ndash(19) are introduced in Nomencla-ture and defined by
119873119905 = 120591119863119879
Δ119879
120572119898
119879infin 119873119887 = 120591119863119861
Δ119862
120572119898
119873119903 =
(120588119875 minus 120588119891infin) Δ119862
120588119891infin
120573 (1 minus 119862infin) Δ119879
119876 =
11987601198712
120572119898(120588119862)119891
119870 =
11989601198712(Δ119862)119899minus1
120572119898
Le =120572119898
119863119861
(20)
3 Lie Group Analysis
We consider the following scaling group of transformationswhich is a special form of Lie group analysis [52]
Γ 119909lowast= 1199091198901205761205721 119910
lowast= 1199101198901205761205722
120595lowast= 1205951198901205761205723 120579
lowast= 1205791198901205761205724
120601lowast= 1206011198901205761205725 119863
lowast1 = 1198631119890
1205761205726
(21)
Here 120576 is the parameter of the group Γ and 120572119894 (119894 =
1 2 3 4 5 6) are arbitrary real numbers whose connec-tion will be determined by our analysis The transfor-mations in (21) can be considered as a point transfor-mation transforming the coordinates (119909 119910 120595 120579 120601 1198631) to
Mathematical Problems in Engineering 5
(119909lowast 119910lowast 120595lowast 120579lowast 120601lowast 119863lowast1 ) We now investigate the relationship
times Δ 119895 (119909 119910 120579 120601 1205973120595
1205971199103)
(119895 = 1 2 3)
(22)
Since this is the requirement that the differential formsΔ 1 Δ 2 and Δ 3 be reformed under the transformation groupin (19) by using (21) (16)ndash(18) are transformed to (see [3538])
The system will remain invariant (structure of the equa-tions same) under the group transformation Γ if we have thefollowing relationship among the exponents
1205723 minus 21205722 = 1205724 minus 1205721 = 1205725 minus 1205721
1205723 + 1205724 minus 1205721 minus 1205722 = 1205724 minus 21205722 = 1205724 + 1205725 minus 21205722
= 21205724 minus 21205722 = 1205724 minus4
3
1205721
1205723 + 1205725 minus 1205721 minus 1205722 = 1205725 minus 21205722 = 1205724 minus 21205722
= constant = 119891 (120578) (say) that is 120595 = 11990913119891 (120578)
(30b)
6 Mathematical Problems in Engineering
where 119891 is an arbitrary function of 120578 From the equations211988911990931205722119909 = 1198891205790 and 211988911990931205722119909 = 1198891206010 we obtain byintegration
Here 120578 is the similarity variable and 119891 120579 120601 are dependentvariables Note that the similarity transformations in (31) areconsistent with the well-known similarity transformationsreported in the paper of Cheng and Chang [21] for 120582 = 0
in their paper Thus the dimensionless velocity components119906 V can be expressed as
119906 =
1198911015840
11990913 V = minus
1
311990923
(119891 minus 21205781198911015840) (32)
where primes indicate differentiation with respect to 120578
32 Similarity Equations On substituting the transforma-tions in (31) into the governing (16)ndash(18) we obtain thefollowing system of ordinary differential equations
where 119887 = (1198631)0119871 is the thermal slip parameter
4 Basic Idea of the HAM
Let us consider the following differential equation
N [119906 (120591)] = 0 (37)
where N is a nonlinear operator 120591 denotes independentvariable and 119906(120591) is an unknown function respectively Forsimplicity we ignore all boundary or initial conditions whichcan be treated in the similar way By means of generalizing
the traditional homotopy method Liao [45] constructs theso-called zero-order deformation equation
(1 minus 119901)L [120593 (120591 119901) minus 1199060 (120591)] = 119901ℏ119867 (120591)N [120593 (120591 119901)] (38)
where 119901 isin [0 1] is the embedding parameter ℏ = 0 is anonzero auxiliary parameter 119867(120591) = 0 is an auxiliary func-tionL is an auxiliary linear operator 1199060(120591) is an initial guessof 119906(120591) and 120593(120591 119901) is an unknown function respectively Itis important that one has great freedom to choose auxiliarythings in the HAM Obviously when 119901 = 0 and 119901 = 1 itholds 120593(120591 0) = 1199060(120591) 120593(120591 1) = 119906(120591) respectively Thus as119901 increases from 0 to 1 the solution 120593(120591 119901) varies from theinitial guess 1199060(120591) to the solution 119906(120591) Expanding 120593(120591 119901) inTaylor series with respect to 119901 we have
If the auxiliary linear operator the initial guess theauxiliary parameter ℏ and the auxiliary function are soproperly chosen the series (39) converges at 119901 = 1 then wehave
119906 (120591) = 1199060 (120591) +
+infin
sum
119898=1
119906119898 (120591) (41)
which must be one of solutions of original nonlinear equa-tion as proved by Liao [45] As ℏ = minus1 and 119867(120591) = 1 (38)becomes
(1 minus 119901)L [120593 (120591 119901) minus 1199060 (120591)] + 119901N [120593 (120591 119901)] = 0 (42)
which is used mostly in the homotopy perturbation methodwhere the solution can be obtained directly without usingTaylor series
According to definition (39) the governing equationcan be deduced from the zero-order deformation equation(37) By defining the vector 119899 = 1199060(120591) 1199061(120591) 119906119899(120591)
and differentiating equation (37) 119898 times with respect tothe embedding parameter 119901 and then setting 119901 = 0 andfinally dividing them by119898 we have the so-called119898th-orderdeformation equation
L [119906119898 (120591) minus 120594119898119906119898minus1 (120591)] = ℏ119867 (120591) 119877119898 (119898minus1) (43)
It should be emphasized that 119906119898(120591) for 119898 ge 1 isgoverned by the linear equation (39) with the linear boundaryconditions that come from original problem which canbe easily solved by symbolic computation software such asMAPLE and MATHEMATICA
Figure 3 The ℏ-curve of 1205791015840(0) given by the various HAM-orderapproximate solution
5 HAM Solution
In this section we applied the HAM to obtain approx-imate analytical solutions of the effect of higher orderchemical reaction internal heat generation and the thermalslip boundary condition on the boundary-layer flow of ananofluid past an upward facing horizontal plate (33)ndash(35)
Figure 4 The ℏ-curve of 1206011015840(0) given by the various HAM-orderapproximate solution
We start with the initial approximation 1198650(120578) = 0 Θ0(120578) =(1(119887 + 1))Exp[minus120578]Φ0(120578) = Exp[minus120578] and the linear operator
L [119865 (120578 119901)] =
1205972119865 (120578 119901)
1205971205783
L [Θ (120578 119901)] =
1205972Θ(120578 119901)
1205971205782
+
120597Θ (120578 119901)
120597120578
L [Φ (120578 119901)] =
1205972Φ(120578 119901)
1205971205782
+
120597Φ (120578 119901)
120597120578
(45)
Furthermore from (33)ndash(35) we define the nonlinearoperators
N [119865 (120578 119901)] =
1205972119865 (120578 119901)
1205971205782
minus
2
3
120578(
120597Θ (120578 119901)
120597120578
minus 119873119903
120597Φ (120578 119901)
120597120578
)
N [Θ (120578 119901)] =
1205972Θ(120578 119901)
1205971205782
+
1
3
119865 (120578 119901)
120597Θ (120578 119901)
120597120578
+ 119873119887
120597Θ (120578 119901)
120597120578
120597Φ (120578 119901)
120597120578
+ 119873119905(
120597Θ (120578 119901)
120597120578
)
2
+ 119876Θ (120578 119901)
N [Φ (120578 119901)] =
1205972Φ(120578 119901)
1205971205782
+
Le3
119865 (120578 119901)
120597Φ (120578 119901)
120597120578
+
119873119905
119873119887
1205972Θ(120578 119901)
1205971205782
minus 119870(Φ (120578 119901))119899
(46)
8 Mathematical Problems in Engineering
Using the above definition with assumption 119867119865(120591) = 1119867Θ(120591) = Exp[minus120578] and 119867Φ(120591) = Exp[minus120578] we construct the119911119890119903119900th-order deformation equation
(1 minus 119901)L [120593 (119909 119901) minus 1199060 (119909)] = 119867 (120591) 119901ℏN [120593 (119909 119901)]
Differentiating the 119911119890119903119900th-order deformation equation(47)119898 times with respect to 119901 and finally dividing by119898 wehave the119898th-order deformation equation
L [119865119898 (119909) minus 120594119898119865119898minus1 (119909)] = ℏ119877119898 (119898minus1)
L [Θ119898 (119909) minus 120594119898Θ119898minus1 (119909)] = Exp [minus120578] ℏ119877119898 (Θ119898minus1)
L [Φ119898 (119909) minus 120594119898Φ119898minus1 (119909)] = Exp [minus120578] ℏ119877119898 (Φ119898minus1)
By using the symbolic software MATHEMATICA tosolve the system of linear equations (49) with the boundaryconditions (46) it can be obtained finally
1198651 (120578) = minus
4
3 (1 + 119887)
ℏ +
4119890minus120578
3 (1 + 119887)
ℏ +
4119873119903
3 (1 + 119887)
ℏ
+
4119887119873119903
3 (1 + 119887)
ℏ minus
4119873119903119890minus120578
3 (1 + 119887)
ℏ minus
4119887119873119903119890minus120578
3 (1 + 119887)
ℏ
+
2119890minus120578120578
3 (1 + 119887)
ℏ minus
2119873119903119890minus120578120578
3 (1 + 119887)
ℏ minus
2119887119873119903119890minus120578120578
3 (1 + 119887)
ℏ
Θ1 (120578) =119890minus2120578
2 (1 + 119887)
ℏ minus
119890minus120578
2(1 + 119887)2ℏ minus
119887119890minus120578
2(1 + 119887)2ℏ
minus
119873119887119890minus3120578
6(1 + 119887)2ℏ +
119887119873119887119890minus3120578
6(1 + 119887)2ℏ minus
119873119887119890minus120578
6(1 + 119887)2ℏ
minus
119887119873119887119890minus120578
6(1 + 119887)2ℏ +
119873119905119890minus3120578
6(1 + 119887)2ℏ minus
119873119905119890minus120578
6(1 + 119887)2ℏ
+
119876119890minus2120578
2 (1 + 119887)
ℏ minus
119876119890minus120578
2(1 + 119887)2ℏ minus
119887119876119890minus120578
2(1 + 119887)2ℏ
Φ1 (120578) =119890minus2120578
2 (1 + 119887)
ℏ +
119887119890minus2120578
2 (1 + 119887)
ℏ minus
119890minus120578
2 (1 + 119887)
ℏ
minus
119887119890minus120578
2 (1 + 119887)
ℏ minus
119870119890minus2120578
2 (1 + 119887)
ℏ minus
119887119870eminus2120578
2 (1 + 119887)
ℏ
+
119870119890minus120578
2 (1 + 119887)
ℏ +
119887119870119890minus120578
2 (1 + 119887)
ℏ +
119873119905
119873119887
119890minus2120578
2 (1 + 119887)
ℏ
minus
119873119905
119873119887
119890minus120578
2 (1 + 119887)
ℏ
(56)
The higher orders solutions of 119865(120578) Θ(120578) and Φ(120578) weretoo long to be mentioned here therefore they are showngraphically
51 Convergence of the HAM Solution As mentioned by Liao[45] HAM provides us with great freedom in choosing thesolution of a nonlinear problem by different base functionsThis has a great effect on the convergence region because theconvergence region and rate of a series are chiefly determinedby the base functions used to express the solution We usedseveral terms in evaluating the approximate solution 119865app asympsum119899119894=0 119865119894Θapp asymp sum
119899119894=0Θ119894Φapp asymp sum
119899119894=0Φ119894 note that the solution
series contains the auxiliary parameter ℏ which provides uswith a simple way to adjust and control the convergence ofthe solution series Generally by means of the so-called ℏ-curve that is a curve of a versus ℏ As pointed by Liao [45]the valid region of ℏ is a horizontal line segment Figures2 3 and 4 show the ℏ-curve with the various order of theHAM for 11989110158401015840(0) 1205791015840(0) 1206011015840(0) respectively when 119873119887 = 01
Figure 6 The residual error of 120579(120578) for (34) using 12th order ofapproximations
119873119905 = 01 119873119903 = 01 119876 = 01 Le = 10 119870 = 10 119899 = 10119887 = 00 It can be seen that when the order of series is 12 thesegment of the horizontal line is more than the other ordersFor example it can be found that for 11986510158401015840(0) the acceptablerange ofℏ is betweenminus05 andminus20 for 12th order of theHAMbut for 8th order of the HAM the acceptable range of ℏ isbetween minus10 and minus20 or minus10 and minus15 for 4th order of theHAM so horizontal line segment of 12th order of the HAM ismore than others Therefore it is straightforward to choosean appropriate range for ℏ which ensures the convergence
10 Mathematical Problems in Engineering
minus00002
120578
= minus10
= minus111 (optimal value)= minus12
Nt = 01Nr = 01
Q = 01Le = 10K = 10n = 10b = 00
0006
0004
0002
0
0 2 4 6 8 10
Nb = 01
Res (120601
(120578))
ℏ
ℏ
ℏ
Figure 7 The residual error of 120601(120578) for (35) using 12th order ofapproximations
of the solution series To choose optimal value of auxiliaryparameter the averaged residual errors (see [53] for moredetails) are defined as
119864119865119898 =1
119870
119870
sum
119894=0
[N119865(119898
sum
119894=0
119865119894 (119894Δ119909))]
2
119864Θ119898 =1
119870
119870
sum
119894=0
[NΘ(119898
sum
119894=0
Θ119894 (119894Δ119909))]
2
119864Φ119898 =1
119870
119870
sum
119894=0
[NΦ(119898
sum
119894=0
Φ119894 (119894Δ119909))]
2
(57)
where Δ119909 = 10119870 and 119870 = 20 For given order ofapproximation 119898 and the optimal values ℏ are given bythe minimum of 119864119898 corresponding to nonlinear algebraicequations
119889119864119865119898
119889ℏ
= 0
119889119864Θ119898
119889ℏ
= 0
119889119864Φ119898
119889ℏ
= 0 (58)
It is noted that the optimal value of ℏ is replaced intothe equations Table 1 shows optimal values obtained for theauxiliary parameter ℏ for various quantities of119873119905 119876 and 119899when119873119887 = 01119873119903 = 01 Le = 10 119870 = 10 119887 = 00 To see
the accuracy of the solutions the residual errors are definedfor the system as (for119898 order approximation)
Res119865119898
=
1198892119865119898
1198891205782minus
2
3
120578(
119889Θ119898
119889120578
minus 119873119903
119889Φ119898
119889120578
) (59)
ResΘ119898
=
1198892Θ119898
1198891205782+
1
3
119865119898
119889Θ119898
d120578+ 119873119887
119889Θ119898
119889120578
119889Φ119898
119889120578
+ 119873119905(
119889Θ119898
119889120578
)
2
+ 119876Θ119898
(60)
ResΦ119898
=
1198892Φ119898
1198891205782+
Le3
119865119898
119889Φ119898
119889120578
+
119873119905
119873119887
1198892Θ119898
1198891205782
minus 119870(
119889Φ119898
119889120578
)
119899
(61)
Figures 5 6 and 7 show the residual errors for 12th-orderdeformation solutions when 119873119887 = 01 119873119905 = 01 119873119903 = 01119876 = 01 Le = 10 119870 = 10 119899 = 10 119887 = 00 For example ℏ =minus145 has the minimum range of residual curve in Figure 5and so on
Graphical representation of results is very useful todemonstrate the efficiency and accuracy of the HAM for theabove problem
6 Comparisons and Verification
It is worth citing that for isothermal plate (119887 = 0) and in theabsence of internal heat generationabsorption (119876 = 0) andchemical reaction (119870 = 0) our problem reduces to Gorla andChamkha [26] and Khan and Pop [27] To verify the accuracyof our results the present results are compared in Table 2with Gorla and Chamkha [26] and are found to be in goodagreement
7 Physical Quantities
Theparameters of physical interest of the present problem arethe local skin friction factor 119862119891119909 the local Nusselt numberNu119909 and the local Sherwood number Sh119909 respectivelyPhysically 119862119891119909 indicates wall shear stress Nu119909 indicates therate of wall heat transfer whilst Sh119909 indicates the rate ofwall nanoparticle volume fractionThe following relations areused to find these quantities
119862119891119909 =2120583
1205881198802119903
(
120597119906
120597119910
)
119910=0
Nu119909 =minus119909
119879119891 minus 119879infin
(
120597119879
120597119910
)
119910=0
Sh119909 =minus119909
119862119908 minus 119862infin
(
120597119862
120597119910
)
119910=0
(62)
Mathematical Problems in Engineering 11
Table 2 Comparison of present solution with Gorla and Chamkha[26] for different values of buoyancy and nanofluid parameters
Present results (HAM) Gorla and Chamkha [26]minus1205791015840(0) minus120601
1015840(0) minus120579
1015840(0) minus120601
1015840(0)
Nr 119873119887 = 03119873119905 = 01 Le = 10 119887 = 119870 = 119876 = 0
01 032578 148242 326119864 minus 01 148416402 032385 146704 325119864 minus 01 146816103 032188 145125 322119864 minus 01 145266404 031985 143503 321119864 minus 01 143639205 031777 141833 319119864 minus 01 1419499Nt 119873119887 = 03119873119903 = 05 Le = 10 119887 = 119870 = 119876 = 0
01 031777 141833 319119864 minus 01 141949902 030486 141491 305119864 minus 01 141653603 02927 141561 293119864 minus 01 141686604 028125 141991 282119864 minus 01 142158205 027046 142737 271119864 minus 01 1429226Nb 119873119905 = 01119873119903 = 05 Le = 10 119887 = 119870 = 119876 = 0
01 03672 132611 368119864 minus 01 132745402 034271 139216 343119864 minus 01 139361503 031777 141833 319119864 minus 01 141949904 029399 143428 294119864 minus 01 143546405 027161 144598 272119864 minus 01 144772
Figure 10 Comparison of dimensionless concentration profile 120601(120578)obtained by various orders of the HAM
Here Ra119909 = 119892119870120573(1minus119862infin)Δ119879119909(120572119898]) is the local Rayleighnumber Pr = ]120572119898 is the Prandtl number and 119880119903 = (1 minus
119862infin)119892119870120573Δ119879120572119898 is the reference velocity
8 Results and Discussion
Graphical representation of results is very useful to discussthe physical features presented by the solutions This sectiondescribes the influence of some interesting parameters on
12 Mathematical Problems in Engineering
0 2 4 6 8 10
02
04
06
08
1
Nr = 01Numerical
Numerical
NumericalNr = 03Nr = 05
f998400 (120578)
120578
Nt = 01Nb = 01Q = 01Le = 10K = 10n = 10b = 00
Figure 11 Effect of buoyancy ratio119873119903 on the dimensionless velocityprofile
0 2 4 6 8 10
02
04
06
08
1
Nt = 01Nb = 01Q = 01Le = 10K = 10n = 10b = 00
Nr = 01Numerical
Numerical
NumericalNr = 03Nr = 05
120578
120579(120578
)
Figure 12 Effect of buoyancy ratio 119873119903 on the dimensionlesstemperature profile
the velocity and temperature fields Equations (33) to (35)with boundary conditions in (36) were solved analytically byHAM and numerically using Runge-Kutta-Fehlberg fourth-fifth order proposed by Aziz [54]
Figures 8 9 and 10 respectively for 1198911015840(120578) 120579(120578) and 120601(120578)show the comparisons between various order approximationof the optimal HAM and the numerical solutions for the case
0 2 4 6 8 10
02
04
06
08
1
Nr = 01Numerical
Numerical
NumericalNr = 03Nr = 05
Nt = 01Nb = 01Q = 01Le = 10K = 10n = 10b = 00
120578
120601(120578
)Figure 13 Effect of buoyancy ratio 119873119903 on the dimensionlessconcentration profile
0 2 4 6 8 10
02
04
06
08
1
12
NumericalNumerical
Numerical
f998400 (120578
)
120578
Nb = 01
Nb = 03
Nb = 05
Nt = 01Nr = 01Q = 01Le = 10K = 10n = 10b = 00
Figure 14 Effect of Brownian motion 119873119887 on the dimensionlessvelocity profile
119873119887 = 01 119873119905 = 01 119873119903 = 01 119876 = 01 Le = 10 119870 = 10119899 = 10 119887 = 00 For better presentation these results arepresented in Tables 3 4 and 5 Also Tables 6 7 and 8 presentthe comparisons between various order approximation of theoptimal HAM and the numerical solutions for the case119873119887 =03 119873119905 = 005 119873119903 = 03 119876 = 01 Le = 20 119870 = 05
Mathematical Problems in Engineering 13
0 2 4 6 8 10
02
04
06
08
1
0
120578
NumericalNumerical
Numerical
Nb = 01
Nb = 03
Nb = 05
Nt = 01Nr = 01Q = 01Le = 10K = 10n = 10b = 00
120579(120578
)
Figure 15 Effect of Brownian motion 119873119887 on the dimensionlesstemperature profile
02
04
06
08
1
00
2 4 6 8 10120578
NumericalNumerical
Numerical
Nb = 01
Nb = 03
Nb = 05
Nt = 01Nr = 01Q = 01Le = 10K = 10n = 10b = 00
120601(120578
)
Figure 16 Effect of Brownian motion 119873119887 on the dimensionlessconcentration profile
119899 = 20 119887 = 01 It is observed that the results of 12th-order approximation of the optimal HAM are very close tothe numerical solutions which confirm the validity of thesemethods
In the following figures effects of various physicalparameters on the dimensionless velocity temperature and
Table 3 Comparison of values of 1198911015840(120578) obtained by various ordersof the HAM with numerical solution when 119873119887 = 01119873119905 =
temperature profiles will be investigated These results havebeen obtained by 12th order of the HAM and have beenvalidated by numerical results Figures 11ndash13 respectivelyrepresent that the comparison of solutions of 1198911015840(120578) 120579(120578)and 120601(120578) for fix values 119873119887 = 01 119873119905 = 01 119876 = 01Le = 10 119870 = 10 119899 = 10 119887 = 00 and different values ofbuoyancy ratio119873119903 In Figure 11 it is clear that velocity of thefluid tremendously decreases in the near of horizontal platewith an increase in the buoyancy ratio It is observed that thetemperature increases slightly but concentration of the fluiddoes not vary sensibly (Figures 12 and 13) The effects of theBrownian motion 119873119887 are depicted in Figures 14 15 and 16when119873119903 = 01119873119905 = 01119876 = 01 Le = 10119870 = 10 119899 = 10
14 Mathematical Problems in Engineering
Table 6 Comparison of values of 1198911015840(120578) obtained by various ordersof the HAM with numerical solution when 119873119887 = 03119873119905 =
119887 = 00 If 119873119887 increases 1198911015840(120578) and 120579(120578) increase lightly and120601(120578) decreases The dimensionless velocity dimensionlesstemperature and dimensionless concentration profiles fordifferent values of thermophoresis 119873119905 with constant values119873119903 = 01 119873119887 = 01 119876 = 01 Le = 10 119870 = 10119899 = 10 119887 = 00 are presented in Figures 17 18 and19 It is observed that the velocity and temperature of thefluid are not impressible from119873119905 but concentration increasesslightly In Figures 20 21 and 22 respectively comparison ofsolutions of 1198911015840(120578) 120579(120578) and 120601(120578) for 119873119903 = 01 119873119887 = 01119873119905 = 01 Le = 10 119870 = 10 119899 = 10 119887 = 00 anddifferent values of heat generationabsorption parameter 119876are shown It is observed that in presence of heat source1198911015840(120578) 120579(120578) both increase while heat sink causes that 1198911015840(120578)
120579(120578) both decrease extremely These effects are more visible
Table 9 Values of reduced Nusselt number and Sherwood numberobtained by HAM for different values of the parameters Le 119870 119899 119887and 119876 when119873119887 = 119873119903 = 119873119905 = 01
Figure 17 Effect of thermophoresis 119873119905 on the dimensionlessvelocity profile
for temperature distribution For 120601(120578) the effect of changing119876 is seen to be almost insignificant This is in agreementwith the physical factThe dimensionless profiles for differentvalues of Lewis number Le with constant parameters 119873119903 =01 119873119887 = 01 119873119905 = 01 119876 = 01 119870 = 10 119899 = 10119887 = 00 are shown in Figures 23 24 and 25 It is clear that forthese constant parameters variation of Lewis number has anyconsiderable effect on the velocity and temperature profilesbut concentration profile decreases with the increase of Lewisnumber What is similar to these results can be detected inFigures 26 27 and 28 for chemical reaction parameter 119870Figures 29 30 and 31 respectively present the comparisonof solutions of 1198911015840(120578) 120579(120578) and 120601(120578) for 119873119903 = 05 119873119887 = 05119873119905 = 01 119876 = 001 Le = 10 119870 = 10 119887 = 00 and differentvalues of order of chemical reaction 119899 It is observed that
Mathematical Problems in Engineering 15
0 2 4 6 80
1
120578
Nt = 00
Nt = 005
Nt = 10NumericalNumerical
Numerical
08
06
04
02
10
Q = 01Le = 10K = 10n = 10b = 00
Nb = 01
120579(120578
)
Nr = 01
Figure 18 Effect of thermophoresis 119873119905 on the dimensionlesstemperature profile
0 2 4 6 80
1
120601(120578
)
120578
Nt = 00
Nt = 005
Nt = 10NumericalNumerical
Numerical
08
06
04
02
10
Q = 01Le = 10K = 10n = 10b = 00
Nb = 01Nr = 01
Figure 19 Effect of thermophoresis 119873119905 on the dimensionlessconcentration profile
for higher order of chemical reaction velocity decreases andtemperature and concentration profiles increase extremelyUltimately Figures 32 33 and 34 depict the effect of thermalslip parameter 119887 on velocity temperature and concentrationfunctions when 119873119903 = 01 119873119887 = 01 119873119905 = 01 119876 = 01Le = 20 119870 = 15 119899 = 10 It is seen that the velocityand temperature of the fluid decrease with the increase of the
0 2 4 6 80
1
12
10
02
04
06
08
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numericalf998400 (120578
)
120578
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 20 Effect of generationabsorption heat parameter119876 on thedimensionless velocity profile
0 2 4 6 8 100
02
04
06
08
1
120579(120578
)
120578
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numerical
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 21 Effect of generationabsorption heat parameter 119876 on thedimensionless temperature profile
thermal slip parameter in the near of horizontal plate whilethe concentration of the fluid does not vary patently
Also for investigation of the parameters of physicalinterest Table 9 is presented In this table numerical valuesof reduced Nusselt number and Sherwood number obtainedby HAM for different values of the parameters 119890 119870 119899 119887 and119876 can be compared when119873119887 = 119873119903 = 119873119905 = 01
16 Mathematical Problems in Engineering
10
02
04
06
08
0 2 4 6 80
1
120578
120601(120578
)
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numerical
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 22 Effect of generationabsorption heat parameter119876 on thedimensionless concentration profile
0 2 4 6 80
1
12
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 23 Effect of Lewis number Le on the dimensionless velocityprofile
9 Conclusions
In this paper we studied the steady laminar incompressiblefree convective flow of a nanofluid past a chemically reactingupward facing horizontal plate in porous medium takinginto account heat generation and the thermal slip boundarycondition
0 2 4 6 80
1
120579(120578
)
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 24 Effect of Lewis number Le on the dimensionlesstemperature profile
0 2 4 6 80
1
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 25 Effect of Lewis number Le on the dimensionlessconcentration profile
The governing partial differential equations have beentransformed by similarity transformations into a system ofordinary differential equations which are solved by OHAMand numerical method (fourth-order Runge-Kutta schemewith the shooting method) Dimensionless velocity temper-ature and concentration functions are presented for various
Mathematical Problems in Engineering 17
0 2 4 6 80
1
12
K = 00
K = 05
K = 10NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 26 Effect of chemical reaction parameter 119870 on the dimen-sionless velocity profile
0 2 4 6 80
1
K = 00
K = 05
K = 10NumericalNumerical
Numerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 27 Effect of chemical reaction parameter 119870 on the dimen-sionless temperature profile
values of parameters for the problem for example buoy-ancy ratio 119873119903 Brownian motion 119873119887 thermophoresis 119873119905heat generationabsorption parameter 119876 Lewis number Lechemical reaction parameter119870 order of chemical reaction 119899and thermal slip parameter 119887 From the present investigationthe following may be concluded
(i) The velocity 1198911015840(120578) increases with increasing 119873119887 anddecreases with increasing119873119903 119899 and 119887
0 2 4 6 80
1
K = 00
K = 05
K = 10NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 28 Effect of chemical reaction parameter 119870 on the dimen-sionless concentration profile
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 29 Effect of order of chemical reaction 119899 on the dimension-less velocity profile
(ii) The temperature 120579(120578) increases with 119873119903 119873119887 119899 anddecreases with 119887
(iii) Parameters of119873119905 Le and119870 have no effect on velocityand temperature profiles
(iv) The concentration distribution 120601(120578) increases withincreasing N119905 119899 and decreases with 119873119887 Le and 119870but119873119903 119876 and 119887 are not affected by it
18 Mathematical Problems in Engineering
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 30 Effect of order of chemical reaction 119899 on the dimension-less temperature profile
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 31 Effect of order of chemical reaction 119899 on the dimension-less concentration profile
(v) In presence of heat source (119876 gt 0) 1198911015840(120578) 120579(120578) bothincrease while heat sink (119876 lt 0) causes that 1198911015840(120578)120579(120578) both decrease extremely These effects are morevisible for temperature distribution
In addition numerical results for the reducedNusselt andSherwood numbers are tabulated It is concluded that thereduced Nusselt number increases with the Lewis numberand reaction parameter whist it decreases with the order of
0 2 4 6 8
1
12
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 32 Effect of order of thermal slip parameter 119887 on thedimensionless velocity profile
0 2 4 6 80
1
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 33 Effect of order of thermal slip parameter 119887 on thedimensionless temperature profile
chemical reaction thermal slip and generation parametersIt is further concluded that the reduced Sherwood numberenhances with the Lewis number generation and reactionparameters whist it suppresses with thermal slip and orderof chemical reaction
The convergence of the solution series and the power ofHAM in controlling and adjusting the convergence regionand rate of solution series were discussed The proper range
Mathematical Problems in Engineering 19
0 2 4 6 80
1
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 34 Effect of order of thermal slip parameter 119887 on thedimensionless concentration profile
of auxiliary parameter ℏ to ensure the convergence of thesolution series was obtained through the so called ℏ-curveThe HAM provides us with a convenient way to control theconvergence of approximation series which is a fundamentalqualitative difference in analysis between the HAM andother methods The numerical results of the above problemsdisplay a fast convergence with minimal calculations Thisshows that the HAM is a very efficient method Finally theagreement between analytical and numerical results of thepresent study with previous published results is excellent
Nomenclature
119887 Thermal slip parameter119862 Dimensional concentration119862119891 Skin friction factor1198631 Thermal slip factor119863119861 Brownian diffusion coefficient119863119879 Thermophoretic diffusion coefficient119891 Dimensionless velocity functions119892 Gravitation acceleration119870 Chemical reaction parameter119896 Permeability of the porous media1198960 The constant reaction rate119896119898 Effective thermal conductivity of the porous
medium119896(119909) Variable reaction rateL Linear operator of the HAM119871 Length of horizontal plateLe Lewis numberN Nonlinear operator of the HAM
119899 Order of chemical reaction119873119887 Brownian motion119873119903 Buoyancy ratio119873119905 ThermophoresisNu Nusselt number119875 PressurePr Prandtl number119876 Heat generationabsorption parameter1198760 Heat generationabsorption constant119902 Internal heat generation rate
Ra Rayleigh numberSh Sherwood number119879 Temperature119905 Time119906 Velocity in 119909-direction119880119903 Reference velocity Velocity vectorV Velocity in 119910-direction119909 Distance along the surface119910 Distance normal to the surface
Greek Letters
120572119898 Thermal diffusivity120573 Volumetric expansion coefficient of nanofluid120576 Porosity of the porous media120601 Dimensionless concentration120578 Similarity variable120583 Dynamic viscosity120579 Dimensionless temperature120588119891 Density of the base fluid120588119901 Density of the nanoparticles(120588119862)119891 Effective heat capacity of the fluid(120588119862)119875 Effective heat capacity of the nanoparticle material120595 Stream function
Subscript Superscript
infin Conditions far away from the surface1015840 Differentiation with respect to 120578
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
M M Rashidi wishes to thank the Centre for DifferentialEquations Continuum Mechanics and Applications Schoolof Computational and Applied Mathematics University ofthe Witwatersrand for the hospitality during his visit EMomoniat thanks theNational Research Foundation of SouthAfrica for their support
20 Mathematical Problems in Engineering
References
[1] E Abu-Nada F O Hakan and I Pop ldquoBuoyancy inducedflow in a nanofluid filled enclosure partially exposed to forcedconvectionrdquo Superlattices and Microstructures vol 51 no 3 pp381ndash395 2012
[2] S Kakac and A Pramuanjaroenkij ldquoReview of convective heattransfer enhancement with nanofluidsrdquo International Journal ofHeat and Mass Transfer vol 52 no 13-14 pp 3187ndash3196 2009
[3] J A Eastman S U S Choi S Li W Yu and L J ThompsonldquoAnomalously increased effective thermal conductivities ofethylene glycol-based nanofluids containing copper nanoparti-clesrdquo Applied Physics Letters vol 78 no 6 pp 718ndash720 2001
[4] Y Xuan and Q Li ldquoHeat transfer enhancement of nanofluidsrdquoInternational Journal of Heat and Fluid Flow vol 21 no 1 pp58ndash64 2000
[5] S K Das N Putra P Thiesen and W Roetzel ldquoTemperaturedependence of thermal conductivity enhancement for nanoflu-idsrdquo Journal of Heat Transfer vol 125 no 4 pp 567ndash574 2003
[6] D H Kumar H E Patel V R R Kumar T SundararajanT Pradeep and S K Das ldquoModel for heat conduction innanofluidsrdquo Physical Review Letters vol 93 no 14 Article ID144301 2004
[7] H E Patel T Sundararajan T Pradeep A Dasgupta NDasgupta and S K Das ldquoA micro-convection model forthermal conductivity of nanofluidsrdquo Pramana vol 65 no 5 pp863ndash869 2005
[8] A V Kuznetsov ldquoNon-oscillatory and oscillatory nanofluidbio-thermal convection in a horizontal layer of finite depthrdquoEuropean Journal of Mechanics vol 30 no 2 pp 156ndash165 2011
[9] M Napoli J C T Eijkel and S Pennathur ldquoNanofluidictechnology for biomolecule applications a critical reviewrdquo Labon a Chip vol 10 no 8 pp 957ndash985 2010
[10] J C Crepeau and R Clarksean ldquoSimilarity solutions of naturalconvection with internal heat generationrdquo Journal of HeatTransfer vol 119 no 1 pp 183ndash185 1997
[11] A Z Sahin ldquoTransient heat conduction in semi-infinite solidwith spatially decaying exponential heat generationrdquo Interna-tional Communications in Heat andMass Transfer vol 19 no 3pp 349ndash358 1992
[12] O D Makind and A Aziz ldquoMixed convection from a con-vectively heated vertical plate to a fluid with internal heatgenerationrdquo Journal of Heat Transfer vol 133 Article ID 1225016 pages 2011
[13] J Buongiorno ldquoConvective transport in nanofluidsrdquo Journal ofHeat Transfer vol 128 no 3 pp 240ndash250 2006
[14] AV Kuznetsov andDANield ldquoNatural convective boundary-layer flow of a nanofluid past a vertical platerdquo InternationalJournal of Thermal Sciences vol 49 no 2 pp 243ndash247 2010
[15] L B Godson D Raja L D Mohan and S WongwisescldquoEnhancement of heat transfer using nanofluids-an overviewrdquoRenewable and Sustainable Energy Reviews vol 14 no 2 pp629ndash641 2010
[16] D A Nield and A BejanConvection in PorousMedia SpringerNew York NY USA 3rd edition 2006
[17] P Vadasz Emerging Topics in Heat and Mass Transfer in PorousMedia Springer New York NY USA 2008
[18] K Vafai Porous Media Applications in Biological Systems andBiotechnology CRC Press New York NY USA 2010
[19] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application to
heat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977
[20] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for natural convective boundary-layer flow in a porousmedium saturated by a nanofluidrdquo International Journal of Heatand Mass Transfer vol 52 no 25-26 pp 5792ndash5795 2009
[21] P Cheng and I D Chang ldquoBuoyancy induced flows in asaturated porous medium adjacent to impermeable horizontalsurfacesrdquo International Journal of Heat and Mass Transfer vol19 no 11 pp 1267ndash1272 1976
[22] I D Chang and P Cheng ldquoMatched asymptotic expansionsfor free convection about an impermeable horizontal surfacein a porous mediumrdquo International Journal of Heat and MassTransfer vol 26 no 2 pp 163ndash174 1983
[23] D S Shiunlin and B Gebhart ldquoBuoyancy-induced flow adja-cent to a horizontal surface submerged in porous mediumsaturated with cold waterrdquo International Journal of Heat andMass Transfer vol 29 no 4 pp 611ndash623 1986
[24] J H Merkin and G Zhang ldquoOn the similarity solutions for freeconvection in a saturated porousmedium adjacent to imperme-able horizontal surfacesrdquoWarme- und Stoffubertragung vol 25no 3 pp 179ndash184 1990
[25] M A Chaudhary J H Merkin and I Pop ldquoNatural convectionfrom a horizontal permeable surface in a porous mediummdashnumerical and asymptotic solutionsrdquo Transport in PorousMedia vol 22 no 3 pp 327ndash344 1996
[26] R S R Gorla and A Chamkha ldquoNatural convective boundarylayer flow over a nonisothermal vertical plate embedded ina porous medium saturated with a nanofluidrdquo Nanoscale andMicroscale Thermophysical Engineering vol 15 no 2 pp 81ndash942011
[27] W A Khan and I Pop ldquoFree convection boundary layer flowpast a horizontal flat plate embedded in a porous medium filledwith a nanofluidrdquo Journal of Heat Transfer vol 133 no 9 ArticleID 094501 2011
[28] A Aziz W A Khan and I Pop ldquoFree convection boundarylayer flow past a horizontal flat plate embedded in porousmedium filled by nanofluid containing gyrotactic microorgan-ismsrdquo International Journal of Thermal Sciences vol 56 pp 48ndash57 2012
[29] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer New York NYUSA 2009
[30] A A Avramenko S G Kobzar I V Shevchuk A V Kuznetsovand L T Iwanisov ldquoSymmetry of turbulent boundary-layerflows investigation of different eddy viscosity modelsrdquo ActaMechanica vol 151 no 1-2 pp 1ndash14 2001
[31] A V Kuznetsov A A Avramenko and P Geng ldquoAnalyticalinvestigation of a falling plume caused by bioconvection ofoxytactic bacteria in a fluid saturated porous mediumrdquo Inter-national Journal of Engineering Science vol 42 no 5-6 pp 557ndash569 2004
[32] M Jalil S Asghar and M Mushtaq ldquoLie group analysis ofmixed convection flow with mass transfer over a stretchingsurface with suction or injectionrdquo Mathematical Problems inEngineering vol 2010 Article ID 264901 14 pages 2010
[33] M A A Hamad M J Uddin and A I M Ismail ldquoRadiationeffects on heat and mass transfer in MHD stagnation-pointflow over a permeable flat plate with thermal convectivesurface boundary condition temperature dependent viscosityand thermal conductivityrdquoNuclear Engineering and Design vol242 pp 194ndash200 2012
Mathematical Problems in Engineering 21
[34] A Aziz M J Uddin M A A Hamad and A I MdIsmail ldquoMHD flow over an inclined radiating plate with thetemperature-dependent thermal conductivity variable reactiveindex and heat generationrdquoHeat Transfer vol 41 no 3 pp 241ndash259 2012
[35] A G Hansen Similarity Analysis of Boundary Layer Problemsin Engineering Prentice Hall Englewood Cliffs NJ USA 1964
[36] W FAmesNonlinear Partial Differential Equations in Engineer-ing Academic Press New York NY USA 1972
[37] R Seshadri and T Y Na Group Invariance in EngineeringBoundary Value Problems Springer New York NY USA 1985
[38] D Shang Theory of Heat Transfer with Forced Convection FilmFlows vol 3 of Heat and Mass Transfer 2010
[40] E Erfani M M Rashidi and A B Parsa ldquoThe modifieddifferential transform method for solving off-centered stag-nation flow toward a rotating discrdquo International Journal ofComputational Methods vol 7 no 4 pp 655ndash670 2010
[41] M M Rashidi N Laraqi and A Basiri Parsa ldquoAnalyticalmodeling of heat convection in magnetized micropolar fluid byusing modified differential transform methodrdquo Heat Transfervol 40 no 3 pp 187ndash204 2011
[42] M M Rashidi and M Keimanesh ldquoUsing differential trans-form method and pade approximant for solving mhd flow ina laminar liquid film from a horizontal stretching surfacerdquoMathematical Problems in Engineering vol 2010 Article ID491319 14 pages 2010
[43] M M Rashidi and G Domairry ldquoNew analytical solution ofthe three-dimensional NavierStokes equationsrdquoModern PhysicsLetters B vol 23 no 26 pp 3147ndash3155 2009
[44] M M Rashidi M Keimanesh O A Beg and T K HungldquoMagnetohydrodynamic biorheological transport phenomenain a porous medium a simulation of magnetic blood flowcontrol and filtrationrdquo International Journal for NumericalMethods in Biomedical Engineering vol 27 no 6 pp 805ndash8212011
[45] S J Liao The proposed homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992
[46] M M Rashidi G Domairry and S Dinarvand ldquoApproximatesolutions for the Burger and regularized long wave equationsby means of the homotopy analysis methodrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 14 no 3 pp708ndash717 2009
[47] Z Ziabakhsh and G Domairry ldquoAnalytic solution of naturalconvection flow of a non-Newtonian fluid between two verticalflat plates using homotopy analysis methodrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 14 no 5pp 1868ndash1880 2009
[48] M M Rashidi S A Mohimanian Pour T Hayat and SObaidat ldquoAnalytic approximate solutions for steady flow over arotating disk in porousmediumwith heat transfer by homotopyanalysis methodrdquo Computers and Fluids vol 54 no 1 pp 1ndash92012
[49] M Sajid T Hayat and S Asghar ldquoComparison between theHAM and HPM solutions of thin film flows of non-Newtonianfluids on a moving beltrdquo Nonlinear Dynamics vol 50 no 1-2pp 27ndash35 2007
[50] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for the double-diffusive natural convective boundary
layer flow in a porous medium saturated by a nanofluidrdquoInternational Journal of Heat and Mass Transfer vol 54 no 1-3pp 374ndash378 2011
[51] K Vajravelu and A Hadjinicolaou ldquoConvective heat transferin an electrically conducting fluid at a stretching surfacewith uniform free streamrdquo International Journal of EngineeringScience vol 35 pp 1237ndash1244 1997
[52] S Mukhopadhyay and G C Layek ldquoEffects of variable fluidviscosity on flow past a heated stretching sheet embedded ina porous medium in presence of heat sourcesinkrdquo Meccanicavol 47 no 4 pp 863ndash876 2012
[53] S J Liao ldquoAn optimal homotopy-analysis approach for stronglynonlinear differential equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 8 pp 2003ndash20162010
[54] A Aziz ldquoA similarity solution for laminar thermal boundarylayer over a flat plate with a convective surface boundary con-ditionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1064ndash1068 2009
With these assumptions along with standard boundary-layer approximation the governing equations can be writtenin dimensional form as
120597119906
120597119909
+
120597V120597119910
= 0 (8)
120597119875
120597119909
= minus
120583
119896
119906 (9)
120597119875
120597119910
= minus
120583
119896
V + [(1 minus 119862infin) 120588119891infin119892120573 (119879 minus 119879infin)
minus (120588119875 minus 120588119891infin) 119892 (119862 minus 119862infin)]
(10)
119906
120597119879
120597119909
+ V120597119879
120597119910
= 120572119898
1205972119879
1205971199102
+ 120591 [119863119861
120597119862
120597119910
120597119879
120597119910
+ (
119863119879
119879infin
)(
120597119879
120597119910
)
2
]
+
1198760
(120588119862)119891
Ra23
11987143
11990943
(119879 minus 119879infin)
(11)
119906
120597119862
120597119909
+ V120597119862
120597119910
= 119863119861
1205972119862
1205971199102+ (
119863119879
119879infin
)
1205972119879
1205971199102
minus
Ra2311989601198714311990943
(119862 minus 119862infin)119899
(12)
where 120572119898 = 119896119898(120588119888119875)119891 is the thermal diffusivity of the fluidand 120591 = 120576(120588119862)119901(120588119862)119891 is a parameter
where 1198631(119909) is the thermal slip factor with dimension(length)minus1 The following new nondimensional variables areintroduced to make (8)ndash(13) dimensionless
119909 =
119909
119871radicRa 119910 =
119910
119871
119906 =
119906119871
120572119898radicRa
V =V119871120572119898
120579 =
119879 minus 119879infin
Δ119879
120601 =
119862 minus 119862infin
Δ119862
Δ119879 = 119879119908 minus 119879infin Δ119862 = 119862119908 minus 119862infin
(14)
where Ra = 119892119896120573(1 minus 119862infin)Δ119879119871(120572119898]) is the Rayleigh numberbased on the characteristic length 119871 A stream function 120595
defined by
119906 =
120597120595
120597119910
V = minus120597 120595
120597119909
(15)
is introduced into (8)ndash(13) to reduce the number of depen-dent variables and the number of equations Note that (8) issatisfied identically Hence we have
The parameters in (16)ndash(19) are introduced in Nomencla-ture and defined by
119873119905 = 120591119863119879
Δ119879
120572119898
119879infin 119873119887 = 120591119863119861
Δ119862
120572119898
119873119903 =
(120588119875 minus 120588119891infin) Δ119862
120588119891infin
120573 (1 minus 119862infin) Δ119879
119876 =
11987601198712
120572119898(120588119862)119891
119870 =
11989601198712(Δ119862)119899minus1
120572119898
Le =120572119898
119863119861
(20)
3 Lie Group Analysis
We consider the following scaling group of transformationswhich is a special form of Lie group analysis [52]
Γ 119909lowast= 1199091198901205761205721 119910
lowast= 1199101198901205761205722
120595lowast= 1205951198901205761205723 120579
lowast= 1205791198901205761205724
120601lowast= 1206011198901205761205725 119863
lowast1 = 1198631119890
1205761205726
(21)
Here 120576 is the parameter of the group Γ and 120572119894 (119894 =
1 2 3 4 5 6) are arbitrary real numbers whose connec-tion will be determined by our analysis The transfor-mations in (21) can be considered as a point transfor-mation transforming the coordinates (119909 119910 120595 120579 120601 1198631) to
Mathematical Problems in Engineering 5
(119909lowast 119910lowast 120595lowast 120579lowast 120601lowast 119863lowast1 ) We now investigate the relationship
times Δ 119895 (119909 119910 120579 120601 1205973120595
1205971199103)
(119895 = 1 2 3)
(22)
Since this is the requirement that the differential formsΔ 1 Δ 2 and Δ 3 be reformed under the transformation groupin (19) by using (21) (16)ndash(18) are transformed to (see [3538])
The system will remain invariant (structure of the equa-tions same) under the group transformation Γ if we have thefollowing relationship among the exponents
1205723 minus 21205722 = 1205724 minus 1205721 = 1205725 minus 1205721
1205723 + 1205724 minus 1205721 minus 1205722 = 1205724 minus 21205722 = 1205724 + 1205725 minus 21205722
= 21205724 minus 21205722 = 1205724 minus4
3
1205721
1205723 + 1205725 minus 1205721 minus 1205722 = 1205725 minus 21205722 = 1205724 minus 21205722
= constant = 119891 (120578) (say) that is 120595 = 11990913119891 (120578)
(30b)
6 Mathematical Problems in Engineering
where 119891 is an arbitrary function of 120578 From the equations211988911990931205722119909 = 1198891205790 and 211988911990931205722119909 = 1198891206010 we obtain byintegration
Here 120578 is the similarity variable and 119891 120579 120601 are dependentvariables Note that the similarity transformations in (31) areconsistent with the well-known similarity transformationsreported in the paper of Cheng and Chang [21] for 120582 = 0
in their paper Thus the dimensionless velocity components119906 V can be expressed as
119906 =
1198911015840
11990913 V = minus
1
311990923
(119891 minus 21205781198911015840) (32)
where primes indicate differentiation with respect to 120578
32 Similarity Equations On substituting the transforma-tions in (31) into the governing (16)ndash(18) we obtain thefollowing system of ordinary differential equations
where 119887 = (1198631)0119871 is the thermal slip parameter
4 Basic Idea of the HAM
Let us consider the following differential equation
N [119906 (120591)] = 0 (37)
where N is a nonlinear operator 120591 denotes independentvariable and 119906(120591) is an unknown function respectively Forsimplicity we ignore all boundary or initial conditions whichcan be treated in the similar way By means of generalizing
the traditional homotopy method Liao [45] constructs theso-called zero-order deformation equation
(1 minus 119901)L [120593 (120591 119901) minus 1199060 (120591)] = 119901ℏ119867 (120591)N [120593 (120591 119901)] (38)
where 119901 isin [0 1] is the embedding parameter ℏ = 0 is anonzero auxiliary parameter 119867(120591) = 0 is an auxiliary func-tionL is an auxiliary linear operator 1199060(120591) is an initial guessof 119906(120591) and 120593(120591 119901) is an unknown function respectively Itis important that one has great freedom to choose auxiliarythings in the HAM Obviously when 119901 = 0 and 119901 = 1 itholds 120593(120591 0) = 1199060(120591) 120593(120591 1) = 119906(120591) respectively Thus as119901 increases from 0 to 1 the solution 120593(120591 119901) varies from theinitial guess 1199060(120591) to the solution 119906(120591) Expanding 120593(120591 119901) inTaylor series with respect to 119901 we have
If the auxiliary linear operator the initial guess theauxiliary parameter ℏ and the auxiliary function are soproperly chosen the series (39) converges at 119901 = 1 then wehave
119906 (120591) = 1199060 (120591) +
+infin
sum
119898=1
119906119898 (120591) (41)
which must be one of solutions of original nonlinear equa-tion as proved by Liao [45] As ℏ = minus1 and 119867(120591) = 1 (38)becomes
(1 minus 119901)L [120593 (120591 119901) minus 1199060 (120591)] + 119901N [120593 (120591 119901)] = 0 (42)
which is used mostly in the homotopy perturbation methodwhere the solution can be obtained directly without usingTaylor series
According to definition (39) the governing equationcan be deduced from the zero-order deformation equation(37) By defining the vector 119899 = 1199060(120591) 1199061(120591) 119906119899(120591)
and differentiating equation (37) 119898 times with respect tothe embedding parameter 119901 and then setting 119901 = 0 andfinally dividing them by119898 we have the so-called119898th-orderdeformation equation
L [119906119898 (120591) minus 120594119898119906119898minus1 (120591)] = ℏ119867 (120591) 119877119898 (119898minus1) (43)
It should be emphasized that 119906119898(120591) for 119898 ge 1 isgoverned by the linear equation (39) with the linear boundaryconditions that come from original problem which canbe easily solved by symbolic computation software such asMAPLE and MATHEMATICA
Figure 3 The ℏ-curve of 1205791015840(0) given by the various HAM-orderapproximate solution
5 HAM Solution
In this section we applied the HAM to obtain approx-imate analytical solutions of the effect of higher orderchemical reaction internal heat generation and the thermalslip boundary condition on the boundary-layer flow of ananofluid past an upward facing horizontal plate (33)ndash(35)
Figure 4 The ℏ-curve of 1206011015840(0) given by the various HAM-orderapproximate solution
We start with the initial approximation 1198650(120578) = 0 Θ0(120578) =(1(119887 + 1))Exp[minus120578]Φ0(120578) = Exp[minus120578] and the linear operator
L [119865 (120578 119901)] =
1205972119865 (120578 119901)
1205971205783
L [Θ (120578 119901)] =
1205972Θ(120578 119901)
1205971205782
+
120597Θ (120578 119901)
120597120578
L [Φ (120578 119901)] =
1205972Φ(120578 119901)
1205971205782
+
120597Φ (120578 119901)
120597120578
(45)
Furthermore from (33)ndash(35) we define the nonlinearoperators
N [119865 (120578 119901)] =
1205972119865 (120578 119901)
1205971205782
minus
2
3
120578(
120597Θ (120578 119901)
120597120578
minus 119873119903
120597Φ (120578 119901)
120597120578
)
N [Θ (120578 119901)] =
1205972Θ(120578 119901)
1205971205782
+
1
3
119865 (120578 119901)
120597Θ (120578 119901)
120597120578
+ 119873119887
120597Θ (120578 119901)
120597120578
120597Φ (120578 119901)
120597120578
+ 119873119905(
120597Θ (120578 119901)
120597120578
)
2
+ 119876Θ (120578 119901)
N [Φ (120578 119901)] =
1205972Φ(120578 119901)
1205971205782
+
Le3
119865 (120578 119901)
120597Φ (120578 119901)
120597120578
+
119873119905
119873119887
1205972Θ(120578 119901)
1205971205782
minus 119870(Φ (120578 119901))119899
(46)
8 Mathematical Problems in Engineering
Using the above definition with assumption 119867119865(120591) = 1119867Θ(120591) = Exp[minus120578] and 119867Φ(120591) = Exp[minus120578] we construct the119911119890119903119900th-order deformation equation
(1 minus 119901)L [120593 (119909 119901) minus 1199060 (119909)] = 119867 (120591) 119901ℏN [120593 (119909 119901)]
Differentiating the 119911119890119903119900th-order deformation equation(47)119898 times with respect to 119901 and finally dividing by119898 wehave the119898th-order deformation equation
L [119865119898 (119909) minus 120594119898119865119898minus1 (119909)] = ℏ119877119898 (119898minus1)
L [Θ119898 (119909) minus 120594119898Θ119898minus1 (119909)] = Exp [minus120578] ℏ119877119898 (Θ119898minus1)
L [Φ119898 (119909) minus 120594119898Φ119898minus1 (119909)] = Exp [minus120578] ℏ119877119898 (Φ119898minus1)
By using the symbolic software MATHEMATICA tosolve the system of linear equations (49) with the boundaryconditions (46) it can be obtained finally
1198651 (120578) = minus
4
3 (1 + 119887)
ℏ +
4119890minus120578
3 (1 + 119887)
ℏ +
4119873119903
3 (1 + 119887)
ℏ
+
4119887119873119903
3 (1 + 119887)
ℏ minus
4119873119903119890minus120578
3 (1 + 119887)
ℏ minus
4119887119873119903119890minus120578
3 (1 + 119887)
ℏ
+
2119890minus120578120578
3 (1 + 119887)
ℏ minus
2119873119903119890minus120578120578
3 (1 + 119887)
ℏ minus
2119887119873119903119890minus120578120578
3 (1 + 119887)
ℏ
Θ1 (120578) =119890minus2120578
2 (1 + 119887)
ℏ minus
119890minus120578
2(1 + 119887)2ℏ minus
119887119890minus120578
2(1 + 119887)2ℏ
minus
119873119887119890minus3120578
6(1 + 119887)2ℏ +
119887119873119887119890minus3120578
6(1 + 119887)2ℏ minus
119873119887119890minus120578
6(1 + 119887)2ℏ
minus
119887119873119887119890minus120578
6(1 + 119887)2ℏ +
119873119905119890minus3120578
6(1 + 119887)2ℏ minus
119873119905119890minus120578
6(1 + 119887)2ℏ
+
119876119890minus2120578
2 (1 + 119887)
ℏ minus
119876119890minus120578
2(1 + 119887)2ℏ minus
119887119876119890minus120578
2(1 + 119887)2ℏ
Φ1 (120578) =119890minus2120578
2 (1 + 119887)
ℏ +
119887119890minus2120578
2 (1 + 119887)
ℏ minus
119890minus120578
2 (1 + 119887)
ℏ
minus
119887119890minus120578
2 (1 + 119887)
ℏ minus
119870119890minus2120578
2 (1 + 119887)
ℏ minus
119887119870eminus2120578
2 (1 + 119887)
ℏ
+
119870119890minus120578
2 (1 + 119887)
ℏ +
119887119870119890minus120578
2 (1 + 119887)
ℏ +
119873119905
119873119887
119890minus2120578
2 (1 + 119887)
ℏ
minus
119873119905
119873119887
119890minus120578
2 (1 + 119887)
ℏ
(56)
The higher orders solutions of 119865(120578) Θ(120578) and Φ(120578) weretoo long to be mentioned here therefore they are showngraphically
51 Convergence of the HAM Solution As mentioned by Liao[45] HAM provides us with great freedom in choosing thesolution of a nonlinear problem by different base functionsThis has a great effect on the convergence region because theconvergence region and rate of a series are chiefly determinedby the base functions used to express the solution We usedseveral terms in evaluating the approximate solution 119865app asympsum119899119894=0 119865119894Θapp asymp sum
119899119894=0Θ119894Φapp asymp sum
119899119894=0Φ119894 note that the solution
series contains the auxiliary parameter ℏ which provides uswith a simple way to adjust and control the convergence ofthe solution series Generally by means of the so-called ℏ-curve that is a curve of a versus ℏ As pointed by Liao [45]the valid region of ℏ is a horizontal line segment Figures2 3 and 4 show the ℏ-curve with the various order of theHAM for 11989110158401015840(0) 1205791015840(0) 1206011015840(0) respectively when 119873119887 = 01
Figure 6 The residual error of 120579(120578) for (34) using 12th order ofapproximations
119873119905 = 01 119873119903 = 01 119876 = 01 Le = 10 119870 = 10 119899 = 10119887 = 00 It can be seen that when the order of series is 12 thesegment of the horizontal line is more than the other ordersFor example it can be found that for 11986510158401015840(0) the acceptablerange ofℏ is betweenminus05 andminus20 for 12th order of theHAMbut for 8th order of the HAM the acceptable range of ℏ isbetween minus10 and minus20 or minus10 and minus15 for 4th order of theHAM so horizontal line segment of 12th order of the HAM ismore than others Therefore it is straightforward to choosean appropriate range for ℏ which ensures the convergence
10 Mathematical Problems in Engineering
minus00002
120578
= minus10
= minus111 (optimal value)= minus12
Nt = 01Nr = 01
Q = 01Le = 10K = 10n = 10b = 00
0006
0004
0002
0
0 2 4 6 8 10
Nb = 01
Res (120601
(120578))
ℏ
ℏ
ℏ
Figure 7 The residual error of 120601(120578) for (35) using 12th order ofapproximations
of the solution series To choose optimal value of auxiliaryparameter the averaged residual errors (see [53] for moredetails) are defined as
119864119865119898 =1
119870
119870
sum
119894=0
[N119865(119898
sum
119894=0
119865119894 (119894Δ119909))]
2
119864Θ119898 =1
119870
119870
sum
119894=0
[NΘ(119898
sum
119894=0
Θ119894 (119894Δ119909))]
2
119864Φ119898 =1
119870
119870
sum
119894=0
[NΦ(119898
sum
119894=0
Φ119894 (119894Δ119909))]
2
(57)
where Δ119909 = 10119870 and 119870 = 20 For given order ofapproximation 119898 and the optimal values ℏ are given bythe minimum of 119864119898 corresponding to nonlinear algebraicequations
119889119864119865119898
119889ℏ
= 0
119889119864Θ119898
119889ℏ
= 0
119889119864Φ119898
119889ℏ
= 0 (58)
It is noted that the optimal value of ℏ is replaced intothe equations Table 1 shows optimal values obtained for theauxiliary parameter ℏ for various quantities of119873119905 119876 and 119899when119873119887 = 01119873119903 = 01 Le = 10 119870 = 10 119887 = 00 To see
the accuracy of the solutions the residual errors are definedfor the system as (for119898 order approximation)
Res119865119898
=
1198892119865119898
1198891205782minus
2
3
120578(
119889Θ119898
119889120578
minus 119873119903
119889Φ119898
119889120578
) (59)
ResΘ119898
=
1198892Θ119898
1198891205782+
1
3
119865119898
119889Θ119898
d120578+ 119873119887
119889Θ119898
119889120578
119889Φ119898
119889120578
+ 119873119905(
119889Θ119898
119889120578
)
2
+ 119876Θ119898
(60)
ResΦ119898
=
1198892Φ119898
1198891205782+
Le3
119865119898
119889Φ119898
119889120578
+
119873119905
119873119887
1198892Θ119898
1198891205782
minus 119870(
119889Φ119898
119889120578
)
119899
(61)
Figures 5 6 and 7 show the residual errors for 12th-orderdeformation solutions when 119873119887 = 01 119873119905 = 01 119873119903 = 01119876 = 01 Le = 10 119870 = 10 119899 = 10 119887 = 00 For example ℏ =minus145 has the minimum range of residual curve in Figure 5and so on
Graphical representation of results is very useful todemonstrate the efficiency and accuracy of the HAM for theabove problem
6 Comparisons and Verification
It is worth citing that for isothermal plate (119887 = 0) and in theabsence of internal heat generationabsorption (119876 = 0) andchemical reaction (119870 = 0) our problem reduces to Gorla andChamkha [26] and Khan and Pop [27] To verify the accuracyof our results the present results are compared in Table 2with Gorla and Chamkha [26] and are found to be in goodagreement
7 Physical Quantities
Theparameters of physical interest of the present problem arethe local skin friction factor 119862119891119909 the local Nusselt numberNu119909 and the local Sherwood number Sh119909 respectivelyPhysically 119862119891119909 indicates wall shear stress Nu119909 indicates therate of wall heat transfer whilst Sh119909 indicates the rate ofwall nanoparticle volume fractionThe following relations areused to find these quantities
119862119891119909 =2120583
1205881198802119903
(
120597119906
120597119910
)
119910=0
Nu119909 =minus119909
119879119891 minus 119879infin
(
120597119879
120597119910
)
119910=0
Sh119909 =minus119909
119862119908 minus 119862infin
(
120597119862
120597119910
)
119910=0
(62)
Mathematical Problems in Engineering 11
Table 2 Comparison of present solution with Gorla and Chamkha[26] for different values of buoyancy and nanofluid parameters
Present results (HAM) Gorla and Chamkha [26]minus1205791015840(0) minus120601
1015840(0) minus120579
1015840(0) minus120601
1015840(0)
Nr 119873119887 = 03119873119905 = 01 Le = 10 119887 = 119870 = 119876 = 0
01 032578 148242 326119864 minus 01 148416402 032385 146704 325119864 minus 01 146816103 032188 145125 322119864 minus 01 145266404 031985 143503 321119864 minus 01 143639205 031777 141833 319119864 minus 01 1419499Nt 119873119887 = 03119873119903 = 05 Le = 10 119887 = 119870 = 119876 = 0
01 031777 141833 319119864 minus 01 141949902 030486 141491 305119864 minus 01 141653603 02927 141561 293119864 minus 01 141686604 028125 141991 282119864 minus 01 142158205 027046 142737 271119864 minus 01 1429226Nb 119873119905 = 01119873119903 = 05 Le = 10 119887 = 119870 = 119876 = 0
01 03672 132611 368119864 minus 01 132745402 034271 139216 343119864 minus 01 139361503 031777 141833 319119864 minus 01 141949904 029399 143428 294119864 minus 01 143546405 027161 144598 272119864 minus 01 144772
Figure 10 Comparison of dimensionless concentration profile 120601(120578)obtained by various orders of the HAM
Here Ra119909 = 119892119870120573(1minus119862infin)Δ119879119909(120572119898]) is the local Rayleighnumber Pr = ]120572119898 is the Prandtl number and 119880119903 = (1 minus
119862infin)119892119870120573Δ119879120572119898 is the reference velocity
8 Results and Discussion
Graphical representation of results is very useful to discussthe physical features presented by the solutions This sectiondescribes the influence of some interesting parameters on
12 Mathematical Problems in Engineering
0 2 4 6 8 10
02
04
06
08
1
Nr = 01Numerical
Numerical
NumericalNr = 03Nr = 05
f998400 (120578)
120578
Nt = 01Nb = 01Q = 01Le = 10K = 10n = 10b = 00
Figure 11 Effect of buoyancy ratio119873119903 on the dimensionless velocityprofile
0 2 4 6 8 10
02
04
06
08
1
Nt = 01Nb = 01Q = 01Le = 10K = 10n = 10b = 00
Nr = 01Numerical
Numerical
NumericalNr = 03Nr = 05
120578
120579(120578
)
Figure 12 Effect of buoyancy ratio 119873119903 on the dimensionlesstemperature profile
the velocity and temperature fields Equations (33) to (35)with boundary conditions in (36) were solved analytically byHAM and numerically using Runge-Kutta-Fehlberg fourth-fifth order proposed by Aziz [54]
Figures 8 9 and 10 respectively for 1198911015840(120578) 120579(120578) and 120601(120578)show the comparisons between various order approximationof the optimal HAM and the numerical solutions for the case
0 2 4 6 8 10
02
04
06
08
1
Nr = 01Numerical
Numerical
NumericalNr = 03Nr = 05
Nt = 01Nb = 01Q = 01Le = 10K = 10n = 10b = 00
120578
120601(120578
)Figure 13 Effect of buoyancy ratio 119873119903 on the dimensionlessconcentration profile
0 2 4 6 8 10
02
04
06
08
1
12
NumericalNumerical
Numerical
f998400 (120578
)
120578
Nb = 01
Nb = 03
Nb = 05
Nt = 01Nr = 01Q = 01Le = 10K = 10n = 10b = 00
Figure 14 Effect of Brownian motion 119873119887 on the dimensionlessvelocity profile
119873119887 = 01 119873119905 = 01 119873119903 = 01 119876 = 01 Le = 10 119870 = 10119899 = 10 119887 = 00 For better presentation these results arepresented in Tables 3 4 and 5 Also Tables 6 7 and 8 presentthe comparisons between various order approximation of theoptimal HAM and the numerical solutions for the case119873119887 =03 119873119905 = 005 119873119903 = 03 119876 = 01 Le = 20 119870 = 05
Mathematical Problems in Engineering 13
0 2 4 6 8 10
02
04
06
08
1
0
120578
NumericalNumerical
Numerical
Nb = 01
Nb = 03
Nb = 05
Nt = 01Nr = 01Q = 01Le = 10K = 10n = 10b = 00
120579(120578
)
Figure 15 Effect of Brownian motion 119873119887 on the dimensionlesstemperature profile
02
04
06
08
1
00
2 4 6 8 10120578
NumericalNumerical
Numerical
Nb = 01
Nb = 03
Nb = 05
Nt = 01Nr = 01Q = 01Le = 10K = 10n = 10b = 00
120601(120578
)
Figure 16 Effect of Brownian motion 119873119887 on the dimensionlessconcentration profile
119899 = 20 119887 = 01 It is observed that the results of 12th-order approximation of the optimal HAM are very close tothe numerical solutions which confirm the validity of thesemethods
In the following figures effects of various physicalparameters on the dimensionless velocity temperature and
Table 3 Comparison of values of 1198911015840(120578) obtained by various ordersof the HAM with numerical solution when 119873119887 = 01119873119905 =
temperature profiles will be investigated These results havebeen obtained by 12th order of the HAM and have beenvalidated by numerical results Figures 11ndash13 respectivelyrepresent that the comparison of solutions of 1198911015840(120578) 120579(120578)and 120601(120578) for fix values 119873119887 = 01 119873119905 = 01 119876 = 01Le = 10 119870 = 10 119899 = 10 119887 = 00 and different values ofbuoyancy ratio119873119903 In Figure 11 it is clear that velocity of thefluid tremendously decreases in the near of horizontal platewith an increase in the buoyancy ratio It is observed that thetemperature increases slightly but concentration of the fluiddoes not vary sensibly (Figures 12 and 13) The effects of theBrownian motion 119873119887 are depicted in Figures 14 15 and 16when119873119903 = 01119873119905 = 01119876 = 01 Le = 10119870 = 10 119899 = 10
14 Mathematical Problems in Engineering
Table 6 Comparison of values of 1198911015840(120578) obtained by various ordersof the HAM with numerical solution when 119873119887 = 03119873119905 =
119887 = 00 If 119873119887 increases 1198911015840(120578) and 120579(120578) increase lightly and120601(120578) decreases The dimensionless velocity dimensionlesstemperature and dimensionless concentration profiles fordifferent values of thermophoresis 119873119905 with constant values119873119903 = 01 119873119887 = 01 119876 = 01 Le = 10 119870 = 10119899 = 10 119887 = 00 are presented in Figures 17 18 and19 It is observed that the velocity and temperature of thefluid are not impressible from119873119905 but concentration increasesslightly In Figures 20 21 and 22 respectively comparison ofsolutions of 1198911015840(120578) 120579(120578) and 120601(120578) for 119873119903 = 01 119873119887 = 01119873119905 = 01 Le = 10 119870 = 10 119899 = 10 119887 = 00 anddifferent values of heat generationabsorption parameter 119876are shown It is observed that in presence of heat source1198911015840(120578) 120579(120578) both increase while heat sink causes that 1198911015840(120578)
120579(120578) both decrease extremely These effects are more visible
Table 9 Values of reduced Nusselt number and Sherwood numberobtained by HAM for different values of the parameters Le 119870 119899 119887and 119876 when119873119887 = 119873119903 = 119873119905 = 01
Figure 17 Effect of thermophoresis 119873119905 on the dimensionlessvelocity profile
for temperature distribution For 120601(120578) the effect of changing119876 is seen to be almost insignificant This is in agreementwith the physical factThe dimensionless profiles for differentvalues of Lewis number Le with constant parameters 119873119903 =01 119873119887 = 01 119873119905 = 01 119876 = 01 119870 = 10 119899 = 10119887 = 00 are shown in Figures 23 24 and 25 It is clear that forthese constant parameters variation of Lewis number has anyconsiderable effect on the velocity and temperature profilesbut concentration profile decreases with the increase of Lewisnumber What is similar to these results can be detected inFigures 26 27 and 28 for chemical reaction parameter 119870Figures 29 30 and 31 respectively present the comparisonof solutions of 1198911015840(120578) 120579(120578) and 120601(120578) for 119873119903 = 05 119873119887 = 05119873119905 = 01 119876 = 001 Le = 10 119870 = 10 119887 = 00 and differentvalues of order of chemical reaction 119899 It is observed that
Mathematical Problems in Engineering 15
0 2 4 6 80
1
120578
Nt = 00
Nt = 005
Nt = 10NumericalNumerical
Numerical
08
06
04
02
10
Q = 01Le = 10K = 10n = 10b = 00
Nb = 01
120579(120578
)
Nr = 01
Figure 18 Effect of thermophoresis 119873119905 on the dimensionlesstemperature profile
0 2 4 6 80
1
120601(120578
)
120578
Nt = 00
Nt = 005
Nt = 10NumericalNumerical
Numerical
08
06
04
02
10
Q = 01Le = 10K = 10n = 10b = 00
Nb = 01Nr = 01
Figure 19 Effect of thermophoresis 119873119905 on the dimensionlessconcentration profile
for higher order of chemical reaction velocity decreases andtemperature and concentration profiles increase extremelyUltimately Figures 32 33 and 34 depict the effect of thermalslip parameter 119887 on velocity temperature and concentrationfunctions when 119873119903 = 01 119873119887 = 01 119873119905 = 01 119876 = 01Le = 20 119870 = 15 119899 = 10 It is seen that the velocityand temperature of the fluid decrease with the increase of the
0 2 4 6 80
1
12
10
02
04
06
08
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numericalf998400 (120578
)
120578
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 20 Effect of generationabsorption heat parameter119876 on thedimensionless velocity profile
0 2 4 6 8 100
02
04
06
08
1
120579(120578
)
120578
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numerical
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 21 Effect of generationabsorption heat parameter 119876 on thedimensionless temperature profile
thermal slip parameter in the near of horizontal plate whilethe concentration of the fluid does not vary patently
Also for investigation of the parameters of physicalinterest Table 9 is presented In this table numerical valuesof reduced Nusselt number and Sherwood number obtainedby HAM for different values of the parameters 119890 119870 119899 119887 and119876 can be compared when119873119887 = 119873119903 = 119873119905 = 01
16 Mathematical Problems in Engineering
10
02
04
06
08
0 2 4 6 80
1
120578
120601(120578
)
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numerical
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 22 Effect of generationabsorption heat parameter119876 on thedimensionless concentration profile
0 2 4 6 80
1
12
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 23 Effect of Lewis number Le on the dimensionless velocityprofile
9 Conclusions
In this paper we studied the steady laminar incompressiblefree convective flow of a nanofluid past a chemically reactingupward facing horizontal plate in porous medium takinginto account heat generation and the thermal slip boundarycondition
0 2 4 6 80
1
120579(120578
)
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 24 Effect of Lewis number Le on the dimensionlesstemperature profile
0 2 4 6 80
1
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 25 Effect of Lewis number Le on the dimensionlessconcentration profile
The governing partial differential equations have beentransformed by similarity transformations into a system ofordinary differential equations which are solved by OHAMand numerical method (fourth-order Runge-Kutta schemewith the shooting method) Dimensionless velocity temper-ature and concentration functions are presented for various
Mathematical Problems in Engineering 17
0 2 4 6 80
1
12
K = 00
K = 05
K = 10NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 26 Effect of chemical reaction parameter 119870 on the dimen-sionless velocity profile
0 2 4 6 80
1
K = 00
K = 05
K = 10NumericalNumerical
Numerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 27 Effect of chemical reaction parameter 119870 on the dimen-sionless temperature profile
values of parameters for the problem for example buoy-ancy ratio 119873119903 Brownian motion 119873119887 thermophoresis 119873119905heat generationabsorption parameter 119876 Lewis number Lechemical reaction parameter119870 order of chemical reaction 119899and thermal slip parameter 119887 From the present investigationthe following may be concluded
(i) The velocity 1198911015840(120578) increases with increasing 119873119887 anddecreases with increasing119873119903 119899 and 119887
0 2 4 6 80
1
K = 00
K = 05
K = 10NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 28 Effect of chemical reaction parameter 119870 on the dimen-sionless concentration profile
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 29 Effect of order of chemical reaction 119899 on the dimension-less velocity profile
(ii) The temperature 120579(120578) increases with 119873119903 119873119887 119899 anddecreases with 119887
(iii) Parameters of119873119905 Le and119870 have no effect on velocityand temperature profiles
(iv) The concentration distribution 120601(120578) increases withincreasing N119905 119899 and decreases with 119873119887 Le and 119870but119873119903 119876 and 119887 are not affected by it
18 Mathematical Problems in Engineering
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 30 Effect of order of chemical reaction 119899 on the dimension-less temperature profile
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 31 Effect of order of chemical reaction 119899 on the dimension-less concentration profile
(v) In presence of heat source (119876 gt 0) 1198911015840(120578) 120579(120578) bothincrease while heat sink (119876 lt 0) causes that 1198911015840(120578)120579(120578) both decrease extremely These effects are morevisible for temperature distribution
In addition numerical results for the reducedNusselt andSherwood numbers are tabulated It is concluded that thereduced Nusselt number increases with the Lewis numberand reaction parameter whist it decreases with the order of
0 2 4 6 8
1
12
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 32 Effect of order of thermal slip parameter 119887 on thedimensionless velocity profile
0 2 4 6 80
1
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 33 Effect of order of thermal slip parameter 119887 on thedimensionless temperature profile
chemical reaction thermal slip and generation parametersIt is further concluded that the reduced Sherwood numberenhances with the Lewis number generation and reactionparameters whist it suppresses with thermal slip and orderof chemical reaction
The convergence of the solution series and the power ofHAM in controlling and adjusting the convergence regionand rate of solution series were discussed The proper range
Mathematical Problems in Engineering 19
0 2 4 6 80
1
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 34 Effect of order of thermal slip parameter 119887 on thedimensionless concentration profile
of auxiliary parameter ℏ to ensure the convergence of thesolution series was obtained through the so called ℏ-curveThe HAM provides us with a convenient way to control theconvergence of approximation series which is a fundamentalqualitative difference in analysis between the HAM andother methods The numerical results of the above problemsdisplay a fast convergence with minimal calculations Thisshows that the HAM is a very efficient method Finally theagreement between analytical and numerical results of thepresent study with previous published results is excellent
Nomenclature
119887 Thermal slip parameter119862 Dimensional concentration119862119891 Skin friction factor1198631 Thermal slip factor119863119861 Brownian diffusion coefficient119863119879 Thermophoretic diffusion coefficient119891 Dimensionless velocity functions119892 Gravitation acceleration119870 Chemical reaction parameter119896 Permeability of the porous media1198960 The constant reaction rate119896119898 Effective thermal conductivity of the porous
medium119896(119909) Variable reaction rateL Linear operator of the HAM119871 Length of horizontal plateLe Lewis numberN Nonlinear operator of the HAM
119899 Order of chemical reaction119873119887 Brownian motion119873119903 Buoyancy ratio119873119905 ThermophoresisNu Nusselt number119875 PressurePr Prandtl number119876 Heat generationabsorption parameter1198760 Heat generationabsorption constant119902 Internal heat generation rate
Ra Rayleigh numberSh Sherwood number119879 Temperature119905 Time119906 Velocity in 119909-direction119880119903 Reference velocity Velocity vectorV Velocity in 119910-direction119909 Distance along the surface119910 Distance normal to the surface
Greek Letters
120572119898 Thermal diffusivity120573 Volumetric expansion coefficient of nanofluid120576 Porosity of the porous media120601 Dimensionless concentration120578 Similarity variable120583 Dynamic viscosity120579 Dimensionless temperature120588119891 Density of the base fluid120588119901 Density of the nanoparticles(120588119862)119891 Effective heat capacity of the fluid(120588119862)119875 Effective heat capacity of the nanoparticle material120595 Stream function
Subscript Superscript
infin Conditions far away from the surface1015840 Differentiation with respect to 120578
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
M M Rashidi wishes to thank the Centre for DifferentialEquations Continuum Mechanics and Applications Schoolof Computational and Applied Mathematics University ofthe Witwatersrand for the hospitality during his visit EMomoniat thanks theNational Research Foundation of SouthAfrica for their support
20 Mathematical Problems in Engineering
References
[1] E Abu-Nada F O Hakan and I Pop ldquoBuoyancy inducedflow in a nanofluid filled enclosure partially exposed to forcedconvectionrdquo Superlattices and Microstructures vol 51 no 3 pp381ndash395 2012
[2] S Kakac and A Pramuanjaroenkij ldquoReview of convective heattransfer enhancement with nanofluidsrdquo International Journal ofHeat and Mass Transfer vol 52 no 13-14 pp 3187ndash3196 2009
[3] J A Eastman S U S Choi S Li W Yu and L J ThompsonldquoAnomalously increased effective thermal conductivities ofethylene glycol-based nanofluids containing copper nanoparti-clesrdquo Applied Physics Letters vol 78 no 6 pp 718ndash720 2001
[4] Y Xuan and Q Li ldquoHeat transfer enhancement of nanofluidsrdquoInternational Journal of Heat and Fluid Flow vol 21 no 1 pp58ndash64 2000
[5] S K Das N Putra P Thiesen and W Roetzel ldquoTemperaturedependence of thermal conductivity enhancement for nanoflu-idsrdquo Journal of Heat Transfer vol 125 no 4 pp 567ndash574 2003
[6] D H Kumar H E Patel V R R Kumar T SundararajanT Pradeep and S K Das ldquoModel for heat conduction innanofluidsrdquo Physical Review Letters vol 93 no 14 Article ID144301 2004
[7] H E Patel T Sundararajan T Pradeep A Dasgupta NDasgupta and S K Das ldquoA micro-convection model forthermal conductivity of nanofluidsrdquo Pramana vol 65 no 5 pp863ndash869 2005
[8] A V Kuznetsov ldquoNon-oscillatory and oscillatory nanofluidbio-thermal convection in a horizontal layer of finite depthrdquoEuropean Journal of Mechanics vol 30 no 2 pp 156ndash165 2011
[9] M Napoli J C T Eijkel and S Pennathur ldquoNanofluidictechnology for biomolecule applications a critical reviewrdquo Labon a Chip vol 10 no 8 pp 957ndash985 2010
[10] J C Crepeau and R Clarksean ldquoSimilarity solutions of naturalconvection with internal heat generationrdquo Journal of HeatTransfer vol 119 no 1 pp 183ndash185 1997
[11] A Z Sahin ldquoTransient heat conduction in semi-infinite solidwith spatially decaying exponential heat generationrdquo Interna-tional Communications in Heat andMass Transfer vol 19 no 3pp 349ndash358 1992
[12] O D Makind and A Aziz ldquoMixed convection from a con-vectively heated vertical plate to a fluid with internal heatgenerationrdquo Journal of Heat Transfer vol 133 Article ID 1225016 pages 2011
[13] J Buongiorno ldquoConvective transport in nanofluidsrdquo Journal ofHeat Transfer vol 128 no 3 pp 240ndash250 2006
[14] AV Kuznetsov andDANield ldquoNatural convective boundary-layer flow of a nanofluid past a vertical platerdquo InternationalJournal of Thermal Sciences vol 49 no 2 pp 243ndash247 2010
[15] L B Godson D Raja L D Mohan and S WongwisescldquoEnhancement of heat transfer using nanofluids-an overviewrdquoRenewable and Sustainable Energy Reviews vol 14 no 2 pp629ndash641 2010
[16] D A Nield and A BejanConvection in PorousMedia SpringerNew York NY USA 3rd edition 2006
[17] P Vadasz Emerging Topics in Heat and Mass Transfer in PorousMedia Springer New York NY USA 2008
[18] K Vafai Porous Media Applications in Biological Systems andBiotechnology CRC Press New York NY USA 2010
[19] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application to
heat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977
[20] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for natural convective boundary-layer flow in a porousmedium saturated by a nanofluidrdquo International Journal of Heatand Mass Transfer vol 52 no 25-26 pp 5792ndash5795 2009
[21] P Cheng and I D Chang ldquoBuoyancy induced flows in asaturated porous medium adjacent to impermeable horizontalsurfacesrdquo International Journal of Heat and Mass Transfer vol19 no 11 pp 1267ndash1272 1976
[22] I D Chang and P Cheng ldquoMatched asymptotic expansionsfor free convection about an impermeable horizontal surfacein a porous mediumrdquo International Journal of Heat and MassTransfer vol 26 no 2 pp 163ndash174 1983
[23] D S Shiunlin and B Gebhart ldquoBuoyancy-induced flow adja-cent to a horizontal surface submerged in porous mediumsaturated with cold waterrdquo International Journal of Heat andMass Transfer vol 29 no 4 pp 611ndash623 1986
[24] J H Merkin and G Zhang ldquoOn the similarity solutions for freeconvection in a saturated porousmedium adjacent to imperme-able horizontal surfacesrdquoWarme- und Stoffubertragung vol 25no 3 pp 179ndash184 1990
[25] M A Chaudhary J H Merkin and I Pop ldquoNatural convectionfrom a horizontal permeable surface in a porous mediummdashnumerical and asymptotic solutionsrdquo Transport in PorousMedia vol 22 no 3 pp 327ndash344 1996
[26] R S R Gorla and A Chamkha ldquoNatural convective boundarylayer flow over a nonisothermal vertical plate embedded ina porous medium saturated with a nanofluidrdquo Nanoscale andMicroscale Thermophysical Engineering vol 15 no 2 pp 81ndash942011
[27] W A Khan and I Pop ldquoFree convection boundary layer flowpast a horizontal flat plate embedded in a porous medium filledwith a nanofluidrdquo Journal of Heat Transfer vol 133 no 9 ArticleID 094501 2011
[28] A Aziz W A Khan and I Pop ldquoFree convection boundarylayer flow past a horizontal flat plate embedded in porousmedium filled by nanofluid containing gyrotactic microorgan-ismsrdquo International Journal of Thermal Sciences vol 56 pp 48ndash57 2012
[29] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer New York NYUSA 2009
[30] A A Avramenko S G Kobzar I V Shevchuk A V Kuznetsovand L T Iwanisov ldquoSymmetry of turbulent boundary-layerflows investigation of different eddy viscosity modelsrdquo ActaMechanica vol 151 no 1-2 pp 1ndash14 2001
[31] A V Kuznetsov A A Avramenko and P Geng ldquoAnalyticalinvestigation of a falling plume caused by bioconvection ofoxytactic bacteria in a fluid saturated porous mediumrdquo Inter-national Journal of Engineering Science vol 42 no 5-6 pp 557ndash569 2004
[32] M Jalil S Asghar and M Mushtaq ldquoLie group analysis ofmixed convection flow with mass transfer over a stretchingsurface with suction or injectionrdquo Mathematical Problems inEngineering vol 2010 Article ID 264901 14 pages 2010
[33] M A A Hamad M J Uddin and A I M Ismail ldquoRadiationeffects on heat and mass transfer in MHD stagnation-pointflow over a permeable flat plate with thermal convectivesurface boundary condition temperature dependent viscosityand thermal conductivityrdquoNuclear Engineering and Design vol242 pp 194ndash200 2012
Mathematical Problems in Engineering 21
[34] A Aziz M J Uddin M A A Hamad and A I MdIsmail ldquoMHD flow over an inclined radiating plate with thetemperature-dependent thermal conductivity variable reactiveindex and heat generationrdquoHeat Transfer vol 41 no 3 pp 241ndash259 2012
[35] A G Hansen Similarity Analysis of Boundary Layer Problemsin Engineering Prentice Hall Englewood Cliffs NJ USA 1964
[36] W FAmesNonlinear Partial Differential Equations in Engineer-ing Academic Press New York NY USA 1972
[37] R Seshadri and T Y Na Group Invariance in EngineeringBoundary Value Problems Springer New York NY USA 1985
[38] D Shang Theory of Heat Transfer with Forced Convection FilmFlows vol 3 of Heat and Mass Transfer 2010
[40] E Erfani M M Rashidi and A B Parsa ldquoThe modifieddifferential transform method for solving off-centered stag-nation flow toward a rotating discrdquo International Journal ofComputational Methods vol 7 no 4 pp 655ndash670 2010
[41] M M Rashidi N Laraqi and A Basiri Parsa ldquoAnalyticalmodeling of heat convection in magnetized micropolar fluid byusing modified differential transform methodrdquo Heat Transfervol 40 no 3 pp 187ndash204 2011
[42] M M Rashidi and M Keimanesh ldquoUsing differential trans-form method and pade approximant for solving mhd flow ina laminar liquid film from a horizontal stretching surfacerdquoMathematical Problems in Engineering vol 2010 Article ID491319 14 pages 2010
[43] M M Rashidi and G Domairry ldquoNew analytical solution ofthe three-dimensional NavierStokes equationsrdquoModern PhysicsLetters B vol 23 no 26 pp 3147ndash3155 2009
[44] M M Rashidi M Keimanesh O A Beg and T K HungldquoMagnetohydrodynamic biorheological transport phenomenain a porous medium a simulation of magnetic blood flowcontrol and filtrationrdquo International Journal for NumericalMethods in Biomedical Engineering vol 27 no 6 pp 805ndash8212011
[45] S J Liao The proposed homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992
[46] M M Rashidi G Domairry and S Dinarvand ldquoApproximatesolutions for the Burger and regularized long wave equationsby means of the homotopy analysis methodrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 14 no 3 pp708ndash717 2009
[47] Z Ziabakhsh and G Domairry ldquoAnalytic solution of naturalconvection flow of a non-Newtonian fluid between two verticalflat plates using homotopy analysis methodrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 14 no 5pp 1868ndash1880 2009
[48] M M Rashidi S A Mohimanian Pour T Hayat and SObaidat ldquoAnalytic approximate solutions for steady flow over arotating disk in porousmediumwith heat transfer by homotopyanalysis methodrdquo Computers and Fluids vol 54 no 1 pp 1ndash92012
[49] M Sajid T Hayat and S Asghar ldquoComparison between theHAM and HPM solutions of thin film flows of non-Newtonianfluids on a moving beltrdquo Nonlinear Dynamics vol 50 no 1-2pp 27ndash35 2007
[50] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for the double-diffusive natural convective boundary
layer flow in a porous medium saturated by a nanofluidrdquoInternational Journal of Heat and Mass Transfer vol 54 no 1-3pp 374ndash378 2011
[51] K Vajravelu and A Hadjinicolaou ldquoConvective heat transferin an electrically conducting fluid at a stretching surfacewith uniform free streamrdquo International Journal of EngineeringScience vol 35 pp 1237ndash1244 1997
[52] S Mukhopadhyay and G C Layek ldquoEffects of variable fluidviscosity on flow past a heated stretching sheet embedded ina porous medium in presence of heat sourcesinkrdquo Meccanicavol 47 no 4 pp 863ndash876 2012
[53] S J Liao ldquoAn optimal homotopy-analysis approach for stronglynonlinear differential equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 8 pp 2003ndash20162010
[54] A Aziz ldquoA similarity solution for laminar thermal boundarylayer over a flat plate with a convective surface boundary con-ditionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1064ndash1068 2009
times Δ 119895 (119909 119910 120579 120601 1205973120595
1205971199103)
(119895 = 1 2 3)
(22)
Since this is the requirement that the differential formsΔ 1 Δ 2 and Δ 3 be reformed under the transformation groupin (19) by using (21) (16)ndash(18) are transformed to (see [3538])
The system will remain invariant (structure of the equa-tions same) under the group transformation Γ if we have thefollowing relationship among the exponents
1205723 minus 21205722 = 1205724 minus 1205721 = 1205725 minus 1205721
1205723 + 1205724 minus 1205721 minus 1205722 = 1205724 minus 21205722 = 1205724 + 1205725 minus 21205722
= 21205724 minus 21205722 = 1205724 minus4
3
1205721
1205723 + 1205725 minus 1205721 minus 1205722 = 1205725 minus 21205722 = 1205724 minus 21205722
= constant = 119891 (120578) (say) that is 120595 = 11990913119891 (120578)
(30b)
6 Mathematical Problems in Engineering
where 119891 is an arbitrary function of 120578 From the equations211988911990931205722119909 = 1198891205790 and 211988911990931205722119909 = 1198891206010 we obtain byintegration
Here 120578 is the similarity variable and 119891 120579 120601 are dependentvariables Note that the similarity transformations in (31) areconsistent with the well-known similarity transformationsreported in the paper of Cheng and Chang [21] for 120582 = 0
in their paper Thus the dimensionless velocity components119906 V can be expressed as
119906 =
1198911015840
11990913 V = minus
1
311990923
(119891 minus 21205781198911015840) (32)
where primes indicate differentiation with respect to 120578
32 Similarity Equations On substituting the transforma-tions in (31) into the governing (16)ndash(18) we obtain thefollowing system of ordinary differential equations
where 119887 = (1198631)0119871 is the thermal slip parameter
4 Basic Idea of the HAM
Let us consider the following differential equation
N [119906 (120591)] = 0 (37)
where N is a nonlinear operator 120591 denotes independentvariable and 119906(120591) is an unknown function respectively Forsimplicity we ignore all boundary or initial conditions whichcan be treated in the similar way By means of generalizing
the traditional homotopy method Liao [45] constructs theso-called zero-order deformation equation
(1 minus 119901)L [120593 (120591 119901) minus 1199060 (120591)] = 119901ℏ119867 (120591)N [120593 (120591 119901)] (38)
where 119901 isin [0 1] is the embedding parameter ℏ = 0 is anonzero auxiliary parameter 119867(120591) = 0 is an auxiliary func-tionL is an auxiliary linear operator 1199060(120591) is an initial guessof 119906(120591) and 120593(120591 119901) is an unknown function respectively Itis important that one has great freedom to choose auxiliarythings in the HAM Obviously when 119901 = 0 and 119901 = 1 itholds 120593(120591 0) = 1199060(120591) 120593(120591 1) = 119906(120591) respectively Thus as119901 increases from 0 to 1 the solution 120593(120591 119901) varies from theinitial guess 1199060(120591) to the solution 119906(120591) Expanding 120593(120591 119901) inTaylor series with respect to 119901 we have
If the auxiliary linear operator the initial guess theauxiliary parameter ℏ and the auxiliary function are soproperly chosen the series (39) converges at 119901 = 1 then wehave
119906 (120591) = 1199060 (120591) +
+infin
sum
119898=1
119906119898 (120591) (41)
which must be one of solutions of original nonlinear equa-tion as proved by Liao [45] As ℏ = minus1 and 119867(120591) = 1 (38)becomes
(1 minus 119901)L [120593 (120591 119901) minus 1199060 (120591)] + 119901N [120593 (120591 119901)] = 0 (42)
which is used mostly in the homotopy perturbation methodwhere the solution can be obtained directly without usingTaylor series
According to definition (39) the governing equationcan be deduced from the zero-order deformation equation(37) By defining the vector 119899 = 1199060(120591) 1199061(120591) 119906119899(120591)
and differentiating equation (37) 119898 times with respect tothe embedding parameter 119901 and then setting 119901 = 0 andfinally dividing them by119898 we have the so-called119898th-orderdeformation equation
L [119906119898 (120591) minus 120594119898119906119898minus1 (120591)] = ℏ119867 (120591) 119877119898 (119898minus1) (43)
It should be emphasized that 119906119898(120591) for 119898 ge 1 isgoverned by the linear equation (39) with the linear boundaryconditions that come from original problem which canbe easily solved by symbolic computation software such asMAPLE and MATHEMATICA
Figure 3 The ℏ-curve of 1205791015840(0) given by the various HAM-orderapproximate solution
5 HAM Solution
In this section we applied the HAM to obtain approx-imate analytical solutions of the effect of higher orderchemical reaction internal heat generation and the thermalslip boundary condition on the boundary-layer flow of ananofluid past an upward facing horizontal plate (33)ndash(35)
Figure 4 The ℏ-curve of 1206011015840(0) given by the various HAM-orderapproximate solution
We start with the initial approximation 1198650(120578) = 0 Θ0(120578) =(1(119887 + 1))Exp[minus120578]Φ0(120578) = Exp[minus120578] and the linear operator
L [119865 (120578 119901)] =
1205972119865 (120578 119901)
1205971205783
L [Θ (120578 119901)] =
1205972Θ(120578 119901)
1205971205782
+
120597Θ (120578 119901)
120597120578
L [Φ (120578 119901)] =
1205972Φ(120578 119901)
1205971205782
+
120597Φ (120578 119901)
120597120578
(45)
Furthermore from (33)ndash(35) we define the nonlinearoperators
N [119865 (120578 119901)] =
1205972119865 (120578 119901)
1205971205782
minus
2
3
120578(
120597Θ (120578 119901)
120597120578
minus 119873119903
120597Φ (120578 119901)
120597120578
)
N [Θ (120578 119901)] =
1205972Θ(120578 119901)
1205971205782
+
1
3
119865 (120578 119901)
120597Θ (120578 119901)
120597120578
+ 119873119887
120597Θ (120578 119901)
120597120578
120597Φ (120578 119901)
120597120578
+ 119873119905(
120597Θ (120578 119901)
120597120578
)
2
+ 119876Θ (120578 119901)
N [Φ (120578 119901)] =
1205972Φ(120578 119901)
1205971205782
+
Le3
119865 (120578 119901)
120597Φ (120578 119901)
120597120578
+
119873119905
119873119887
1205972Θ(120578 119901)
1205971205782
minus 119870(Φ (120578 119901))119899
(46)
8 Mathematical Problems in Engineering
Using the above definition with assumption 119867119865(120591) = 1119867Θ(120591) = Exp[minus120578] and 119867Φ(120591) = Exp[minus120578] we construct the119911119890119903119900th-order deformation equation
(1 minus 119901)L [120593 (119909 119901) minus 1199060 (119909)] = 119867 (120591) 119901ℏN [120593 (119909 119901)]
Differentiating the 119911119890119903119900th-order deformation equation(47)119898 times with respect to 119901 and finally dividing by119898 wehave the119898th-order deformation equation
L [119865119898 (119909) minus 120594119898119865119898minus1 (119909)] = ℏ119877119898 (119898minus1)
L [Θ119898 (119909) minus 120594119898Θ119898minus1 (119909)] = Exp [minus120578] ℏ119877119898 (Θ119898minus1)
L [Φ119898 (119909) minus 120594119898Φ119898minus1 (119909)] = Exp [minus120578] ℏ119877119898 (Φ119898minus1)
By using the symbolic software MATHEMATICA tosolve the system of linear equations (49) with the boundaryconditions (46) it can be obtained finally
1198651 (120578) = minus
4
3 (1 + 119887)
ℏ +
4119890minus120578
3 (1 + 119887)
ℏ +
4119873119903
3 (1 + 119887)
ℏ
+
4119887119873119903
3 (1 + 119887)
ℏ minus
4119873119903119890minus120578
3 (1 + 119887)
ℏ minus
4119887119873119903119890minus120578
3 (1 + 119887)
ℏ
+
2119890minus120578120578
3 (1 + 119887)
ℏ minus
2119873119903119890minus120578120578
3 (1 + 119887)
ℏ minus
2119887119873119903119890minus120578120578
3 (1 + 119887)
ℏ
Θ1 (120578) =119890minus2120578
2 (1 + 119887)
ℏ minus
119890minus120578
2(1 + 119887)2ℏ minus
119887119890minus120578
2(1 + 119887)2ℏ
minus
119873119887119890minus3120578
6(1 + 119887)2ℏ +
119887119873119887119890minus3120578
6(1 + 119887)2ℏ minus
119873119887119890minus120578
6(1 + 119887)2ℏ
minus
119887119873119887119890minus120578
6(1 + 119887)2ℏ +
119873119905119890minus3120578
6(1 + 119887)2ℏ minus
119873119905119890minus120578
6(1 + 119887)2ℏ
+
119876119890minus2120578
2 (1 + 119887)
ℏ minus
119876119890minus120578
2(1 + 119887)2ℏ minus
119887119876119890minus120578
2(1 + 119887)2ℏ
Φ1 (120578) =119890minus2120578
2 (1 + 119887)
ℏ +
119887119890minus2120578
2 (1 + 119887)
ℏ minus
119890minus120578
2 (1 + 119887)
ℏ
minus
119887119890minus120578
2 (1 + 119887)
ℏ minus
119870119890minus2120578
2 (1 + 119887)
ℏ minus
119887119870eminus2120578
2 (1 + 119887)
ℏ
+
119870119890minus120578
2 (1 + 119887)
ℏ +
119887119870119890minus120578
2 (1 + 119887)
ℏ +
119873119905
119873119887
119890minus2120578
2 (1 + 119887)
ℏ
minus
119873119905
119873119887
119890minus120578
2 (1 + 119887)
ℏ
(56)
The higher orders solutions of 119865(120578) Θ(120578) and Φ(120578) weretoo long to be mentioned here therefore they are showngraphically
51 Convergence of the HAM Solution As mentioned by Liao[45] HAM provides us with great freedom in choosing thesolution of a nonlinear problem by different base functionsThis has a great effect on the convergence region because theconvergence region and rate of a series are chiefly determinedby the base functions used to express the solution We usedseveral terms in evaluating the approximate solution 119865app asympsum119899119894=0 119865119894Θapp asymp sum
119899119894=0Θ119894Φapp asymp sum
119899119894=0Φ119894 note that the solution
series contains the auxiliary parameter ℏ which provides uswith a simple way to adjust and control the convergence ofthe solution series Generally by means of the so-called ℏ-curve that is a curve of a versus ℏ As pointed by Liao [45]the valid region of ℏ is a horizontal line segment Figures2 3 and 4 show the ℏ-curve with the various order of theHAM for 11989110158401015840(0) 1205791015840(0) 1206011015840(0) respectively when 119873119887 = 01
Figure 6 The residual error of 120579(120578) for (34) using 12th order ofapproximations
119873119905 = 01 119873119903 = 01 119876 = 01 Le = 10 119870 = 10 119899 = 10119887 = 00 It can be seen that when the order of series is 12 thesegment of the horizontal line is more than the other ordersFor example it can be found that for 11986510158401015840(0) the acceptablerange ofℏ is betweenminus05 andminus20 for 12th order of theHAMbut for 8th order of the HAM the acceptable range of ℏ isbetween minus10 and minus20 or minus10 and minus15 for 4th order of theHAM so horizontal line segment of 12th order of the HAM ismore than others Therefore it is straightforward to choosean appropriate range for ℏ which ensures the convergence
10 Mathematical Problems in Engineering
minus00002
120578
= minus10
= minus111 (optimal value)= minus12
Nt = 01Nr = 01
Q = 01Le = 10K = 10n = 10b = 00
0006
0004
0002
0
0 2 4 6 8 10
Nb = 01
Res (120601
(120578))
ℏ
ℏ
ℏ
Figure 7 The residual error of 120601(120578) for (35) using 12th order ofapproximations
of the solution series To choose optimal value of auxiliaryparameter the averaged residual errors (see [53] for moredetails) are defined as
119864119865119898 =1
119870
119870
sum
119894=0
[N119865(119898
sum
119894=0
119865119894 (119894Δ119909))]
2
119864Θ119898 =1
119870
119870
sum
119894=0
[NΘ(119898
sum
119894=0
Θ119894 (119894Δ119909))]
2
119864Φ119898 =1
119870
119870
sum
119894=0
[NΦ(119898
sum
119894=0
Φ119894 (119894Δ119909))]
2
(57)
where Δ119909 = 10119870 and 119870 = 20 For given order ofapproximation 119898 and the optimal values ℏ are given bythe minimum of 119864119898 corresponding to nonlinear algebraicequations
119889119864119865119898
119889ℏ
= 0
119889119864Θ119898
119889ℏ
= 0
119889119864Φ119898
119889ℏ
= 0 (58)
It is noted that the optimal value of ℏ is replaced intothe equations Table 1 shows optimal values obtained for theauxiliary parameter ℏ for various quantities of119873119905 119876 and 119899when119873119887 = 01119873119903 = 01 Le = 10 119870 = 10 119887 = 00 To see
the accuracy of the solutions the residual errors are definedfor the system as (for119898 order approximation)
Res119865119898
=
1198892119865119898
1198891205782minus
2
3
120578(
119889Θ119898
119889120578
minus 119873119903
119889Φ119898
119889120578
) (59)
ResΘ119898
=
1198892Θ119898
1198891205782+
1
3
119865119898
119889Θ119898
d120578+ 119873119887
119889Θ119898
119889120578
119889Φ119898
119889120578
+ 119873119905(
119889Θ119898
119889120578
)
2
+ 119876Θ119898
(60)
ResΦ119898
=
1198892Φ119898
1198891205782+
Le3
119865119898
119889Φ119898
119889120578
+
119873119905
119873119887
1198892Θ119898
1198891205782
minus 119870(
119889Φ119898
119889120578
)
119899
(61)
Figures 5 6 and 7 show the residual errors for 12th-orderdeformation solutions when 119873119887 = 01 119873119905 = 01 119873119903 = 01119876 = 01 Le = 10 119870 = 10 119899 = 10 119887 = 00 For example ℏ =minus145 has the minimum range of residual curve in Figure 5and so on
Graphical representation of results is very useful todemonstrate the efficiency and accuracy of the HAM for theabove problem
6 Comparisons and Verification
It is worth citing that for isothermal plate (119887 = 0) and in theabsence of internal heat generationabsorption (119876 = 0) andchemical reaction (119870 = 0) our problem reduces to Gorla andChamkha [26] and Khan and Pop [27] To verify the accuracyof our results the present results are compared in Table 2with Gorla and Chamkha [26] and are found to be in goodagreement
7 Physical Quantities
Theparameters of physical interest of the present problem arethe local skin friction factor 119862119891119909 the local Nusselt numberNu119909 and the local Sherwood number Sh119909 respectivelyPhysically 119862119891119909 indicates wall shear stress Nu119909 indicates therate of wall heat transfer whilst Sh119909 indicates the rate ofwall nanoparticle volume fractionThe following relations areused to find these quantities
119862119891119909 =2120583
1205881198802119903
(
120597119906
120597119910
)
119910=0
Nu119909 =minus119909
119879119891 minus 119879infin
(
120597119879
120597119910
)
119910=0
Sh119909 =minus119909
119862119908 minus 119862infin
(
120597119862
120597119910
)
119910=0
(62)
Mathematical Problems in Engineering 11
Table 2 Comparison of present solution with Gorla and Chamkha[26] for different values of buoyancy and nanofluid parameters
Present results (HAM) Gorla and Chamkha [26]minus1205791015840(0) minus120601
1015840(0) minus120579
1015840(0) minus120601
1015840(0)
Nr 119873119887 = 03119873119905 = 01 Le = 10 119887 = 119870 = 119876 = 0
01 032578 148242 326119864 minus 01 148416402 032385 146704 325119864 minus 01 146816103 032188 145125 322119864 minus 01 145266404 031985 143503 321119864 minus 01 143639205 031777 141833 319119864 minus 01 1419499Nt 119873119887 = 03119873119903 = 05 Le = 10 119887 = 119870 = 119876 = 0
01 031777 141833 319119864 minus 01 141949902 030486 141491 305119864 minus 01 141653603 02927 141561 293119864 minus 01 141686604 028125 141991 282119864 minus 01 142158205 027046 142737 271119864 minus 01 1429226Nb 119873119905 = 01119873119903 = 05 Le = 10 119887 = 119870 = 119876 = 0
01 03672 132611 368119864 minus 01 132745402 034271 139216 343119864 minus 01 139361503 031777 141833 319119864 minus 01 141949904 029399 143428 294119864 minus 01 143546405 027161 144598 272119864 minus 01 144772
Figure 10 Comparison of dimensionless concentration profile 120601(120578)obtained by various orders of the HAM
Here Ra119909 = 119892119870120573(1minus119862infin)Δ119879119909(120572119898]) is the local Rayleighnumber Pr = ]120572119898 is the Prandtl number and 119880119903 = (1 minus
119862infin)119892119870120573Δ119879120572119898 is the reference velocity
8 Results and Discussion
Graphical representation of results is very useful to discussthe physical features presented by the solutions This sectiondescribes the influence of some interesting parameters on
12 Mathematical Problems in Engineering
0 2 4 6 8 10
02
04
06
08
1
Nr = 01Numerical
Numerical
NumericalNr = 03Nr = 05
f998400 (120578)
120578
Nt = 01Nb = 01Q = 01Le = 10K = 10n = 10b = 00
Figure 11 Effect of buoyancy ratio119873119903 on the dimensionless velocityprofile
0 2 4 6 8 10
02
04
06
08
1
Nt = 01Nb = 01Q = 01Le = 10K = 10n = 10b = 00
Nr = 01Numerical
Numerical
NumericalNr = 03Nr = 05
120578
120579(120578
)
Figure 12 Effect of buoyancy ratio 119873119903 on the dimensionlesstemperature profile
the velocity and temperature fields Equations (33) to (35)with boundary conditions in (36) were solved analytically byHAM and numerically using Runge-Kutta-Fehlberg fourth-fifth order proposed by Aziz [54]
Figures 8 9 and 10 respectively for 1198911015840(120578) 120579(120578) and 120601(120578)show the comparisons between various order approximationof the optimal HAM and the numerical solutions for the case
0 2 4 6 8 10
02
04
06
08
1
Nr = 01Numerical
Numerical
NumericalNr = 03Nr = 05
Nt = 01Nb = 01Q = 01Le = 10K = 10n = 10b = 00
120578
120601(120578
)Figure 13 Effect of buoyancy ratio 119873119903 on the dimensionlessconcentration profile
0 2 4 6 8 10
02
04
06
08
1
12
NumericalNumerical
Numerical
f998400 (120578
)
120578
Nb = 01
Nb = 03
Nb = 05
Nt = 01Nr = 01Q = 01Le = 10K = 10n = 10b = 00
Figure 14 Effect of Brownian motion 119873119887 on the dimensionlessvelocity profile
119873119887 = 01 119873119905 = 01 119873119903 = 01 119876 = 01 Le = 10 119870 = 10119899 = 10 119887 = 00 For better presentation these results arepresented in Tables 3 4 and 5 Also Tables 6 7 and 8 presentthe comparisons between various order approximation of theoptimal HAM and the numerical solutions for the case119873119887 =03 119873119905 = 005 119873119903 = 03 119876 = 01 Le = 20 119870 = 05
Mathematical Problems in Engineering 13
0 2 4 6 8 10
02
04
06
08
1
0
120578
NumericalNumerical
Numerical
Nb = 01
Nb = 03
Nb = 05
Nt = 01Nr = 01Q = 01Le = 10K = 10n = 10b = 00
120579(120578
)
Figure 15 Effect of Brownian motion 119873119887 on the dimensionlesstemperature profile
02
04
06
08
1
00
2 4 6 8 10120578
NumericalNumerical
Numerical
Nb = 01
Nb = 03
Nb = 05
Nt = 01Nr = 01Q = 01Le = 10K = 10n = 10b = 00
120601(120578
)
Figure 16 Effect of Brownian motion 119873119887 on the dimensionlessconcentration profile
119899 = 20 119887 = 01 It is observed that the results of 12th-order approximation of the optimal HAM are very close tothe numerical solutions which confirm the validity of thesemethods
In the following figures effects of various physicalparameters on the dimensionless velocity temperature and
Table 3 Comparison of values of 1198911015840(120578) obtained by various ordersof the HAM with numerical solution when 119873119887 = 01119873119905 =
temperature profiles will be investigated These results havebeen obtained by 12th order of the HAM and have beenvalidated by numerical results Figures 11ndash13 respectivelyrepresent that the comparison of solutions of 1198911015840(120578) 120579(120578)and 120601(120578) for fix values 119873119887 = 01 119873119905 = 01 119876 = 01Le = 10 119870 = 10 119899 = 10 119887 = 00 and different values ofbuoyancy ratio119873119903 In Figure 11 it is clear that velocity of thefluid tremendously decreases in the near of horizontal platewith an increase in the buoyancy ratio It is observed that thetemperature increases slightly but concentration of the fluiddoes not vary sensibly (Figures 12 and 13) The effects of theBrownian motion 119873119887 are depicted in Figures 14 15 and 16when119873119903 = 01119873119905 = 01119876 = 01 Le = 10119870 = 10 119899 = 10
14 Mathematical Problems in Engineering
Table 6 Comparison of values of 1198911015840(120578) obtained by various ordersof the HAM with numerical solution when 119873119887 = 03119873119905 =
119887 = 00 If 119873119887 increases 1198911015840(120578) and 120579(120578) increase lightly and120601(120578) decreases The dimensionless velocity dimensionlesstemperature and dimensionless concentration profiles fordifferent values of thermophoresis 119873119905 with constant values119873119903 = 01 119873119887 = 01 119876 = 01 Le = 10 119870 = 10119899 = 10 119887 = 00 are presented in Figures 17 18 and19 It is observed that the velocity and temperature of thefluid are not impressible from119873119905 but concentration increasesslightly In Figures 20 21 and 22 respectively comparison ofsolutions of 1198911015840(120578) 120579(120578) and 120601(120578) for 119873119903 = 01 119873119887 = 01119873119905 = 01 Le = 10 119870 = 10 119899 = 10 119887 = 00 anddifferent values of heat generationabsorption parameter 119876are shown It is observed that in presence of heat source1198911015840(120578) 120579(120578) both increase while heat sink causes that 1198911015840(120578)
120579(120578) both decrease extremely These effects are more visible
Table 9 Values of reduced Nusselt number and Sherwood numberobtained by HAM for different values of the parameters Le 119870 119899 119887and 119876 when119873119887 = 119873119903 = 119873119905 = 01
Figure 17 Effect of thermophoresis 119873119905 on the dimensionlessvelocity profile
for temperature distribution For 120601(120578) the effect of changing119876 is seen to be almost insignificant This is in agreementwith the physical factThe dimensionless profiles for differentvalues of Lewis number Le with constant parameters 119873119903 =01 119873119887 = 01 119873119905 = 01 119876 = 01 119870 = 10 119899 = 10119887 = 00 are shown in Figures 23 24 and 25 It is clear that forthese constant parameters variation of Lewis number has anyconsiderable effect on the velocity and temperature profilesbut concentration profile decreases with the increase of Lewisnumber What is similar to these results can be detected inFigures 26 27 and 28 for chemical reaction parameter 119870Figures 29 30 and 31 respectively present the comparisonof solutions of 1198911015840(120578) 120579(120578) and 120601(120578) for 119873119903 = 05 119873119887 = 05119873119905 = 01 119876 = 001 Le = 10 119870 = 10 119887 = 00 and differentvalues of order of chemical reaction 119899 It is observed that
Mathematical Problems in Engineering 15
0 2 4 6 80
1
120578
Nt = 00
Nt = 005
Nt = 10NumericalNumerical
Numerical
08
06
04
02
10
Q = 01Le = 10K = 10n = 10b = 00
Nb = 01
120579(120578
)
Nr = 01
Figure 18 Effect of thermophoresis 119873119905 on the dimensionlesstemperature profile
0 2 4 6 80
1
120601(120578
)
120578
Nt = 00
Nt = 005
Nt = 10NumericalNumerical
Numerical
08
06
04
02
10
Q = 01Le = 10K = 10n = 10b = 00
Nb = 01Nr = 01
Figure 19 Effect of thermophoresis 119873119905 on the dimensionlessconcentration profile
for higher order of chemical reaction velocity decreases andtemperature and concentration profiles increase extremelyUltimately Figures 32 33 and 34 depict the effect of thermalslip parameter 119887 on velocity temperature and concentrationfunctions when 119873119903 = 01 119873119887 = 01 119873119905 = 01 119876 = 01Le = 20 119870 = 15 119899 = 10 It is seen that the velocityand temperature of the fluid decrease with the increase of the
0 2 4 6 80
1
12
10
02
04
06
08
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numericalf998400 (120578
)
120578
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 20 Effect of generationabsorption heat parameter119876 on thedimensionless velocity profile
0 2 4 6 8 100
02
04
06
08
1
120579(120578
)
120578
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numerical
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 21 Effect of generationabsorption heat parameter 119876 on thedimensionless temperature profile
thermal slip parameter in the near of horizontal plate whilethe concentration of the fluid does not vary patently
Also for investigation of the parameters of physicalinterest Table 9 is presented In this table numerical valuesof reduced Nusselt number and Sherwood number obtainedby HAM for different values of the parameters 119890 119870 119899 119887 and119876 can be compared when119873119887 = 119873119903 = 119873119905 = 01
16 Mathematical Problems in Engineering
10
02
04
06
08
0 2 4 6 80
1
120578
120601(120578
)
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numerical
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 22 Effect of generationabsorption heat parameter119876 on thedimensionless concentration profile
0 2 4 6 80
1
12
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 23 Effect of Lewis number Le on the dimensionless velocityprofile
9 Conclusions
In this paper we studied the steady laminar incompressiblefree convective flow of a nanofluid past a chemically reactingupward facing horizontal plate in porous medium takinginto account heat generation and the thermal slip boundarycondition
0 2 4 6 80
1
120579(120578
)
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 24 Effect of Lewis number Le on the dimensionlesstemperature profile
0 2 4 6 80
1
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 25 Effect of Lewis number Le on the dimensionlessconcentration profile
The governing partial differential equations have beentransformed by similarity transformations into a system ofordinary differential equations which are solved by OHAMand numerical method (fourth-order Runge-Kutta schemewith the shooting method) Dimensionless velocity temper-ature and concentration functions are presented for various
Mathematical Problems in Engineering 17
0 2 4 6 80
1
12
K = 00
K = 05
K = 10NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 26 Effect of chemical reaction parameter 119870 on the dimen-sionless velocity profile
0 2 4 6 80
1
K = 00
K = 05
K = 10NumericalNumerical
Numerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 27 Effect of chemical reaction parameter 119870 on the dimen-sionless temperature profile
values of parameters for the problem for example buoy-ancy ratio 119873119903 Brownian motion 119873119887 thermophoresis 119873119905heat generationabsorption parameter 119876 Lewis number Lechemical reaction parameter119870 order of chemical reaction 119899and thermal slip parameter 119887 From the present investigationthe following may be concluded
(i) The velocity 1198911015840(120578) increases with increasing 119873119887 anddecreases with increasing119873119903 119899 and 119887
0 2 4 6 80
1
K = 00
K = 05
K = 10NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 28 Effect of chemical reaction parameter 119870 on the dimen-sionless concentration profile
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 29 Effect of order of chemical reaction 119899 on the dimension-less velocity profile
(ii) The temperature 120579(120578) increases with 119873119903 119873119887 119899 anddecreases with 119887
(iii) Parameters of119873119905 Le and119870 have no effect on velocityand temperature profiles
(iv) The concentration distribution 120601(120578) increases withincreasing N119905 119899 and decreases with 119873119887 Le and 119870but119873119903 119876 and 119887 are not affected by it
18 Mathematical Problems in Engineering
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 30 Effect of order of chemical reaction 119899 on the dimension-less temperature profile
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 31 Effect of order of chemical reaction 119899 on the dimension-less concentration profile
(v) In presence of heat source (119876 gt 0) 1198911015840(120578) 120579(120578) bothincrease while heat sink (119876 lt 0) causes that 1198911015840(120578)120579(120578) both decrease extremely These effects are morevisible for temperature distribution
In addition numerical results for the reducedNusselt andSherwood numbers are tabulated It is concluded that thereduced Nusselt number increases with the Lewis numberand reaction parameter whist it decreases with the order of
0 2 4 6 8
1
12
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 32 Effect of order of thermal slip parameter 119887 on thedimensionless velocity profile
0 2 4 6 80
1
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 33 Effect of order of thermal slip parameter 119887 on thedimensionless temperature profile
chemical reaction thermal slip and generation parametersIt is further concluded that the reduced Sherwood numberenhances with the Lewis number generation and reactionparameters whist it suppresses with thermal slip and orderof chemical reaction
The convergence of the solution series and the power ofHAM in controlling and adjusting the convergence regionand rate of solution series were discussed The proper range
Mathematical Problems in Engineering 19
0 2 4 6 80
1
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 34 Effect of order of thermal slip parameter 119887 on thedimensionless concentration profile
of auxiliary parameter ℏ to ensure the convergence of thesolution series was obtained through the so called ℏ-curveThe HAM provides us with a convenient way to control theconvergence of approximation series which is a fundamentalqualitative difference in analysis between the HAM andother methods The numerical results of the above problemsdisplay a fast convergence with minimal calculations Thisshows that the HAM is a very efficient method Finally theagreement between analytical and numerical results of thepresent study with previous published results is excellent
Nomenclature
119887 Thermal slip parameter119862 Dimensional concentration119862119891 Skin friction factor1198631 Thermal slip factor119863119861 Brownian diffusion coefficient119863119879 Thermophoretic diffusion coefficient119891 Dimensionless velocity functions119892 Gravitation acceleration119870 Chemical reaction parameter119896 Permeability of the porous media1198960 The constant reaction rate119896119898 Effective thermal conductivity of the porous
medium119896(119909) Variable reaction rateL Linear operator of the HAM119871 Length of horizontal plateLe Lewis numberN Nonlinear operator of the HAM
119899 Order of chemical reaction119873119887 Brownian motion119873119903 Buoyancy ratio119873119905 ThermophoresisNu Nusselt number119875 PressurePr Prandtl number119876 Heat generationabsorption parameter1198760 Heat generationabsorption constant119902 Internal heat generation rate
Ra Rayleigh numberSh Sherwood number119879 Temperature119905 Time119906 Velocity in 119909-direction119880119903 Reference velocity Velocity vectorV Velocity in 119910-direction119909 Distance along the surface119910 Distance normal to the surface
Greek Letters
120572119898 Thermal diffusivity120573 Volumetric expansion coefficient of nanofluid120576 Porosity of the porous media120601 Dimensionless concentration120578 Similarity variable120583 Dynamic viscosity120579 Dimensionless temperature120588119891 Density of the base fluid120588119901 Density of the nanoparticles(120588119862)119891 Effective heat capacity of the fluid(120588119862)119875 Effective heat capacity of the nanoparticle material120595 Stream function
Subscript Superscript
infin Conditions far away from the surface1015840 Differentiation with respect to 120578
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
M M Rashidi wishes to thank the Centre for DifferentialEquations Continuum Mechanics and Applications Schoolof Computational and Applied Mathematics University ofthe Witwatersrand for the hospitality during his visit EMomoniat thanks theNational Research Foundation of SouthAfrica for their support
20 Mathematical Problems in Engineering
References
[1] E Abu-Nada F O Hakan and I Pop ldquoBuoyancy inducedflow in a nanofluid filled enclosure partially exposed to forcedconvectionrdquo Superlattices and Microstructures vol 51 no 3 pp381ndash395 2012
[2] S Kakac and A Pramuanjaroenkij ldquoReview of convective heattransfer enhancement with nanofluidsrdquo International Journal ofHeat and Mass Transfer vol 52 no 13-14 pp 3187ndash3196 2009
[3] J A Eastman S U S Choi S Li W Yu and L J ThompsonldquoAnomalously increased effective thermal conductivities ofethylene glycol-based nanofluids containing copper nanoparti-clesrdquo Applied Physics Letters vol 78 no 6 pp 718ndash720 2001
[4] Y Xuan and Q Li ldquoHeat transfer enhancement of nanofluidsrdquoInternational Journal of Heat and Fluid Flow vol 21 no 1 pp58ndash64 2000
[5] S K Das N Putra P Thiesen and W Roetzel ldquoTemperaturedependence of thermal conductivity enhancement for nanoflu-idsrdquo Journal of Heat Transfer vol 125 no 4 pp 567ndash574 2003
[6] D H Kumar H E Patel V R R Kumar T SundararajanT Pradeep and S K Das ldquoModel for heat conduction innanofluidsrdquo Physical Review Letters vol 93 no 14 Article ID144301 2004
[7] H E Patel T Sundararajan T Pradeep A Dasgupta NDasgupta and S K Das ldquoA micro-convection model forthermal conductivity of nanofluidsrdquo Pramana vol 65 no 5 pp863ndash869 2005
[8] A V Kuznetsov ldquoNon-oscillatory and oscillatory nanofluidbio-thermal convection in a horizontal layer of finite depthrdquoEuropean Journal of Mechanics vol 30 no 2 pp 156ndash165 2011
[9] M Napoli J C T Eijkel and S Pennathur ldquoNanofluidictechnology for biomolecule applications a critical reviewrdquo Labon a Chip vol 10 no 8 pp 957ndash985 2010
[10] J C Crepeau and R Clarksean ldquoSimilarity solutions of naturalconvection with internal heat generationrdquo Journal of HeatTransfer vol 119 no 1 pp 183ndash185 1997
[11] A Z Sahin ldquoTransient heat conduction in semi-infinite solidwith spatially decaying exponential heat generationrdquo Interna-tional Communications in Heat andMass Transfer vol 19 no 3pp 349ndash358 1992
[12] O D Makind and A Aziz ldquoMixed convection from a con-vectively heated vertical plate to a fluid with internal heatgenerationrdquo Journal of Heat Transfer vol 133 Article ID 1225016 pages 2011
[13] J Buongiorno ldquoConvective transport in nanofluidsrdquo Journal ofHeat Transfer vol 128 no 3 pp 240ndash250 2006
[14] AV Kuznetsov andDANield ldquoNatural convective boundary-layer flow of a nanofluid past a vertical platerdquo InternationalJournal of Thermal Sciences vol 49 no 2 pp 243ndash247 2010
[15] L B Godson D Raja L D Mohan and S WongwisescldquoEnhancement of heat transfer using nanofluids-an overviewrdquoRenewable and Sustainable Energy Reviews vol 14 no 2 pp629ndash641 2010
[16] D A Nield and A BejanConvection in PorousMedia SpringerNew York NY USA 3rd edition 2006
[17] P Vadasz Emerging Topics in Heat and Mass Transfer in PorousMedia Springer New York NY USA 2008
[18] K Vafai Porous Media Applications in Biological Systems andBiotechnology CRC Press New York NY USA 2010
[19] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application to
heat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977
[20] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for natural convective boundary-layer flow in a porousmedium saturated by a nanofluidrdquo International Journal of Heatand Mass Transfer vol 52 no 25-26 pp 5792ndash5795 2009
[21] P Cheng and I D Chang ldquoBuoyancy induced flows in asaturated porous medium adjacent to impermeable horizontalsurfacesrdquo International Journal of Heat and Mass Transfer vol19 no 11 pp 1267ndash1272 1976
[22] I D Chang and P Cheng ldquoMatched asymptotic expansionsfor free convection about an impermeable horizontal surfacein a porous mediumrdquo International Journal of Heat and MassTransfer vol 26 no 2 pp 163ndash174 1983
[23] D S Shiunlin and B Gebhart ldquoBuoyancy-induced flow adja-cent to a horizontal surface submerged in porous mediumsaturated with cold waterrdquo International Journal of Heat andMass Transfer vol 29 no 4 pp 611ndash623 1986
[24] J H Merkin and G Zhang ldquoOn the similarity solutions for freeconvection in a saturated porousmedium adjacent to imperme-able horizontal surfacesrdquoWarme- und Stoffubertragung vol 25no 3 pp 179ndash184 1990
[25] M A Chaudhary J H Merkin and I Pop ldquoNatural convectionfrom a horizontal permeable surface in a porous mediummdashnumerical and asymptotic solutionsrdquo Transport in PorousMedia vol 22 no 3 pp 327ndash344 1996
[26] R S R Gorla and A Chamkha ldquoNatural convective boundarylayer flow over a nonisothermal vertical plate embedded ina porous medium saturated with a nanofluidrdquo Nanoscale andMicroscale Thermophysical Engineering vol 15 no 2 pp 81ndash942011
[27] W A Khan and I Pop ldquoFree convection boundary layer flowpast a horizontal flat plate embedded in a porous medium filledwith a nanofluidrdquo Journal of Heat Transfer vol 133 no 9 ArticleID 094501 2011
[28] A Aziz W A Khan and I Pop ldquoFree convection boundarylayer flow past a horizontal flat plate embedded in porousmedium filled by nanofluid containing gyrotactic microorgan-ismsrdquo International Journal of Thermal Sciences vol 56 pp 48ndash57 2012
[29] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer New York NYUSA 2009
[30] A A Avramenko S G Kobzar I V Shevchuk A V Kuznetsovand L T Iwanisov ldquoSymmetry of turbulent boundary-layerflows investigation of different eddy viscosity modelsrdquo ActaMechanica vol 151 no 1-2 pp 1ndash14 2001
[31] A V Kuznetsov A A Avramenko and P Geng ldquoAnalyticalinvestigation of a falling plume caused by bioconvection ofoxytactic bacteria in a fluid saturated porous mediumrdquo Inter-national Journal of Engineering Science vol 42 no 5-6 pp 557ndash569 2004
[32] M Jalil S Asghar and M Mushtaq ldquoLie group analysis ofmixed convection flow with mass transfer over a stretchingsurface with suction or injectionrdquo Mathematical Problems inEngineering vol 2010 Article ID 264901 14 pages 2010
[33] M A A Hamad M J Uddin and A I M Ismail ldquoRadiationeffects on heat and mass transfer in MHD stagnation-pointflow over a permeable flat plate with thermal convectivesurface boundary condition temperature dependent viscosityand thermal conductivityrdquoNuclear Engineering and Design vol242 pp 194ndash200 2012
Mathematical Problems in Engineering 21
[34] A Aziz M J Uddin M A A Hamad and A I MdIsmail ldquoMHD flow over an inclined radiating plate with thetemperature-dependent thermal conductivity variable reactiveindex and heat generationrdquoHeat Transfer vol 41 no 3 pp 241ndash259 2012
[35] A G Hansen Similarity Analysis of Boundary Layer Problemsin Engineering Prentice Hall Englewood Cliffs NJ USA 1964
[36] W FAmesNonlinear Partial Differential Equations in Engineer-ing Academic Press New York NY USA 1972
[37] R Seshadri and T Y Na Group Invariance in EngineeringBoundary Value Problems Springer New York NY USA 1985
[38] D Shang Theory of Heat Transfer with Forced Convection FilmFlows vol 3 of Heat and Mass Transfer 2010
[40] E Erfani M M Rashidi and A B Parsa ldquoThe modifieddifferential transform method for solving off-centered stag-nation flow toward a rotating discrdquo International Journal ofComputational Methods vol 7 no 4 pp 655ndash670 2010
[41] M M Rashidi N Laraqi and A Basiri Parsa ldquoAnalyticalmodeling of heat convection in magnetized micropolar fluid byusing modified differential transform methodrdquo Heat Transfervol 40 no 3 pp 187ndash204 2011
[42] M M Rashidi and M Keimanesh ldquoUsing differential trans-form method and pade approximant for solving mhd flow ina laminar liquid film from a horizontal stretching surfacerdquoMathematical Problems in Engineering vol 2010 Article ID491319 14 pages 2010
[43] M M Rashidi and G Domairry ldquoNew analytical solution ofthe three-dimensional NavierStokes equationsrdquoModern PhysicsLetters B vol 23 no 26 pp 3147ndash3155 2009
[44] M M Rashidi M Keimanesh O A Beg and T K HungldquoMagnetohydrodynamic biorheological transport phenomenain a porous medium a simulation of magnetic blood flowcontrol and filtrationrdquo International Journal for NumericalMethods in Biomedical Engineering vol 27 no 6 pp 805ndash8212011
[45] S J Liao The proposed homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992
[46] M M Rashidi G Domairry and S Dinarvand ldquoApproximatesolutions for the Burger and regularized long wave equationsby means of the homotopy analysis methodrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 14 no 3 pp708ndash717 2009
[47] Z Ziabakhsh and G Domairry ldquoAnalytic solution of naturalconvection flow of a non-Newtonian fluid between two verticalflat plates using homotopy analysis methodrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 14 no 5pp 1868ndash1880 2009
[48] M M Rashidi S A Mohimanian Pour T Hayat and SObaidat ldquoAnalytic approximate solutions for steady flow over arotating disk in porousmediumwith heat transfer by homotopyanalysis methodrdquo Computers and Fluids vol 54 no 1 pp 1ndash92012
[49] M Sajid T Hayat and S Asghar ldquoComparison between theHAM and HPM solutions of thin film flows of non-Newtonianfluids on a moving beltrdquo Nonlinear Dynamics vol 50 no 1-2pp 27ndash35 2007
[50] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for the double-diffusive natural convective boundary
layer flow in a porous medium saturated by a nanofluidrdquoInternational Journal of Heat and Mass Transfer vol 54 no 1-3pp 374ndash378 2011
[51] K Vajravelu and A Hadjinicolaou ldquoConvective heat transferin an electrically conducting fluid at a stretching surfacewith uniform free streamrdquo International Journal of EngineeringScience vol 35 pp 1237ndash1244 1997
[52] S Mukhopadhyay and G C Layek ldquoEffects of variable fluidviscosity on flow past a heated stretching sheet embedded ina porous medium in presence of heat sourcesinkrdquo Meccanicavol 47 no 4 pp 863ndash876 2012
[53] S J Liao ldquoAn optimal homotopy-analysis approach for stronglynonlinear differential equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 8 pp 2003ndash20162010
[54] A Aziz ldquoA similarity solution for laminar thermal boundarylayer over a flat plate with a convective surface boundary con-ditionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1064ndash1068 2009
where 119891 is an arbitrary function of 120578 From the equations211988911990931205722119909 = 1198891205790 and 211988911990931205722119909 = 1198891206010 we obtain byintegration
Here 120578 is the similarity variable and 119891 120579 120601 are dependentvariables Note that the similarity transformations in (31) areconsistent with the well-known similarity transformationsreported in the paper of Cheng and Chang [21] for 120582 = 0
in their paper Thus the dimensionless velocity components119906 V can be expressed as
119906 =
1198911015840
11990913 V = minus
1
311990923
(119891 minus 21205781198911015840) (32)
where primes indicate differentiation with respect to 120578
32 Similarity Equations On substituting the transforma-tions in (31) into the governing (16)ndash(18) we obtain thefollowing system of ordinary differential equations
where 119887 = (1198631)0119871 is the thermal slip parameter
4 Basic Idea of the HAM
Let us consider the following differential equation
N [119906 (120591)] = 0 (37)
where N is a nonlinear operator 120591 denotes independentvariable and 119906(120591) is an unknown function respectively Forsimplicity we ignore all boundary or initial conditions whichcan be treated in the similar way By means of generalizing
the traditional homotopy method Liao [45] constructs theso-called zero-order deformation equation
(1 minus 119901)L [120593 (120591 119901) minus 1199060 (120591)] = 119901ℏ119867 (120591)N [120593 (120591 119901)] (38)
where 119901 isin [0 1] is the embedding parameter ℏ = 0 is anonzero auxiliary parameter 119867(120591) = 0 is an auxiliary func-tionL is an auxiliary linear operator 1199060(120591) is an initial guessof 119906(120591) and 120593(120591 119901) is an unknown function respectively Itis important that one has great freedom to choose auxiliarythings in the HAM Obviously when 119901 = 0 and 119901 = 1 itholds 120593(120591 0) = 1199060(120591) 120593(120591 1) = 119906(120591) respectively Thus as119901 increases from 0 to 1 the solution 120593(120591 119901) varies from theinitial guess 1199060(120591) to the solution 119906(120591) Expanding 120593(120591 119901) inTaylor series with respect to 119901 we have
If the auxiliary linear operator the initial guess theauxiliary parameter ℏ and the auxiliary function are soproperly chosen the series (39) converges at 119901 = 1 then wehave
119906 (120591) = 1199060 (120591) +
+infin
sum
119898=1
119906119898 (120591) (41)
which must be one of solutions of original nonlinear equa-tion as proved by Liao [45] As ℏ = minus1 and 119867(120591) = 1 (38)becomes
(1 minus 119901)L [120593 (120591 119901) minus 1199060 (120591)] + 119901N [120593 (120591 119901)] = 0 (42)
which is used mostly in the homotopy perturbation methodwhere the solution can be obtained directly without usingTaylor series
According to definition (39) the governing equationcan be deduced from the zero-order deformation equation(37) By defining the vector 119899 = 1199060(120591) 1199061(120591) 119906119899(120591)
and differentiating equation (37) 119898 times with respect tothe embedding parameter 119901 and then setting 119901 = 0 andfinally dividing them by119898 we have the so-called119898th-orderdeformation equation
L [119906119898 (120591) minus 120594119898119906119898minus1 (120591)] = ℏ119867 (120591) 119877119898 (119898minus1) (43)
It should be emphasized that 119906119898(120591) for 119898 ge 1 isgoverned by the linear equation (39) with the linear boundaryconditions that come from original problem which canbe easily solved by symbolic computation software such asMAPLE and MATHEMATICA
Figure 3 The ℏ-curve of 1205791015840(0) given by the various HAM-orderapproximate solution
5 HAM Solution
In this section we applied the HAM to obtain approx-imate analytical solutions of the effect of higher orderchemical reaction internal heat generation and the thermalslip boundary condition on the boundary-layer flow of ananofluid past an upward facing horizontal plate (33)ndash(35)
Figure 4 The ℏ-curve of 1206011015840(0) given by the various HAM-orderapproximate solution
We start with the initial approximation 1198650(120578) = 0 Θ0(120578) =(1(119887 + 1))Exp[minus120578]Φ0(120578) = Exp[minus120578] and the linear operator
L [119865 (120578 119901)] =
1205972119865 (120578 119901)
1205971205783
L [Θ (120578 119901)] =
1205972Θ(120578 119901)
1205971205782
+
120597Θ (120578 119901)
120597120578
L [Φ (120578 119901)] =
1205972Φ(120578 119901)
1205971205782
+
120597Φ (120578 119901)
120597120578
(45)
Furthermore from (33)ndash(35) we define the nonlinearoperators
N [119865 (120578 119901)] =
1205972119865 (120578 119901)
1205971205782
minus
2
3
120578(
120597Θ (120578 119901)
120597120578
minus 119873119903
120597Φ (120578 119901)
120597120578
)
N [Θ (120578 119901)] =
1205972Θ(120578 119901)
1205971205782
+
1
3
119865 (120578 119901)
120597Θ (120578 119901)
120597120578
+ 119873119887
120597Θ (120578 119901)
120597120578
120597Φ (120578 119901)
120597120578
+ 119873119905(
120597Θ (120578 119901)
120597120578
)
2
+ 119876Θ (120578 119901)
N [Φ (120578 119901)] =
1205972Φ(120578 119901)
1205971205782
+
Le3
119865 (120578 119901)
120597Φ (120578 119901)
120597120578
+
119873119905
119873119887
1205972Θ(120578 119901)
1205971205782
minus 119870(Φ (120578 119901))119899
(46)
8 Mathematical Problems in Engineering
Using the above definition with assumption 119867119865(120591) = 1119867Θ(120591) = Exp[minus120578] and 119867Φ(120591) = Exp[minus120578] we construct the119911119890119903119900th-order deformation equation
(1 minus 119901)L [120593 (119909 119901) minus 1199060 (119909)] = 119867 (120591) 119901ℏN [120593 (119909 119901)]
Differentiating the 119911119890119903119900th-order deformation equation(47)119898 times with respect to 119901 and finally dividing by119898 wehave the119898th-order deformation equation
L [119865119898 (119909) minus 120594119898119865119898minus1 (119909)] = ℏ119877119898 (119898minus1)
L [Θ119898 (119909) minus 120594119898Θ119898minus1 (119909)] = Exp [minus120578] ℏ119877119898 (Θ119898minus1)
L [Φ119898 (119909) minus 120594119898Φ119898minus1 (119909)] = Exp [minus120578] ℏ119877119898 (Φ119898minus1)
By using the symbolic software MATHEMATICA tosolve the system of linear equations (49) with the boundaryconditions (46) it can be obtained finally
1198651 (120578) = minus
4
3 (1 + 119887)
ℏ +
4119890minus120578
3 (1 + 119887)
ℏ +
4119873119903
3 (1 + 119887)
ℏ
+
4119887119873119903
3 (1 + 119887)
ℏ minus
4119873119903119890minus120578
3 (1 + 119887)
ℏ minus
4119887119873119903119890minus120578
3 (1 + 119887)
ℏ
+
2119890minus120578120578
3 (1 + 119887)
ℏ minus
2119873119903119890minus120578120578
3 (1 + 119887)
ℏ minus
2119887119873119903119890minus120578120578
3 (1 + 119887)
ℏ
Θ1 (120578) =119890minus2120578
2 (1 + 119887)
ℏ minus
119890minus120578
2(1 + 119887)2ℏ minus
119887119890minus120578
2(1 + 119887)2ℏ
minus
119873119887119890minus3120578
6(1 + 119887)2ℏ +
119887119873119887119890minus3120578
6(1 + 119887)2ℏ minus
119873119887119890minus120578
6(1 + 119887)2ℏ
minus
119887119873119887119890minus120578
6(1 + 119887)2ℏ +
119873119905119890minus3120578
6(1 + 119887)2ℏ minus
119873119905119890minus120578
6(1 + 119887)2ℏ
+
119876119890minus2120578
2 (1 + 119887)
ℏ minus
119876119890minus120578
2(1 + 119887)2ℏ minus
119887119876119890minus120578
2(1 + 119887)2ℏ
Φ1 (120578) =119890minus2120578
2 (1 + 119887)
ℏ +
119887119890minus2120578
2 (1 + 119887)
ℏ minus
119890minus120578
2 (1 + 119887)
ℏ
minus
119887119890minus120578
2 (1 + 119887)
ℏ minus
119870119890minus2120578
2 (1 + 119887)
ℏ minus
119887119870eminus2120578
2 (1 + 119887)
ℏ
+
119870119890minus120578
2 (1 + 119887)
ℏ +
119887119870119890minus120578
2 (1 + 119887)
ℏ +
119873119905
119873119887
119890minus2120578
2 (1 + 119887)
ℏ
minus
119873119905
119873119887
119890minus120578
2 (1 + 119887)
ℏ
(56)
The higher orders solutions of 119865(120578) Θ(120578) and Φ(120578) weretoo long to be mentioned here therefore they are showngraphically
51 Convergence of the HAM Solution As mentioned by Liao[45] HAM provides us with great freedom in choosing thesolution of a nonlinear problem by different base functionsThis has a great effect on the convergence region because theconvergence region and rate of a series are chiefly determinedby the base functions used to express the solution We usedseveral terms in evaluating the approximate solution 119865app asympsum119899119894=0 119865119894Θapp asymp sum
119899119894=0Θ119894Φapp asymp sum
119899119894=0Φ119894 note that the solution
series contains the auxiliary parameter ℏ which provides uswith a simple way to adjust and control the convergence ofthe solution series Generally by means of the so-called ℏ-curve that is a curve of a versus ℏ As pointed by Liao [45]the valid region of ℏ is a horizontal line segment Figures2 3 and 4 show the ℏ-curve with the various order of theHAM for 11989110158401015840(0) 1205791015840(0) 1206011015840(0) respectively when 119873119887 = 01
Figure 6 The residual error of 120579(120578) for (34) using 12th order ofapproximations
119873119905 = 01 119873119903 = 01 119876 = 01 Le = 10 119870 = 10 119899 = 10119887 = 00 It can be seen that when the order of series is 12 thesegment of the horizontal line is more than the other ordersFor example it can be found that for 11986510158401015840(0) the acceptablerange ofℏ is betweenminus05 andminus20 for 12th order of theHAMbut for 8th order of the HAM the acceptable range of ℏ isbetween minus10 and minus20 or minus10 and minus15 for 4th order of theHAM so horizontal line segment of 12th order of the HAM ismore than others Therefore it is straightforward to choosean appropriate range for ℏ which ensures the convergence
10 Mathematical Problems in Engineering
minus00002
120578
= minus10
= minus111 (optimal value)= minus12
Nt = 01Nr = 01
Q = 01Le = 10K = 10n = 10b = 00
0006
0004
0002
0
0 2 4 6 8 10
Nb = 01
Res (120601
(120578))
ℏ
ℏ
ℏ
Figure 7 The residual error of 120601(120578) for (35) using 12th order ofapproximations
of the solution series To choose optimal value of auxiliaryparameter the averaged residual errors (see [53] for moredetails) are defined as
119864119865119898 =1
119870
119870
sum
119894=0
[N119865(119898
sum
119894=0
119865119894 (119894Δ119909))]
2
119864Θ119898 =1
119870
119870
sum
119894=0
[NΘ(119898
sum
119894=0
Θ119894 (119894Δ119909))]
2
119864Φ119898 =1
119870
119870
sum
119894=0
[NΦ(119898
sum
119894=0
Φ119894 (119894Δ119909))]
2
(57)
where Δ119909 = 10119870 and 119870 = 20 For given order ofapproximation 119898 and the optimal values ℏ are given bythe minimum of 119864119898 corresponding to nonlinear algebraicequations
119889119864119865119898
119889ℏ
= 0
119889119864Θ119898
119889ℏ
= 0
119889119864Φ119898
119889ℏ
= 0 (58)
It is noted that the optimal value of ℏ is replaced intothe equations Table 1 shows optimal values obtained for theauxiliary parameter ℏ for various quantities of119873119905 119876 and 119899when119873119887 = 01119873119903 = 01 Le = 10 119870 = 10 119887 = 00 To see
the accuracy of the solutions the residual errors are definedfor the system as (for119898 order approximation)
Res119865119898
=
1198892119865119898
1198891205782minus
2
3
120578(
119889Θ119898
119889120578
minus 119873119903
119889Φ119898
119889120578
) (59)
ResΘ119898
=
1198892Θ119898
1198891205782+
1
3
119865119898
119889Θ119898
d120578+ 119873119887
119889Θ119898
119889120578
119889Φ119898
119889120578
+ 119873119905(
119889Θ119898
119889120578
)
2
+ 119876Θ119898
(60)
ResΦ119898
=
1198892Φ119898
1198891205782+
Le3
119865119898
119889Φ119898
119889120578
+
119873119905
119873119887
1198892Θ119898
1198891205782
minus 119870(
119889Φ119898
119889120578
)
119899
(61)
Figures 5 6 and 7 show the residual errors for 12th-orderdeformation solutions when 119873119887 = 01 119873119905 = 01 119873119903 = 01119876 = 01 Le = 10 119870 = 10 119899 = 10 119887 = 00 For example ℏ =minus145 has the minimum range of residual curve in Figure 5and so on
Graphical representation of results is very useful todemonstrate the efficiency and accuracy of the HAM for theabove problem
6 Comparisons and Verification
It is worth citing that for isothermal plate (119887 = 0) and in theabsence of internal heat generationabsorption (119876 = 0) andchemical reaction (119870 = 0) our problem reduces to Gorla andChamkha [26] and Khan and Pop [27] To verify the accuracyof our results the present results are compared in Table 2with Gorla and Chamkha [26] and are found to be in goodagreement
7 Physical Quantities
Theparameters of physical interest of the present problem arethe local skin friction factor 119862119891119909 the local Nusselt numberNu119909 and the local Sherwood number Sh119909 respectivelyPhysically 119862119891119909 indicates wall shear stress Nu119909 indicates therate of wall heat transfer whilst Sh119909 indicates the rate ofwall nanoparticle volume fractionThe following relations areused to find these quantities
119862119891119909 =2120583
1205881198802119903
(
120597119906
120597119910
)
119910=0
Nu119909 =minus119909
119879119891 minus 119879infin
(
120597119879
120597119910
)
119910=0
Sh119909 =minus119909
119862119908 minus 119862infin
(
120597119862
120597119910
)
119910=0
(62)
Mathematical Problems in Engineering 11
Table 2 Comparison of present solution with Gorla and Chamkha[26] for different values of buoyancy and nanofluid parameters
Present results (HAM) Gorla and Chamkha [26]minus1205791015840(0) minus120601
1015840(0) minus120579
1015840(0) minus120601
1015840(0)
Nr 119873119887 = 03119873119905 = 01 Le = 10 119887 = 119870 = 119876 = 0
01 032578 148242 326119864 minus 01 148416402 032385 146704 325119864 minus 01 146816103 032188 145125 322119864 minus 01 145266404 031985 143503 321119864 minus 01 143639205 031777 141833 319119864 minus 01 1419499Nt 119873119887 = 03119873119903 = 05 Le = 10 119887 = 119870 = 119876 = 0
01 031777 141833 319119864 minus 01 141949902 030486 141491 305119864 minus 01 141653603 02927 141561 293119864 minus 01 141686604 028125 141991 282119864 minus 01 142158205 027046 142737 271119864 minus 01 1429226Nb 119873119905 = 01119873119903 = 05 Le = 10 119887 = 119870 = 119876 = 0
01 03672 132611 368119864 minus 01 132745402 034271 139216 343119864 minus 01 139361503 031777 141833 319119864 minus 01 141949904 029399 143428 294119864 minus 01 143546405 027161 144598 272119864 minus 01 144772
Figure 10 Comparison of dimensionless concentration profile 120601(120578)obtained by various orders of the HAM
Here Ra119909 = 119892119870120573(1minus119862infin)Δ119879119909(120572119898]) is the local Rayleighnumber Pr = ]120572119898 is the Prandtl number and 119880119903 = (1 minus
119862infin)119892119870120573Δ119879120572119898 is the reference velocity
8 Results and Discussion
Graphical representation of results is very useful to discussthe physical features presented by the solutions This sectiondescribes the influence of some interesting parameters on
12 Mathematical Problems in Engineering
0 2 4 6 8 10
02
04
06
08
1
Nr = 01Numerical
Numerical
NumericalNr = 03Nr = 05
f998400 (120578)
120578
Nt = 01Nb = 01Q = 01Le = 10K = 10n = 10b = 00
Figure 11 Effect of buoyancy ratio119873119903 on the dimensionless velocityprofile
0 2 4 6 8 10
02
04
06
08
1
Nt = 01Nb = 01Q = 01Le = 10K = 10n = 10b = 00
Nr = 01Numerical
Numerical
NumericalNr = 03Nr = 05
120578
120579(120578
)
Figure 12 Effect of buoyancy ratio 119873119903 on the dimensionlesstemperature profile
the velocity and temperature fields Equations (33) to (35)with boundary conditions in (36) were solved analytically byHAM and numerically using Runge-Kutta-Fehlberg fourth-fifth order proposed by Aziz [54]
Figures 8 9 and 10 respectively for 1198911015840(120578) 120579(120578) and 120601(120578)show the comparisons between various order approximationof the optimal HAM and the numerical solutions for the case
0 2 4 6 8 10
02
04
06
08
1
Nr = 01Numerical
Numerical
NumericalNr = 03Nr = 05
Nt = 01Nb = 01Q = 01Le = 10K = 10n = 10b = 00
120578
120601(120578
)Figure 13 Effect of buoyancy ratio 119873119903 on the dimensionlessconcentration profile
0 2 4 6 8 10
02
04
06
08
1
12
NumericalNumerical
Numerical
f998400 (120578
)
120578
Nb = 01
Nb = 03
Nb = 05
Nt = 01Nr = 01Q = 01Le = 10K = 10n = 10b = 00
Figure 14 Effect of Brownian motion 119873119887 on the dimensionlessvelocity profile
119873119887 = 01 119873119905 = 01 119873119903 = 01 119876 = 01 Le = 10 119870 = 10119899 = 10 119887 = 00 For better presentation these results arepresented in Tables 3 4 and 5 Also Tables 6 7 and 8 presentthe comparisons between various order approximation of theoptimal HAM and the numerical solutions for the case119873119887 =03 119873119905 = 005 119873119903 = 03 119876 = 01 Le = 20 119870 = 05
Mathematical Problems in Engineering 13
0 2 4 6 8 10
02
04
06
08
1
0
120578
NumericalNumerical
Numerical
Nb = 01
Nb = 03
Nb = 05
Nt = 01Nr = 01Q = 01Le = 10K = 10n = 10b = 00
120579(120578
)
Figure 15 Effect of Brownian motion 119873119887 on the dimensionlesstemperature profile
02
04
06
08
1
00
2 4 6 8 10120578
NumericalNumerical
Numerical
Nb = 01
Nb = 03
Nb = 05
Nt = 01Nr = 01Q = 01Le = 10K = 10n = 10b = 00
120601(120578
)
Figure 16 Effect of Brownian motion 119873119887 on the dimensionlessconcentration profile
119899 = 20 119887 = 01 It is observed that the results of 12th-order approximation of the optimal HAM are very close tothe numerical solutions which confirm the validity of thesemethods
In the following figures effects of various physicalparameters on the dimensionless velocity temperature and
Table 3 Comparison of values of 1198911015840(120578) obtained by various ordersof the HAM with numerical solution when 119873119887 = 01119873119905 =
temperature profiles will be investigated These results havebeen obtained by 12th order of the HAM and have beenvalidated by numerical results Figures 11ndash13 respectivelyrepresent that the comparison of solutions of 1198911015840(120578) 120579(120578)and 120601(120578) for fix values 119873119887 = 01 119873119905 = 01 119876 = 01Le = 10 119870 = 10 119899 = 10 119887 = 00 and different values ofbuoyancy ratio119873119903 In Figure 11 it is clear that velocity of thefluid tremendously decreases in the near of horizontal platewith an increase in the buoyancy ratio It is observed that thetemperature increases slightly but concentration of the fluiddoes not vary sensibly (Figures 12 and 13) The effects of theBrownian motion 119873119887 are depicted in Figures 14 15 and 16when119873119903 = 01119873119905 = 01119876 = 01 Le = 10119870 = 10 119899 = 10
14 Mathematical Problems in Engineering
Table 6 Comparison of values of 1198911015840(120578) obtained by various ordersof the HAM with numerical solution when 119873119887 = 03119873119905 =
119887 = 00 If 119873119887 increases 1198911015840(120578) and 120579(120578) increase lightly and120601(120578) decreases The dimensionless velocity dimensionlesstemperature and dimensionless concentration profiles fordifferent values of thermophoresis 119873119905 with constant values119873119903 = 01 119873119887 = 01 119876 = 01 Le = 10 119870 = 10119899 = 10 119887 = 00 are presented in Figures 17 18 and19 It is observed that the velocity and temperature of thefluid are not impressible from119873119905 but concentration increasesslightly In Figures 20 21 and 22 respectively comparison ofsolutions of 1198911015840(120578) 120579(120578) and 120601(120578) for 119873119903 = 01 119873119887 = 01119873119905 = 01 Le = 10 119870 = 10 119899 = 10 119887 = 00 anddifferent values of heat generationabsorption parameter 119876are shown It is observed that in presence of heat source1198911015840(120578) 120579(120578) both increase while heat sink causes that 1198911015840(120578)
120579(120578) both decrease extremely These effects are more visible
Table 9 Values of reduced Nusselt number and Sherwood numberobtained by HAM for different values of the parameters Le 119870 119899 119887and 119876 when119873119887 = 119873119903 = 119873119905 = 01
Figure 17 Effect of thermophoresis 119873119905 on the dimensionlessvelocity profile
for temperature distribution For 120601(120578) the effect of changing119876 is seen to be almost insignificant This is in agreementwith the physical factThe dimensionless profiles for differentvalues of Lewis number Le with constant parameters 119873119903 =01 119873119887 = 01 119873119905 = 01 119876 = 01 119870 = 10 119899 = 10119887 = 00 are shown in Figures 23 24 and 25 It is clear that forthese constant parameters variation of Lewis number has anyconsiderable effect on the velocity and temperature profilesbut concentration profile decreases with the increase of Lewisnumber What is similar to these results can be detected inFigures 26 27 and 28 for chemical reaction parameter 119870Figures 29 30 and 31 respectively present the comparisonof solutions of 1198911015840(120578) 120579(120578) and 120601(120578) for 119873119903 = 05 119873119887 = 05119873119905 = 01 119876 = 001 Le = 10 119870 = 10 119887 = 00 and differentvalues of order of chemical reaction 119899 It is observed that
Mathematical Problems in Engineering 15
0 2 4 6 80
1
120578
Nt = 00
Nt = 005
Nt = 10NumericalNumerical
Numerical
08
06
04
02
10
Q = 01Le = 10K = 10n = 10b = 00
Nb = 01
120579(120578
)
Nr = 01
Figure 18 Effect of thermophoresis 119873119905 on the dimensionlesstemperature profile
0 2 4 6 80
1
120601(120578
)
120578
Nt = 00
Nt = 005
Nt = 10NumericalNumerical
Numerical
08
06
04
02
10
Q = 01Le = 10K = 10n = 10b = 00
Nb = 01Nr = 01
Figure 19 Effect of thermophoresis 119873119905 on the dimensionlessconcentration profile
for higher order of chemical reaction velocity decreases andtemperature and concentration profiles increase extremelyUltimately Figures 32 33 and 34 depict the effect of thermalslip parameter 119887 on velocity temperature and concentrationfunctions when 119873119903 = 01 119873119887 = 01 119873119905 = 01 119876 = 01Le = 20 119870 = 15 119899 = 10 It is seen that the velocityand temperature of the fluid decrease with the increase of the
0 2 4 6 80
1
12
10
02
04
06
08
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numericalf998400 (120578
)
120578
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 20 Effect of generationabsorption heat parameter119876 on thedimensionless velocity profile
0 2 4 6 8 100
02
04
06
08
1
120579(120578
)
120578
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numerical
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 21 Effect of generationabsorption heat parameter 119876 on thedimensionless temperature profile
thermal slip parameter in the near of horizontal plate whilethe concentration of the fluid does not vary patently
Also for investigation of the parameters of physicalinterest Table 9 is presented In this table numerical valuesof reduced Nusselt number and Sherwood number obtainedby HAM for different values of the parameters 119890 119870 119899 119887 and119876 can be compared when119873119887 = 119873119903 = 119873119905 = 01
16 Mathematical Problems in Engineering
10
02
04
06
08
0 2 4 6 80
1
120578
120601(120578
)
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numerical
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 22 Effect of generationabsorption heat parameter119876 on thedimensionless concentration profile
0 2 4 6 80
1
12
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 23 Effect of Lewis number Le on the dimensionless velocityprofile
9 Conclusions
In this paper we studied the steady laminar incompressiblefree convective flow of a nanofluid past a chemically reactingupward facing horizontal plate in porous medium takinginto account heat generation and the thermal slip boundarycondition
0 2 4 6 80
1
120579(120578
)
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 24 Effect of Lewis number Le on the dimensionlesstemperature profile
0 2 4 6 80
1
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 25 Effect of Lewis number Le on the dimensionlessconcentration profile
The governing partial differential equations have beentransformed by similarity transformations into a system ofordinary differential equations which are solved by OHAMand numerical method (fourth-order Runge-Kutta schemewith the shooting method) Dimensionless velocity temper-ature and concentration functions are presented for various
Mathematical Problems in Engineering 17
0 2 4 6 80
1
12
K = 00
K = 05
K = 10NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 26 Effect of chemical reaction parameter 119870 on the dimen-sionless velocity profile
0 2 4 6 80
1
K = 00
K = 05
K = 10NumericalNumerical
Numerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 27 Effect of chemical reaction parameter 119870 on the dimen-sionless temperature profile
values of parameters for the problem for example buoy-ancy ratio 119873119903 Brownian motion 119873119887 thermophoresis 119873119905heat generationabsorption parameter 119876 Lewis number Lechemical reaction parameter119870 order of chemical reaction 119899and thermal slip parameter 119887 From the present investigationthe following may be concluded
(i) The velocity 1198911015840(120578) increases with increasing 119873119887 anddecreases with increasing119873119903 119899 and 119887
0 2 4 6 80
1
K = 00
K = 05
K = 10NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 28 Effect of chemical reaction parameter 119870 on the dimen-sionless concentration profile
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 29 Effect of order of chemical reaction 119899 on the dimension-less velocity profile
(ii) The temperature 120579(120578) increases with 119873119903 119873119887 119899 anddecreases with 119887
(iii) Parameters of119873119905 Le and119870 have no effect on velocityand temperature profiles
(iv) The concentration distribution 120601(120578) increases withincreasing N119905 119899 and decreases with 119873119887 Le and 119870but119873119903 119876 and 119887 are not affected by it
18 Mathematical Problems in Engineering
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 30 Effect of order of chemical reaction 119899 on the dimension-less temperature profile
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 31 Effect of order of chemical reaction 119899 on the dimension-less concentration profile
(v) In presence of heat source (119876 gt 0) 1198911015840(120578) 120579(120578) bothincrease while heat sink (119876 lt 0) causes that 1198911015840(120578)120579(120578) both decrease extremely These effects are morevisible for temperature distribution
In addition numerical results for the reducedNusselt andSherwood numbers are tabulated It is concluded that thereduced Nusselt number increases with the Lewis numberand reaction parameter whist it decreases with the order of
0 2 4 6 8
1
12
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 32 Effect of order of thermal slip parameter 119887 on thedimensionless velocity profile
0 2 4 6 80
1
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 33 Effect of order of thermal slip parameter 119887 on thedimensionless temperature profile
chemical reaction thermal slip and generation parametersIt is further concluded that the reduced Sherwood numberenhances with the Lewis number generation and reactionparameters whist it suppresses with thermal slip and orderof chemical reaction
The convergence of the solution series and the power ofHAM in controlling and adjusting the convergence regionand rate of solution series were discussed The proper range
Mathematical Problems in Engineering 19
0 2 4 6 80
1
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 34 Effect of order of thermal slip parameter 119887 on thedimensionless concentration profile
of auxiliary parameter ℏ to ensure the convergence of thesolution series was obtained through the so called ℏ-curveThe HAM provides us with a convenient way to control theconvergence of approximation series which is a fundamentalqualitative difference in analysis between the HAM andother methods The numerical results of the above problemsdisplay a fast convergence with minimal calculations Thisshows that the HAM is a very efficient method Finally theagreement between analytical and numerical results of thepresent study with previous published results is excellent
Nomenclature
119887 Thermal slip parameter119862 Dimensional concentration119862119891 Skin friction factor1198631 Thermal slip factor119863119861 Brownian diffusion coefficient119863119879 Thermophoretic diffusion coefficient119891 Dimensionless velocity functions119892 Gravitation acceleration119870 Chemical reaction parameter119896 Permeability of the porous media1198960 The constant reaction rate119896119898 Effective thermal conductivity of the porous
medium119896(119909) Variable reaction rateL Linear operator of the HAM119871 Length of horizontal plateLe Lewis numberN Nonlinear operator of the HAM
119899 Order of chemical reaction119873119887 Brownian motion119873119903 Buoyancy ratio119873119905 ThermophoresisNu Nusselt number119875 PressurePr Prandtl number119876 Heat generationabsorption parameter1198760 Heat generationabsorption constant119902 Internal heat generation rate
Ra Rayleigh numberSh Sherwood number119879 Temperature119905 Time119906 Velocity in 119909-direction119880119903 Reference velocity Velocity vectorV Velocity in 119910-direction119909 Distance along the surface119910 Distance normal to the surface
Greek Letters
120572119898 Thermal diffusivity120573 Volumetric expansion coefficient of nanofluid120576 Porosity of the porous media120601 Dimensionless concentration120578 Similarity variable120583 Dynamic viscosity120579 Dimensionless temperature120588119891 Density of the base fluid120588119901 Density of the nanoparticles(120588119862)119891 Effective heat capacity of the fluid(120588119862)119875 Effective heat capacity of the nanoparticle material120595 Stream function
Subscript Superscript
infin Conditions far away from the surface1015840 Differentiation with respect to 120578
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
M M Rashidi wishes to thank the Centre for DifferentialEquations Continuum Mechanics and Applications Schoolof Computational and Applied Mathematics University ofthe Witwatersrand for the hospitality during his visit EMomoniat thanks theNational Research Foundation of SouthAfrica for their support
20 Mathematical Problems in Engineering
References
[1] E Abu-Nada F O Hakan and I Pop ldquoBuoyancy inducedflow in a nanofluid filled enclosure partially exposed to forcedconvectionrdquo Superlattices and Microstructures vol 51 no 3 pp381ndash395 2012
[2] S Kakac and A Pramuanjaroenkij ldquoReview of convective heattransfer enhancement with nanofluidsrdquo International Journal ofHeat and Mass Transfer vol 52 no 13-14 pp 3187ndash3196 2009
[3] J A Eastman S U S Choi S Li W Yu and L J ThompsonldquoAnomalously increased effective thermal conductivities ofethylene glycol-based nanofluids containing copper nanoparti-clesrdquo Applied Physics Letters vol 78 no 6 pp 718ndash720 2001
[4] Y Xuan and Q Li ldquoHeat transfer enhancement of nanofluidsrdquoInternational Journal of Heat and Fluid Flow vol 21 no 1 pp58ndash64 2000
[5] S K Das N Putra P Thiesen and W Roetzel ldquoTemperaturedependence of thermal conductivity enhancement for nanoflu-idsrdquo Journal of Heat Transfer vol 125 no 4 pp 567ndash574 2003
[6] D H Kumar H E Patel V R R Kumar T SundararajanT Pradeep and S K Das ldquoModel for heat conduction innanofluidsrdquo Physical Review Letters vol 93 no 14 Article ID144301 2004
[7] H E Patel T Sundararajan T Pradeep A Dasgupta NDasgupta and S K Das ldquoA micro-convection model forthermal conductivity of nanofluidsrdquo Pramana vol 65 no 5 pp863ndash869 2005
[8] A V Kuznetsov ldquoNon-oscillatory and oscillatory nanofluidbio-thermal convection in a horizontal layer of finite depthrdquoEuropean Journal of Mechanics vol 30 no 2 pp 156ndash165 2011
[9] M Napoli J C T Eijkel and S Pennathur ldquoNanofluidictechnology for biomolecule applications a critical reviewrdquo Labon a Chip vol 10 no 8 pp 957ndash985 2010
[10] J C Crepeau and R Clarksean ldquoSimilarity solutions of naturalconvection with internal heat generationrdquo Journal of HeatTransfer vol 119 no 1 pp 183ndash185 1997
[11] A Z Sahin ldquoTransient heat conduction in semi-infinite solidwith spatially decaying exponential heat generationrdquo Interna-tional Communications in Heat andMass Transfer vol 19 no 3pp 349ndash358 1992
[12] O D Makind and A Aziz ldquoMixed convection from a con-vectively heated vertical plate to a fluid with internal heatgenerationrdquo Journal of Heat Transfer vol 133 Article ID 1225016 pages 2011
[13] J Buongiorno ldquoConvective transport in nanofluidsrdquo Journal ofHeat Transfer vol 128 no 3 pp 240ndash250 2006
[14] AV Kuznetsov andDANield ldquoNatural convective boundary-layer flow of a nanofluid past a vertical platerdquo InternationalJournal of Thermal Sciences vol 49 no 2 pp 243ndash247 2010
[15] L B Godson D Raja L D Mohan and S WongwisescldquoEnhancement of heat transfer using nanofluids-an overviewrdquoRenewable and Sustainable Energy Reviews vol 14 no 2 pp629ndash641 2010
[16] D A Nield and A BejanConvection in PorousMedia SpringerNew York NY USA 3rd edition 2006
[17] P Vadasz Emerging Topics in Heat and Mass Transfer in PorousMedia Springer New York NY USA 2008
[18] K Vafai Porous Media Applications in Biological Systems andBiotechnology CRC Press New York NY USA 2010
[19] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application to
heat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977
[20] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for natural convective boundary-layer flow in a porousmedium saturated by a nanofluidrdquo International Journal of Heatand Mass Transfer vol 52 no 25-26 pp 5792ndash5795 2009
[21] P Cheng and I D Chang ldquoBuoyancy induced flows in asaturated porous medium adjacent to impermeable horizontalsurfacesrdquo International Journal of Heat and Mass Transfer vol19 no 11 pp 1267ndash1272 1976
[22] I D Chang and P Cheng ldquoMatched asymptotic expansionsfor free convection about an impermeable horizontal surfacein a porous mediumrdquo International Journal of Heat and MassTransfer vol 26 no 2 pp 163ndash174 1983
[23] D S Shiunlin and B Gebhart ldquoBuoyancy-induced flow adja-cent to a horizontal surface submerged in porous mediumsaturated with cold waterrdquo International Journal of Heat andMass Transfer vol 29 no 4 pp 611ndash623 1986
[24] J H Merkin and G Zhang ldquoOn the similarity solutions for freeconvection in a saturated porousmedium adjacent to imperme-able horizontal surfacesrdquoWarme- und Stoffubertragung vol 25no 3 pp 179ndash184 1990
[25] M A Chaudhary J H Merkin and I Pop ldquoNatural convectionfrom a horizontal permeable surface in a porous mediummdashnumerical and asymptotic solutionsrdquo Transport in PorousMedia vol 22 no 3 pp 327ndash344 1996
[26] R S R Gorla and A Chamkha ldquoNatural convective boundarylayer flow over a nonisothermal vertical plate embedded ina porous medium saturated with a nanofluidrdquo Nanoscale andMicroscale Thermophysical Engineering vol 15 no 2 pp 81ndash942011
[27] W A Khan and I Pop ldquoFree convection boundary layer flowpast a horizontal flat plate embedded in a porous medium filledwith a nanofluidrdquo Journal of Heat Transfer vol 133 no 9 ArticleID 094501 2011
[28] A Aziz W A Khan and I Pop ldquoFree convection boundarylayer flow past a horizontal flat plate embedded in porousmedium filled by nanofluid containing gyrotactic microorgan-ismsrdquo International Journal of Thermal Sciences vol 56 pp 48ndash57 2012
[29] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer New York NYUSA 2009
[30] A A Avramenko S G Kobzar I V Shevchuk A V Kuznetsovand L T Iwanisov ldquoSymmetry of turbulent boundary-layerflows investigation of different eddy viscosity modelsrdquo ActaMechanica vol 151 no 1-2 pp 1ndash14 2001
[31] A V Kuznetsov A A Avramenko and P Geng ldquoAnalyticalinvestigation of a falling plume caused by bioconvection ofoxytactic bacteria in a fluid saturated porous mediumrdquo Inter-national Journal of Engineering Science vol 42 no 5-6 pp 557ndash569 2004
[32] M Jalil S Asghar and M Mushtaq ldquoLie group analysis ofmixed convection flow with mass transfer over a stretchingsurface with suction or injectionrdquo Mathematical Problems inEngineering vol 2010 Article ID 264901 14 pages 2010
[33] M A A Hamad M J Uddin and A I M Ismail ldquoRadiationeffects on heat and mass transfer in MHD stagnation-pointflow over a permeable flat plate with thermal convectivesurface boundary condition temperature dependent viscosityand thermal conductivityrdquoNuclear Engineering and Design vol242 pp 194ndash200 2012
Mathematical Problems in Engineering 21
[34] A Aziz M J Uddin M A A Hamad and A I MdIsmail ldquoMHD flow over an inclined radiating plate with thetemperature-dependent thermal conductivity variable reactiveindex and heat generationrdquoHeat Transfer vol 41 no 3 pp 241ndash259 2012
[35] A G Hansen Similarity Analysis of Boundary Layer Problemsin Engineering Prentice Hall Englewood Cliffs NJ USA 1964
[36] W FAmesNonlinear Partial Differential Equations in Engineer-ing Academic Press New York NY USA 1972
[37] R Seshadri and T Y Na Group Invariance in EngineeringBoundary Value Problems Springer New York NY USA 1985
[38] D Shang Theory of Heat Transfer with Forced Convection FilmFlows vol 3 of Heat and Mass Transfer 2010
[40] E Erfani M M Rashidi and A B Parsa ldquoThe modifieddifferential transform method for solving off-centered stag-nation flow toward a rotating discrdquo International Journal ofComputational Methods vol 7 no 4 pp 655ndash670 2010
[41] M M Rashidi N Laraqi and A Basiri Parsa ldquoAnalyticalmodeling of heat convection in magnetized micropolar fluid byusing modified differential transform methodrdquo Heat Transfervol 40 no 3 pp 187ndash204 2011
[42] M M Rashidi and M Keimanesh ldquoUsing differential trans-form method and pade approximant for solving mhd flow ina laminar liquid film from a horizontal stretching surfacerdquoMathematical Problems in Engineering vol 2010 Article ID491319 14 pages 2010
[43] M M Rashidi and G Domairry ldquoNew analytical solution ofthe three-dimensional NavierStokes equationsrdquoModern PhysicsLetters B vol 23 no 26 pp 3147ndash3155 2009
[44] M M Rashidi M Keimanesh O A Beg and T K HungldquoMagnetohydrodynamic biorheological transport phenomenain a porous medium a simulation of magnetic blood flowcontrol and filtrationrdquo International Journal for NumericalMethods in Biomedical Engineering vol 27 no 6 pp 805ndash8212011
[45] S J Liao The proposed homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992
[46] M M Rashidi G Domairry and S Dinarvand ldquoApproximatesolutions for the Burger and regularized long wave equationsby means of the homotopy analysis methodrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 14 no 3 pp708ndash717 2009
[47] Z Ziabakhsh and G Domairry ldquoAnalytic solution of naturalconvection flow of a non-Newtonian fluid between two verticalflat plates using homotopy analysis methodrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 14 no 5pp 1868ndash1880 2009
[48] M M Rashidi S A Mohimanian Pour T Hayat and SObaidat ldquoAnalytic approximate solutions for steady flow over arotating disk in porousmediumwith heat transfer by homotopyanalysis methodrdquo Computers and Fluids vol 54 no 1 pp 1ndash92012
[49] M Sajid T Hayat and S Asghar ldquoComparison between theHAM and HPM solutions of thin film flows of non-Newtonianfluids on a moving beltrdquo Nonlinear Dynamics vol 50 no 1-2pp 27ndash35 2007
[50] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for the double-diffusive natural convective boundary
layer flow in a porous medium saturated by a nanofluidrdquoInternational Journal of Heat and Mass Transfer vol 54 no 1-3pp 374ndash378 2011
[51] K Vajravelu and A Hadjinicolaou ldquoConvective heat transferin an electrically conducting fluid at a stretching surfacewith uniform free streamrdquo International Journal of EngineeringScience vol 35 pp 1237ndash1244 1997
[52] S Mukhopadhyay and G C Layek ldquoEffects of variable fluidviscosity on flow past a heated stretching sheet embedded ina porous medium in presence of heat sourcesinkrdquo Meccanicavol 47 no 4 pp 863ndash876 2012
[53] S J Liao ldquoAn optimal homotopy-analysis approach for stronglynonlinear differential equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 8 pp 2003ndash20162010
[54] A Aziz ldquoA similarity solution for laminar thermal boundarylayer over a flat plate with a convective surface boundary con-ditionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1064ndash1068 2009
Figure 3 The ℏ-curve of 1205791015840(0) given by the various HAM-orderapproximate solution
5 HAM Solution
In this section we applied the HAM to obtain approx-imate analytical solutions of the effect of higher orderchemical reaction internal heat generation and the thermalslip boundary condition on the boundary-layer flow of ananofluid past an upward facing horizontal plate (33)ndash(35)
Figure 4 The ℏ-curve of 1206011015840(0) given by the various HAM-orderapproximate solution
We start with the initial approximation 1198650(120578) = 0 Θ0(120578) =(1(119887 + 1))Exp[minus120578]Φ0(120578) = Exp[minus120578] and the linear operator
L [119865 (120578 119901)] =
1205972119865 (120578 119901)
1205971205783
L [Θ (120578 119901)] =
1205972Θ(120578 119901)
1205971205782
+
120597Θ (120578 119901)
120597120578
L [Φ (120578 119901)] =
1205972Φ(120578 119901)
1205971205782
+
120597Φ (120578 119901)
120597120578
(45)
Furthermore from (33)ndash(35) we define the nonlinearoperators
N [119865 (120578 119901)] =
1205972119865 (120578 119901)
1205971205782
minus
2
3
120578(
120597Θ (120578 119901)
120597120578
minus 119873119903
120597Φ (120578 119901)
120597120578
)
N [Θ (120578 119901)] =
1205972Θ(120578 119901)
1205971205782
+
1
3
119865 (120578 119901)
120597Θ (120578 119901)
120597120578
+ 119873119887
120597Θ (120578 119901)
120597120578
120597Φ (120578 119901)
120597120578
+ 119873119905(
120597Θ (120578 119901)
120597120578
)
2
+ 119876Θ (120578 119901)
N [Φ (120578 119901)] =
1205972Φ(120578 119901)
1205971205782
+
Le3
119865 (120578 119901)
120597Φ (120578 119901)
120597120578
+
119873119905
119873119887
1205972Θ(120578 119901)
1205971205782
minus 119870(Φ (120578 119901))119899
(46)
8 Mathematical Problems in Engineering
Using the above definition with assumption 119867119865(120591) = 1119867Θ(120591) = Exp[minus120578] and 119867Φ(120591) = Exp[minus120578] we construct the119911119890119903119900th-order deformation equation
(1 minus 119901)L [120593 (119909 119901) minus 1199060 (119909)] = 119867 (120591) 119901ℏN [120593 (119909 119901)]
Differentiating the 119911119890119903119900th-order deformation equation(47)119898 times with respect to 119901 and finally dividing by119898 wehave the119898th-order deformation equation
L [119865119898 (119909) minus 120594119898119865119898minus1 (119909)] = ℏ119877119898 (119898minus1)
L [Θ119898 (119909) minus 120594119898Θ119898minus1 (119909)] = Exp [minus120578] ℏ119877119898 (Θ119898minus1)
L [Φ119898 (119909) minus 120594119898Φ119898minus1 (119909)] = Exp [minus120578] ℏ119877119898 (Φ119898minus1)
By using the symbolic software MATHEMATICA tosolve the system of linear equations (49) with the boundaryconditions (46) it can be obtained finally
1198651 (120578) = minus
4
3 (1 + 119887)
ℏ +
4119890minus120578
3 (1 + 119887)
ℏ +
4119873119903
3 (1 + 119887)
ℏ
+
4119887119873119903
3 (1 + 119887)
ℏ minus
4119873119903119890minus120578
3 (1 + 119887)
ℏ minus
4119887119873119903119890minus120578
3 (1 + 119887)
ℏ
+
2119890minus120578120578
3 (1 + 119887)
ℏ minus
2119873119903119890minus120578120578
3 (1 + 119887)
ℏ minus
2119887119873119903119890minus120578120578
3 (1 + 119887)
ℏ
Θ1 (120578) =119890minus2120578
2 (1 + 119887)
ℏ minus
119890minus120578
2(1 + 119887)2ℏ minus
119887119890minus120578
2(1 + 119887)2ℏ
minus
119873119887119890minus3120578
6(1 + 119887)2ℏ +
119887119873119887119890minus3120578
6(1 + 119887)2ℏ minus
119873119887119890minus120578
6(1 + 119887)2ℏ
minus
119887119873119887119890minus120578
6(1 + 119887)2ℏ +
119873119905119890minus3120578
6(1 + 119887)2ℏ minus
119873119905119890minus120578
6(1 + 119887)2ℏ
+
119876119890minus2120578
2 (1 + 119887)
ℏ minus
119876119890minus120578
2(1 + 119887)2ℏ minus
119887119876119890minus120578
2(1 + 119887)2ℏ
Φ1 (120578) =119890minus2120578
2 (1 + 119887)
ℏ +
119887119890minus2120578
2 (1 + 119887)
ℏ minus
119890minus120578
2 (1 + 119887)
ℏ
minus
119887119890minus120578
2 (1 + 119887)
ℏ minus
119870119890minus2120578
2 (1 + 119887)
ℏ minus
119887119870eminus2120578
2 (1 + 119887)
ℏ
+
119870119890minus120578
2 (1 + 119887)
ℏ +
119887119870119890minus120578
2 (1 + 119887)
ℏ +
119873119905
119873119887
119890minus2120578
2 (1 + 119887)
ℏ
minus
119873119905
119873119887
119890minus120578
2 (1 + 119887)
ℏ
(56)
The higher orders solutions of 119865(120578) Θ(120578) and Φ(120578) weretoo long to be mentioned here therefore they are showngraphically
51 Convergence of the HAM Solution As mentioned by Liao[45] HAM provides us with great freedom in choosing thesolution of a nonlinear problem by different base functionsThis has a great effect on the convergence region because theconvergence region and rate of a series are chiefly determinedby the base functions used to express the solution We usedseveral terms in evaluating the approximate solution 119865app asympsum119899119894=0 119865119894Θapp asymp sum
119899119894=0Θ119894Φapp asymp sum
119899119894=0Φ119894 note that the solution
series contains the auxiliary parameter ℏ which provides uswith a simple way to adjust and control the convergence ofthe solution series Generally by means of the so-called ℏ-curve that is a curve of a versus ℏ As pointed by Liao [45]the valid region of ℏ is a horizontal line segment Figures2 3 and 4 show the ℏ-curve with the various order of theHAM for 11989110158401015840(0) 1205791015840(0) 1206011015840(0) respectively when 119873119887 = 01
Figure 6 The residual error of 120579(120578) for (34) using 12th order ofapproximations
119873119905 = 01 119873119903 = 01 119876 = 01 Le = 10 119870 = 10 119899 = 10119887 = 00 It can be seen that when the order of series is 12 thesegment of the horizontal line is more than the other ordersFor example it can be found that for 11986510158401015840(0) the acceptablerange ofℏ is betweenminus05 andminus20 for 12th order of theHAMbut for 8th order of the HAM the acceptable range of ℏ isbetween minus10 and minus20 or minus10 and minus15 for 4th order of theHAM so horizontal line segment of 12th order of the HAM ismore than others Therefore it is straightforward to choosean appropriate range for ℏ which ensures the convergence
10 Mathematical Problems in Engineering
minus00002
120578
= minus10
= minus111 (optimal value)= minus12
Nt = 01Nr = 01
Q = 01Le = 10K = 10n = 10b = 00
0006
0004
0002
0
0 2 4 6 8 10
Nb = 01
Res (120601
(120578))
ℏ
ℏ
ℏ
Figure 7 The residual error of 120601(120578) for (35) using 12th order ofapproximations
of the solution series To choose optimal value of auxiliaryparameter the averaged residual errors (see [53] for moredetails) are defined as
119864119865119898 =1
119870
119870
sum
119894=0
[N119865(119898
sum
119894=0
119865119894 (119894Δ119909))]
2
119864Θ119898 =1
119870
119870
sum
119894=0
[NΘ(119898
sum
119894=0
Θ119894 (119894Δ119909))]
2
119864Φ119898 =1
119870
119870
sum
119894=0
[NΦ(119898
sum
119894=0
Φ119894 (119894Δ119909))]
2
(57)
where Δ119909 = 10119870 and 119870 = 20 For given order ofapproximation 119898 and the optimal values ℏ are given bythe minimum of 119864119898 corresponding to nonlinear algebraicequations
119889119864119865119898
119889ℏ
= 0
119889119864Θ119898
119889ℏ
= 0
119889119864Φ119898
119889ℏ
= 0 (58)
It is noted that the optimal value of ℏ is replaced intothe equations Table 1 shows optimal values obtained for theauxiliary parameter ℏ for various quantities of119873119905 119876 and 119899when119873119887 = 01119873119903 = 01 Le = 10 119870 = 10 119887 = 00 To see
the accuracy of the solutions the residual errors are definedfor the system as (for119898 order approximation)
Res119865119898
=
1198892119865119898
1198891205782minus
2
3
120578(
119889Θ119898
119889120578
minus 119873119903
119889Φ119898
119889120578
) (59)
ResΘ119898
=
1198892Θ119898
1198891205782+
1
3
119865119898
119889Θ119898
d120578+ 119873119887
119889Θ119898
119889120578
119889Φ119898
119889120578
+ 119873119905(
119889Θ119898
119889120578
)
2
+ 119876Θ119898
(60)
ResΦ119898
=
1198892Φ119898
1198891205782+
Le3
119865119898
119889Φ119898
119889120578
+
119873119905
119873119887
1198892Θ119898
1198891205782
minus 119870(
119889Φ119898
119889120578
)
119899
(61)
Figures 5 6 and 7 show the residual errors for 12th-orderdeformation solutions when 119873119887 = 01 119873119905 = 01 119873119903 = 01119876 = 01 Le = 10 119870 = 10 119899 = 10 119887 = 00 For example ℏ =minus145 has the minimum range of residual curve in Figure 5and so on
Graphical representation of results is very useful todemonstrate the efficiency and accuracy of the HAM for theabove problem
6 Comparisons and Verification
It is worth citing that for isothermal plate (119887 = 0) and in theabsence of internal heat generationabsorption (119876 = 0) andchemical reaction (119870 = 0) our problem reduces to Gorla andChamkha [26] and Khan and Pop [27] To verify the accuracyof our results the present results are compared in Table 2with Gorla and Chamkha [26] and are found to be in goodagreement
7 Physical Quantities
Theparameters of physical interest of the present problem arethe local skin friction factor 119862119891119909 the local Nusselt numberNu119909 and the local Sherwood number Sh119909 respectivelyPhysically 119862119891119909 indicates wall shear stress Nu119909 indicates therate of wall heat transfer whilst Sh119909 indicates the rate ofwall nanoparticle volume fractionThe following relations areused to find these quantities
119862119891119909 =2120583
1205881198802119903
(
120597119906
120597119910
)
119910=0
Nu119909 =minus119909
119879119891 minus 119879infin
(
120597119879
120597119910
)
119910=0
Sh119909 =minus119909
119862119908 minus 119862infin
(
120597119862
120597119910
)
119910=0
(62)
Mathematical Problems in Engineering 11
Table 2 Comparison of present solution with Gorla and Chamkha[26] for different values of buoyancy and nanofluid parameters
Present results (HAM) Gorla and Chamkha [26]minus1205791015840(0) minus120601
1015840(0) minus120579
1015840(0) minus120601
1015840(0)
Nr 119873119887 = 03119873119905 = 01 Le = 10 119887 = 119870 = 119876 = 0
01 032578 148242 326119864 minus 01 148416402 032385 146704 325119864 minus 01 146816103 032188 145125 322119864 minus 01 145266404 031985 143503 321119864 minus 01 143639205 031777 141833 319119864 minus 01 1419499Nt 119873119887 = 03119873119903 = 05 Le = 10 119887 = 119870 = 119876 = 0
01 031777 141833 319119864 minus 01 141949902 030486 141491 305119864 minus 01 141653603 02927 141561 293119864 minus 01 141686604 028125 141991 282119864 minus 01 142158205 027046 142737 271119864 minus 01 1429226Nb 119873119905 = 01119873119903 = 05 Le = 10 119887 = 119870 = 119876 = 0
01 03672 132611 368119864 minus 01 132745402 034271 139216 343119864 minus 01 139361503 031777 141833 319119864 minus 01 141949904 029399 143428 294119864 minus 01 143546405 027161 144598 272119864 minus 01 144772
Figure 10 Comparison of dimensionless concentration profile 120601(120578)obtained by various orders of the HAM
Here Ra119909 = 119892119870120573(1minus119862infin)Δ119879119909(120572119898]) is the local Rayleighnumber Pr = ]120572119898 is the Prandtl number and 119880119903 = (1 minus
119862infin)119892119870120573Δ119879120572119898 is the reference velocity
8 Results and Discussion
Graphical representation of results is very useful to discussthe physical features presented by the solutions This sectiondescribes the influence of some interesting parameters on
12 Mathematical Problems in Engineering
0 2 4 6 8 10
02
04
06
08
1
Nr = 01Numerical
Numerical
NumericalNr = 03Nr = 05
f998400 (120578)
120578
Nt = 01Nb = 01Q = 01Le = 10K = 10n = 10b = 00
Figure 11 Effect of buoyancy ratio119873119903 on the dimensionless velocityprofile
0 2 4 6 8 10
02
04
06
08
1
Nt = 01Nb = 01Q = 01Le = 10K = 10n = 10b = 00
Nr = 01Numerical
Numerical
NumericalNr = 03Nr = 05
120578
120579(120578
)
Figure 12 Effect of buoyancy ratio 119873119903 on the dimensionlesstemperature profile
the velocity and temperature fields Equations (33) to (35)with boundary conditions in (36) were solved analytically byHAM and numerically using Runge-Kutta-Fehlberg fourth-fifth order proposed by Aziz [54]
Figures 8 9 and 10 respectively for 1198911015840(120578) 120579(120578) and 120601(120578)show the comparisons between various order approximationof the optimal HAM and the numerical solutions for the case
0 2 4 6 8 10
02
04
06
08
1
Nr = 01Numerical
Numerical
NumericalNr = 03Nr = 05
Nt = 01Nb = 01Q = 01Le = 10K = 10n = 10b = 00
120578
120601(120578
)Figure 13 Effect of buoyancy ratio 119873119903 on the dimensionlessconcentration profile
0 2 4 6 8 10
02
04
06
08
1
12
NumericalNumerical
Numerical
f998400 (120578
)
120578
Nb = 01
Nb = 03
Nb = 05
Nt = 01Nr = 01Q = 01Le = 10K = 10n = 10b = 00
Figure 14 Effect of Brownian motion 119873119887 on the dimensionlessvelocity profile
119873119887 = 01 119873119905 = 01 119873119903 = 01 119876 = 01 Le = 10 119870 = 10119899 = 10 119887 = 00 For better presentation these results arepresented in Tables 3 4 and 5 Also Tables 6 7 and 8 presentthe comparisons between various order approximation of theoptimal HAM and the numerical solutions for the case119873119887 =03 119873119905 = 005 119873119903 = 03 119876 = 01 Le = 20 119870 = 05
Mathematical Problems in Engineering 13
0 2 4 6 8 10
02
04
06
08
1
0
120578
NumericalNumerical
Numerical
Nb = 01
Nb = 03
Nb = 05
Nt = 01Nr = 01Q = 01Le = 10K = 10n = 10b = 00
120579(120578
)
Figure 15 Effect of Brownian motion 119873119887 on the dimensionlesstemperature profile
02
04
06
08
1
00
2 4 6 8 10120578
NumericalNumerical
Numerical
Nb = 01
Nb = 03
Nb = 05
Nt = 01Nr = 01Q = 01Le = 10K = 10n = 10b = 00
120601(120578
)
Figure 16 Effect of Brownian motion 119873119887 on the dimensionlessconcentration profile
119899 = 20 119887 = 01 It is observed that the results of 12th-order approximation of the optimal HAM are very close tothe numerical solutions which confirm the validity of thesemethods
In the following figures effects of various physicalparameters on the dimensionless velocity temperature and
Table 3 Comparison of values of 1198911015840(120578) obtained by various ordersof the HAM with numerical solution when 119873119887 = 01119873119905 =
temperature profiles will be investigated These results havebeen obtained by 12th order of the HAM and have beenvalidated by numerical results Figures 11ndash13 respectivelyrepresent that the comparison of solutions of 1198911015840(120578) 120579(120578)and 120601(120578) for fix values 119873119887 = 01 119873119905 = 01 119876 = 01Le = 10 119870 = 10 119899 = 10 119887 = 00 and different values ofbuoyancy ratio119873119903 In Figure 11 it is clear that velocity of thefluid tremendously decreases in the near of horizontal platewith an increase in the buoyancy ratio It is observed that thetemperature increases slightly but concentration of the fluiddoes not vary sensibly (Figures 12 and 13) The effects of theBrownian motion 119873119887 are depicted in Figures 14 15 and 16when119873119903 = 01119873119905 = 01119876 = 01 Le = 10119870 = 10 119899 = 10
14 Mathematical Problems in Engineering
Table 6 Comparison of values of 1198911015840(120578) obtained by various ordersof the HAM with numerical solution when 119873119887 = 03119873119905 =
119887 = 00 If 119873119887 increases 1198911015840(120578) and 120579(120578) increase lightly and120601(120578) decreases The dimensionless velocity dimensionlesstemperature and dimensionless concentration profiles fordifferent values of thermophoresis 119873119905 with constant values119873119903 = 01 119873119887 = 01 119876 = 01 Le = 10 119870 = 10119899 = 10 119887 = 00 are presented in Figures 17 18 and19 It is observed that the velocity and temperature of thefluid are not impressible from119873119905 but concentration increasesslightly In Figures 20 21 and 22 respectively comparison ofsolutions of 1198911015840(120578) 120579(120578) and 120601(120578) for 119873119903 = 01 119873119887 = 01119873119905 = 01 Le = 10 119870 = 10 119899 = 10 119887 = 00 anddifferent values of heat generationabsorption parameter 119876are shown It is observed that in presence of heat source1198911015840(120578) 120579(120578) both increase while heat sink causes that 1198911015840(120578)
120579(120578) both decrease extremely These effects are more visible
Table 9 Values of reduced Nusselt number and Sherwood numberobtained by HAM for different values of the parameters Le 119870 119899 119887and 119876 when119873119887 = 119873119903 = 119873119905 = 01
Figure 17 Effect of thermophoresis 119873119905 on the dimensionlessvelocity profile
for temperature distribution For 120601(120578) the effect of changing119876 is seen to be almost insignificant This is in agreementwith the physical factThe dimensionless profiles for differentvalues of Lewis number Le with constant parameters 119873119903 =01 119873119887 = 01 119873119905 = 01 119876 = 01 119870 = 10 119899 = 10119887 = 00 are shown in Figures 23 24 and 25 It is clear that forthese constant parameters variation of Lewis number has anyconsiderable effect on the velocity and temperature profilesbut concentration profile decreases with the increase of Lewisnumber What is similar to these results can be detected inFigures 26 27 and 28 for chemical reaction parameter 119870Figures 29 30 and 31 respectively present the comparisonof solutions of 1198911015840(120578) 120579(120578) and 120601(120578) for 119873119903 = 05 119873119887 = 05119873119905 = 01 119876 = 001 Le = 10 119870 = 10 119887 = 00 and differentvalues of order of chemical reaction 119899 It is observed that
Mathematical Problems in Engineering 15
0 2 4 6 80
1
120578
Nt = 00
Nt = 005
Nt = 10NumericalNumerical
Numerical
08
06
04
02
10
Q = 01Le = 10K = 10n = 10b = 00
Nb = 01
120579(120578
)
Nr = 01
Figure 18 Effect of thermophoresis 119873119905 on the dimensionlesstemperature profile
0 2 4 6 80
1
120601(120578
)
120578
Nt = 00
Nt = 005
Nt = 10NumericalNumerical
Numerical
08
06
04
02
10
Q = 01Le = 10K = 10n = 10b = 00
Nb = 01Nr = 01
Figure 19 Effect of thermophoresis 119873119905 on the dimensionlessconcentration profile
for higher order of chemical reaction velocity decreases andtemperature and concentration profiles increase extremelyUltimately Figures 32 33 and 34 depict the effect of thermalslip parameter 119887 on velocity temperature and concentrationfunctions when 119873119903 = 01 119873119887 = 01 119873119905 = 01 119876 = 01Le = 20 119870 = 15 119899 = 10 It is seen that the velocityand temperature of the fluid decrease with the increase of the
0 2 4 6 80
1
12
10
02
04
06
08
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numericalf998400 (120578
)
120578
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 20 Effect of generationabsorption heat parameter119876 on thedimensionless velocity profile
0 2 4 6 8 100
02
04
06
08
1
120579(120578
)
120578
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numerical
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 21 Effect of generationabsorption heat parameter 119876 on thedimensionless temperature profile
thermal slip parameter in the near of horizontal plate whilethe concentration of the fluid does not vary patently
Also for investigation of the parameters of physicalinterest Table 9 is presented In this table numerical valuesof reduced Nusselt number and Sherwood number obtainedby HAM for different values of the parameters 119890 119870 119899 119887 and119876 can be compared when119873119887 = 119873119903 = 119873119905 = 01
16 Mathematical Problems in Engineering
10
02
04
06
08
0 2 4 6 80
1
120578
120601(120578
)
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numerical
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 22 Effect of generationabsorption heat parameter119876 on thedimensionless concentration profile
0 2 4 6 80
1
12
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 23 Effect of Lewis number Le on the dimensionless velocityprofile
9 Conclusions
In this paper we studied the steady laminar incompressiblefree convective flow of a nanofluid past a chemically reactingupward facing horizontal plate in porous medium takinginto account heat generation and the thermal slip boundarycondition
0 2 4 6 80
1
120579(120578
)
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 24 Effect of Lewis number Le on the dimensionlesstemperature profile
0 2 4 6 80
1
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 25 Effect of Lewis number Le on the dimensionlessconcentration profile
The governing partial differential equations have beentransformed by similarity transformations into a system ofordinary differential equations which are solved by OHAMand numerical method (fourth-order Runge-Kutta schemewith the shooting method) Dimensionless velocity temper-ature and concentration functions are presented for various
Mathematical Problems in Engineering 17
0 2 4 6 80
1
12
K = 00
K = 05
K = 10NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 26 Effect of chemical reaction parameter 119870 on the dimen-sionless velocity profile
0 2 4 6 80
1
K = 00
K = 05
K = 10NumericalNumerical
Numerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 27 Effect of chemical reaction parameter 119870 on the dimen-sionless temperature profile
values of parameters for the problem for example buoy-ancy ratio 119873119903 Brownian motion 119873119887 thermophoresis 119873119905heat generationabsorption parameter 119876 Lewis number Lechemical reaction parameter119870 order of chemical reaction 119899and thermal slip parameter 119887 From the present investigationthe following may be concluded
(i) The velocity 1198911015840(120578) increases with increasing 119873119887 anddecreases with increasing119873119903 119899 and 119887
0 2 4 6 80
1
K = 00
K = 05
K = 10NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 28 Effect of chemical reaction parameter 119870 on the dimen-sionless concentration profile
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 29 Effect of order of chemical reaction 119899 on the dimension-less velocity profile
(ii) The temperature 120579(120578) increases with 119873119903 119873119887 119899 anddecreases with 119887
(iii) Parameters of119873119905 Le and119870 have no effect on velocityand temperature profiles
(iv) The concentration distribution 120601(120578) increases withincreasing N119905 119899 and decreases with 119873119887 Le and 119870but119873119903 119876 and 119887 are not affected by it
18 Mathematical Problems in Engineering
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 30 Effect of order of chemical reaction 119899 on the dimension-less temperature profile
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 31 Effect of order of chemical reaction 119899 on the dimension-less concentration profile
(v) In presence of heat source (119876 gt 0) 1198911015840(120578) 120579(120578) bothincrease while heat sink (119876 lt 0) causes that 1198911015840(120578)120579(120578) both decrease extremely These effects are morevisible for temperature distribution
In addition numerical results for the reducedNusselt andSherwood numbers are tabulated It is concluded that thereduced Nusselt number increases with the Lewis numberand reaction parameter whist it decreases with the order of
0 2 4 6 8
1
12
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 32 Effect of order of thermal slip parameter 119887 on thedimensionless velocity profile
0 2 4 6 80
1
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 33 Effect of order of thermal slip parameter 119887 on thedimensionless temperature profile
chemical reaction thermal slip and generation parametersIt is further concluded that the reduced Sherwood numberenhances with the Lewis number generation and reactionparameters whist it suppresses with thermal slip and orderof chemical reaction
The convergence of the solution series and the power ofHAM in controlling and adjusting the convergence regionand rate of solution series were discussed The proper range
Mathematical Problems in Engineering 19
0 2 4 6 80
1
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 34 Effect of order of thermal slip parameter 119887 on thedimensionless concentration profile
of auxiliary parameter ℏ to ensure the convergence of thesolution series was obtained through the so called ℏ-curveThe HAM provides us with a convenient way to control theconvergence of approximation series which is a fundamentalqualitative difference in analysis between the HAM andother methods The numerical results of the above problemsdisplay a fast convergence with minimal calculations Thisshows that the HAM is a very efficient method Finally theagreement between analytical and numerical results of thepresent study with previous published results is excellent
Nomenclature
119887 Thermal slip parameter119862 Dimensional concentration119862119891 Skin friction factor1198631 Thermal slip factor119863119861 Brownian diffusion coefficient119863119879 Thermophoretic diffusion coefficient119891 Dimensionless velocity functions119892 Gravitation acceleration119870 Chemical reaction parameter119896 Permeability of the porous media1198960 The constant reaction rate119896119898 Effective thermal conductivity of the porous
medium119896(119909) Variable reaction rateL Linear operator of the HAM119871 Length of horizontal plateLe Lewis numberN Nonlinear operator of the HAM
119899 Order of chemical reaction119873119887 Brownian motion119873119903 Buoyancy ratio119873119905 ThermophoresisNu Nusselt number119875 PressurePr Prandtl number119876 Heat generationabsorption parameter1198760 Heat generationabsorption constant119902 Internal heat generation rate
Ra Rayleigh numberSh Sherwood number119879 Temperature119905 Time119906 Velocity in 119909-direction119880119903 Reference velocity Velocity vectorV Velocity in 119910-direction119909 Distance along the surface119910 Distance normal to the surface
Greek Letters
120572119898 Thermal diffusivity120573 Volumetric expansion coefficient of nanofluid120576 Porosity of the porous media120601 Dimensionless concentration120578 Similarity variable120583 Dynamic viscosity120579 Dimensionless temperature120588119891 Density of the base fluid120588119901 Density of the nanoparticles(120588119862)119891 Effective heat capacity of the fluid(120588119862)119875 Effective heat capacity of the nanoparticle material120595 Stream function
Subscript Superscript
infin Conditions far away from the surface1015840 Differentiation with respect to 120578
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
M M Rashidi wishes to thank the Centre for DifferentialEquations Continuum Mechanics and Applications Schoolof Computational and Applied Mathematics University ofthe Witwatersrand for the hospitality during his visit EMomoniat thanks theNational Research Foundation of SouthAfrica for their support
20 Mathematical Problems in Engineering
References
[1] E Abu-Nada F O Hakan and I Pop ldquoBuoyancy inducedflow in a nanofluid filled enclosure partially exposed to forcedconvectionrdquo Superlattices and Microstructures vol 51 no 3 pp381ndash395 2012
[2] S Kakac and A Pramuanjaroenkij ldquoReview of convective heattransfer enhancement with nanofluidsrdquo International Journal ofHeat and Mass Transfer vol 52 no 13-14 pp 3187ndash3196 2009
[3] J A Eastman S U S Choi S Li W Yu and L J ThompsonldquoAnomalously increased effective thermal conductivities ofethylene glycol-based nanofluids containing copper nanoparti-clesrdquo Applied Physics Letters vol 78 no 6 pp 718ndash720 2001
[4] Y Xuan and Q Li ldquoHeat transfer enhancement of nanofluidsrdquoInternational Journal of Heat and Fluid Flow vol 21 no 1 pp58ndash64 2000
[5] S K Das N Putra P Thiesen and W Roetzel ldquoTemperaturedependence of thermal conductivity enhancement for nanoflu-idsrdquo Journal of Heat Transfer vol 125 no 4 pp 567ndash574 2003
[6] D H Kumar H E Patel V R R Kumar T SundararajanT Pradeep and S K Das ldquoModel for heat conduction innanofluidsrdquo Physical Review Letters vol 93 no 14 Article ID144301 2004
[7] H E Patel T Sundararajan T Pradeep A Dasgupta NDasgupta and S K Das ldquoA micro-convection model forthermal conductivity of nanofluidsrdquo Pramana vol 65 no 5 pp863ndash869 2005
[8] A V Kuznetsov ldquoNon-oscillatory and oscillatory nanofluidbio-thermal convection in a horizontal layer of finite depthrdquoEuropean Journal of Mechanics vol 30 no 2 pp 156ndash165 2011
[9] M Napoli J C T Eijkel and S Pennathur ldquoNanofluidictechnology for biomolecule applications a critical reviewrdquo Labon a Chip vol 10 no 8 pp 957ndash985 2010
[10] J C Crepeau and R Clarksean ldquoSimilarity solutions of naturalconvection with internal heat generationrdquo Journal of HeatTransfer vol 119 no 1 pp 183ndash185 1997
[11] A Z Sahin ldquoTransient heat conduction in semi-infinite solidwith spatially decaying exponential heat generationrdquo Interna-tional Communications in Heat andMass Transfer vol 19 no 3pp 349ndash358 1992
[12] O D Makind and A Aziz ldquoMixed convection from a con-vectively heated vertical plate to a fluid with internal heatgenerationrdquo Journal of Heat Transfer vol 133 Article ID 1225016 pages 2011
[13] J Buongiorno ldquoConvective transport in nanofluidsrdquo Journal ofHeat Transfer vol 128 no 3 pp 240ndash250 2006
[14] AV Kuznetsov andDANield ldquoNatural convective boundary-layer flow of a nanofluid past a vertical platerdquo InternationalJournal of Thermal Sciences vol 49 no 2 pp 243ndash247 2010
[15] L B Godson D Raja L D Mohan and S WongwisescldquoEnhancement of heat transfer using nanofluids-an overviewrdquoRenewable and Sustainable Energy Reviews vol 14 no 2 pp629ndash641 2010
[16] D A Nield and A BejanConvection in PorousMedia SpringerNew York NY USA 3rd edition 2006
[17] P Vadasz Emerging Topics in Heat and Mass Transfer in PorousMedia Springer New York NY USA 2008
[18] K Vafai Porous Media Applications in Biological Systems andBiotechnology CRC Press New York NY USA 2010
[19] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application to
heat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977
[20] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for natural convective boundary-layer flow in a porousmedium saturated by a nanofluidrdquo International Journal of Heatand Mass Transfer vol 52 no 25-26 pp 5792ndash5795 2009
[21] P Cheng and I D Chang ldquoBuoyancy induced flows in asaturated porous medium adjacent to impermeable horizontalsurfacesrdquo International Journal of Heat and Mass Transfer vol19 no 11 pp 1267ndash1272 1976
[22] I D Chang and P Cheng ldquoMatched asymptotic expansionsfor free convection about an impermeable horizontal surfacein a porous mediumrdquo International Journal of Heat and MassTransfer vol 26 no 2 pp 163ndash174 1983
[23] D S Shiunlin and B Gebhart ldquoBuoyancy-induced flow adja-cent to a horizontal surface submerged in porous mediumsaturated with cold waterrdquo International Journal of Heat andMass Transfer vol 29 no 4 pp 611ndash623 1986
[24] J H Merkin and G Zhang ldquoOn the similarity solutions for freeconvection in a saturated porousmedium adjacent to imperme-able horizontal surfacesrdquoWarme- und Stoffubertragung vol 25no 3 pp 179ndash184 1990
[25] M A Chaudhary J H Merkin and I Pop ldquoNatural convectionfrom a horizontal permeable surface in a porous mediummdashnumerical and asymptotic solutionsrdquo Transport in PorousMedia vol 22 no 3 pp 327ndash344 1996
[26] R S R Gorla and A Chamkha ldquoNatural convective boundarylayer flow over a nonisothermal vertical plate embedded ina porous medium saturated with a nanofluidrdquo Nanoscale andMicroscale Thermophysical Engineering vol 15 no 2 pp 81ndash942011
[27] W A Khan and I Pop ldquoFree convection boundary layer flowpast a horizontal flat plate embedded in a porous medium filledwith a nanofluidrdquo Journal of Heat Transfer vol 133 no 9 ArticleID 094501 2011
[28] A Aziz W A Khan and I Pop ldquoFree convection boundarylayer flow past a horizontal flat plate embedded in porousmedium filled by nanofluid containing gyrotactic microorgan-ismsrdquo International Journal of Thermal Sciences vol 56 pp 48ndash57 2012
[29] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer New York NYUSA 2009
[30] A A Avramenko S G Kobzar I V Shevchuk A V Kuznetsovand L T Iwanisov ldquoSymmetry of turbulent boundary-layerflows investigation of different eddy viscosity modelsrdquo ActaMechanica vol 151 no 1-2 pp 1ndash14 2001
[31] A V Kuznetsov A A Avramenko and P Geng ldquoAnalyticalinvestigation of a falling plume caused by bioconvection ofoxytactic bacteria in a fluid saturated porous mediumrdquo Inter-national Journal of Engineering Science vol 42 no 5-6 pp 557ndash569 2004
[32] M Jalil S Asghar and M Mushtaq ldquoLie group analysis ofmixed convection flow with mass transfer over a stretchingsurface with suction or injectionrdquo Mathematical Problems inEngineering vol 2010 Article ID 264901 14 pages 2010
[33] M A A Hamad M J Uddin and A I M Ismail ldquoRadiationeffects on heat and mass transfer in MHD stagnation-pointflow over a permeable flat plate with thermal convectivesurface boundary condition temperature dependent viscosityand thermal conductivityrdquoNuclear Engineering and Design vol242 pp 194ndash200 2012
Mathematical Problems in Engineering 21
[34] A Aziz M J Uddin M A A Hamad and A I MdIsmail ldquoMHD flow over an inclined radiating plate with thetemperature-dependent thermal conductivity variable reactiveindex and heat generationrdquoHeat Transfer vol 41 no 3 pp 241ndash259 2012
[35] A G Hansen Similarity Analysis of Boundary Layer Problemsin Engineering Prentice Hall Englewood Cliffs NJ USA 1964
[36] W FAmesNonlinear Partial Differential Equations in Engineer-ing Academic Press New York NY USA 1972
[37] R Seshadri and T Y Na Group Invariance in EngineeringBoundary Value Problems Springer New York NY USA 1985
[38] D Shang Theory of Heat Transfer with Forced Convection FilmFlows vol 3 of Heat and Mass Transfer 2010
[40] E Erfani M M Rashidi and A B Parsa ldquoThe modifieddifferential transform method for solving off-centered stag-nation flow toward a rotating discrdquo International Journal ofComputational Methods vol 7 no 4 pp 655ndash670 2010
[41] M M Rashidi N Laraqi and A Basiri Parsa ldquoAnalyticalmodeling of heat convection in magnetized micropolar fluid byusing modified differential transform methodrdquo Heat Transfervol 40 no 3 pp 187ndash204 2011
[42] M M Rashidi and M Keimanesh ldquoUsing differential trans-form method and pade approximant for solving mhd flow ina laminar liquid film from a horizontal stretching surfacerdquoMathematical Problems in Engineering vol 2010 Article ID491319 14 pages 2010
[43] M M Rashidi and G Domairry ldquoNew analytical solution ofthe three-dimensional NavierStokes equationsrdquoModern PhysicsLetters B vol 23 no 26 pp 3147ndash3155 2009
[44] M M Rashidi M Keimanesh O A Beg and T K HungldquoMagnetohydrodynamic biorheological transport phenomenain a porous medium a simulation of magnetic blood flowcontrol and filtrationrdquo International Journal for NumericalMethods in Biomedical Engineering vol 27 no 6 pp 805ndash8212011
[45] S J Liao The proposed homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992
[46] M M Rashidi G Domairry and S Dinarvand ldquoApproximatesolutions for the Burger and regularized long wave equationsby means of the homotopy analysis methodrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 14 no 3 pp708ndash717 2009
[47] Z Ziabakhsh and G Domairry ldquoAnalytic solution of naturalconvection flow of a non-Newtonian fluid between two verticalflat plates using homotopy analysis methodrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 14 no 5pp 1868ndash1880 2009
[48] M M Rashidi S A Mohimanian Pour T Hayat and SObaidat ldquoAnalytic approximate solutions for steady flow over arotating disk in porousmediumwith heat transfer by homotopyanalysis methodrdquo Computers and Fluids vol 54 no 1 pp 1ndash92012
[49] M Sajid T Hayat and S Asghar ldquoComparison between theHAM and HPM solutions of thin film flows of non-Newtonianfluids on a moving beltrdquo Nonlinear Dynamics vol 50 no 1-2pp 27ndash35 2007
[50] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for the double-diffusive natural convective boundary
layer flow in a porous medium saturated by a nanofluidrdquoInternational Journal of Heat and Mass Transfer vol 54 no 1-3pp 374ndash378 2011
[51] K Vajravelu and A Hadjinicolaou ldquoConvective heat transferin an electrically conducting fluid at a stretching surfacewith uniform free streamrdquo International Journal of EngineeringScience vol 35 pp 1237ndash1244 1997
[52] S Mukhopadhyay and G C Layek ldquoEffects of variable fluidviscosity on flow past a heated stretching sheet embedded ina porous medium in presence of heat sourcesinkrdquo Meccanicavol 47 no 4 pp 863ndash876 2012
[53] S J Liao ldquoAn optimal homotopy-analysis approach for stronglynonlinear differential equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 8 pp 2003ndash20162010
[54] A Aziz ldquoA similarity solution for laminar thermal boundarylayer over a flat plate with a convective surface boundary con-ditionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1064ndash1068 2009
Using the above definition with assumption 119867119865(120591) = 1119867Θ(120591) = Exp[minus120578] and 119867Φ(120591) = Exp[minus120578] we construct the119911119890119903119900th-order deformation equation
(1 minus 119901)L [120593 (119909 119901) minus 1199060 (119909)] = 119867 (120591) 119901ℏN [120593 (119909 119901)]
Differentiating the 119911119890119903119900th-order deformation equation(47)119898 times with respect to 119901 and finally dividing by119898 wehave the119898th-order deformation equation
L [119865119898 (119909) minus 120594119898119865119898minus1 (119909)] = ℏ119877119898 (119898minus1)
L [Θ119898 (119909) minus 120594119898Θ119898minus1 (119909)] = Exp [minus120578] ℏ119877119898 (Θ119898minus1)
L [Φ119898 (119909) minus 120594119898Φ119898minus1 (119909)] = Exp [minus120578] ℏ119877119898 (Φ119898minus1)
By using the symbolic software MATHEMATICA tosolve the system of linear equations (49) with the boundaryconditions (46) it can be obtained finally
1198651 (120578) = minus
4
3 (1 + 119887)
ℏ +
4119890minus120578
3 (1 + 119887)
ℏ +
4119873119903
3 (1 + 119887)
ℏ
+
4119887119873119903
3 (1 + 119887)
ℏ minus
4119873119903119890minus120578
3 (1 + 119887)
ℏ minus
4119887119873119903119890minus120578
3 (1 + 119887)
ℏ
+
2119890minus120578120578
3 (1 + 119887)
ℏ minus
2119873119903119890minus120578120578
3 (1 + 119887)
ℏ minus
2119887119873119903119890minus120578120578
3 (1 + 119887)
ℏ
Θ1 (120578) =119890minus2120578
2 (1 + 119887)
ℏ minus
119890minus120578
2(1 + 119887)2ℏ minus
119887119890minus120578
2(1 + 119887)2ℏ
minus
119873119887119890minus3120578
6(1 + 119887)2ℏ +
119887119873119887119890minus3120578
6(1 + 119887)2ℏ minus
119873119887119890minus120578
6(1 + 119887)2ℏ
minus
119887119873119887119890minus120578
6(1 + 119887)2ℏ +
119873119905119890minus3120578
6(1 + 119887)2ℏ minus
119873119905119890minus120578
6(1 + 119887)2ℏ
+
119876119890minus2120578
2 (1 + 119887)
ℏ minus
119876119890minus120578
2(1 + 119887)2ℏ minus
119887119876119890minus120578
2(1 + 119887)2ℏ
Φ1 (120578) =119890minus2120578
2 (1 + 119887)
ℏ +
119887119890minus2120578
2 (1 + 119887)
ℏ minus
119890minus120578
2 (1 + 119887)
ℏ
minus
119887119890minus120578
2 (1 + 119887)
ℏ minus
119870119890minus2120578
2 (1 + 119887)
ℏ minus
119887119870eminus2120578
2 (1 + 119887)
ℏ
+
119870119890minus120578
2 (1 + 119887)
ℏ +
119887119870119890minus120578
2 (1 + 119887)
ℏ +
119873119905
119873119887
119890minus2120578
2 (1 + 119887)
ℏ
minus
119873119905
119873119887
119890minus120578
2 (1 + 119887)
ℏ
(56)
The higher orders solutions of 119865(120578) Θ(120578) and Φ(120578) weretoo long to be mentioned here therefore they are showngraphically
51 Convergence of the HAM Solution As mentioned by Liao[45] HAM provides us with great freedom in choosing thesolution of a nonlinear problem by different base functionsThis has a great effect on the convergence region because theconvergence region and rate of a series are chiefly determinedby the base functions used to express the solution We usedseveral terms in evaluating the approximate solution 119865app asympsum119899119894=0 119865119894Θapp asymp sum
119899119894=0Θ119894Φapp asymp sum
119899119894=0Φ119894 note that the solution
series contains the auxiliary parameter ℏ which provides uswith a simple way to adjust and control the convergence ofthe solution series Generally by means of the so-called ℏ-curve that is a curve of a versus ℏ As pointed by Liao [45]the valid region of ℏ is a horizontal line segment Figures2 3 and 4 show the ℏ-curve with the various order of theHAM for 11989110158401015840(0) 1205791015840(0) 1206011015840(0) respectively when 119873119887 = 01
Figure 6 The residual error of 120579(120578) for (34) using 12th order ofapproximations
119873119905 = 01 119873119903 = 01 119876 = 01 Le = 10 119870 = 10 119899 = 10119887 = 00 It can be seen that when the order of series is 12 thesegment of the horizontal line is more than the other ordersFor example it can be found that for 11986510158401015840(0) the acceptablerange ofℏ is betweenminus05 andminus20 for 12th order of theHAMbut for 8th order of the HAM the acceptable range of ℏ isbetween minus10 and minus20 or minus10 and minus15 for 4th order of theHAM so horizontal line segment of 12th order of the HAM ismore than others Therefore it is straightforward to choosean appropriate range for ℏ which ensures the convergence
10 Mathematical Problems in Engineering
minus00002
120578
= minus10
= minus111 (optimal value)= minus12
Nt = 01Nr = 01
Q = 01Le = 10K = 10n = 10b = 00
0006
0004
0002
0
0 2 4 6 8 10
Nb = 01
Res (120601
(120578))
ℏ
ℏ
ℏ
Figure 7 The residual error of 120601(120578) for (35) using 12th order ofapproximations
of the solution series To choose optimal value of auxiliaryparameter the averaged residual errors (see [53] for moredetails) are defined as
119864119865119898 =1
119870
119870
sum
119894=0
[N119865(119898
sum
119894=0
119865119894 (119894Δ119909))]
2
119864Θ119898 =1
119870
119870
sum
119894=0
[NΘ(119898
sum
119894=0
Θ119894 (119894Δ119909))]
2
119864Φ119898 =1
119870
119870
sum
119894=0
[NΦ(119898
sum
119894=0
Φ119894 (119894Δ119909))]
2
(57)
where Δ119909 = 10119870 and 119870 = 20 For given order ofapproximation 119898 and the optimal values ℏ are given bythe minimum of 119864119898 corresponding to nonlinear algebraicequations
119889119864119865119898
119889ℏ
= 0
119889119864Θ119898
119889ℏ
= 0
119889119864Φ119898
119889ℏ
= 0 (58)
It is noted that the optimal value of ℏ is replaced intothe equations Table 1 shows optimal values obtained for theauxiliary parameter ℏ for various quantities of119873119905 119876 and 119899when119873119887 = 01119873119903 = 01 Le = 10 119870 = 10 119887 = 00 To see
the accuracy of the solutions the residual errors are definedfor the system as (for119898 order approximation)
Res119865119898
=
1198892119865119898
1198891205782minus
2
3
120578(
119889Θ119898
119889120578
minus 119873119903
119889Φ119898
119889120578
) (59)
ResΘ119898
=
1198892Θ119898
1198891205782+
1
3
119865119898
119889Θ119898
d120578+ 119873119887
119889Θ119898
119889120578
119889Φ119898
119889120578
+ 119873119905(
119889Θ119898
119889120578
)
2
+ 119876Θ119898
(60)
ResΦ119898
=
1198892Φ119898
1198891205782+
Le3
119865119898
119889Φ119898
119889120578
+
119873119905
119873119887
1198892Θ119898
1198891205782
minus 119870(
119889Φ119898
119889120578
)
119899
(61)
Figures 5 6 and 7 show the residual errors for 12th-orderdeformation solutions when 119873119887 = 01 119873119905 = 01 119873119903 = 01119876 = 01 Le = 10 119870 = 10 119899 = 10 119887 = 00 For example ℏ =minus145 has the minimum range of residual curve in Figure 5and so on
Graphical representation of results is very useful todemonstrate the efficiency and accuracy of the HAM for theabove problem
6 Comparisons and Verification
It is worth citing that for isothermal plate (119887 = 0) and in theabsence of internal heat generationabsorption (119876 = 0) andchemical reaction (119870 = 0) our problem reduces to Gorla andChamkha [26] and Khan and Pop [27] To verify the accuracyof our results the present results are compared in Table 2with Gorla and Chamkha [26] and are found to be in goodagreement
7 Physical Quantities
Theparameters of physical interest of the present problem arethe local skin friction factor 119862119891119909 the local Nusselt numberNu119909 and the local Sherwood number Sh119909 respectivelyPhysically 119862119891119909 indicates wall shear stress Nu119909 indicates therate of wall heat transfer whilst Sh119909 indicates the rate ofwall nanoparticle volume fractionThe following relations areused to find these quantities
119862119891119909 =2120583
1205881198802119903
(
120597119906
120597119910
)
119910=0
Nu119909 =minus119909
119879119891 minus 119879infin
(
120597119879
120597119910
)
119910=0
Sh119909 =minus119909
119862119908 minus 119862infin
(
120597119862
120597119910
)
119910=0
(62)
Mathematical Problems in Engineering 11
Table 2 Comparison of present solution with Gorla and Chamkha[26] for different values of buoyancy and nanofluid parameters
Present results (HAM) Gorla and Chamkha [26]minus1205791015840(0) minus120601
1015840(0) minus120579
1015840(0) minus120601
1015840(0)
Nr 119873119887 = 03119873119905 = 01 Le = 10 119887 = 119870 = 119876 = 0
01 032578 148242 326119864 minus 01 148416402 032385 146704 325119864 minus 01 146816103 032188 145125 322119864 minus 01 145266404 031985 143503 321119864 minus 01 143639205 031777 141833 319119864 minus 01 1419499Nt 119873119887 = 03119873119903 = 05 Le = 10 119887 = 119870 = 119876 = 0
01 031777 141833 319119864 minus 01 141949902 030486 141491 305119864 minus 01 141653603 02927 141561 293119864 minus 01 141686604 028125 141991 282119864 minus 01 142158205 027046 142737 271119864 minus 01 1429226Nb 119873119905 = 01119873119903 = 05 Le = 10 119887 = 119870 = 119876 = 0
01 03672 132611 368119864 minus 01 132745402 034271 139216 343119864 minus 01 139361503 031777 141833 319119864 minus 01 141949904 029399 143428 294119864 minus 01 143546405 027161 144598 272119864 minus 01 144772
Figure 10 Comparison of dimensionless concentration profile 120601(120578)obtained by various orders of the HAM
Here Ra119909 = 119892119870120573(1minus119862infin)Δ119879119909(120572119898]) is the local Rayleighnumber Pr = ]120572119898 is the Prandtl number and 119880119903 = (1 minus
119862infin)119892119870120573Δ119879120572119898 is the reference velocity
8 Results and Discussion
Graphical representation of results is very useful to discussthe physical features presented by the solutions This sectiondescribes the influence of some interesting parameters on
12 Mathematical Problems in Engineering
0 2 4 6 8 10
02
04
06
08
1
Nr = 01Numerical
Numerical
NumericalNr = 03Nr = 05
f998400 (120578)
120578
Nt = 01Nb = 01Q = 01Le = 10K = 10n = 10b = 00
Figure 11 Effect of buoyancy ratio119873119903 on the dimensionless velocityprofile
0 2 4 6 8 10
02
04
06
08
1
Nt = 01Nb = 01Q = 01Le = 10K = 10n = 10b = 00
Nr = 01Numerical
Numerical
NumericalNr = 03Nr = 05
120578
120579(120578
)
Figure 12 Effect of buoyancy ratio 119873119903 on the dimensionlesstemperature profile
the velocity and temperature fields Equations (33) to (35)with boundary conditions in (36) were solved analytically byHAM and numerically using Runge-Kutta-Fehlberg fourth-fifth order proposed by Aziz [54]
Figures 8 9 and 10 respectively for 1198911015840(120578) 120579(120578) and 120601(120578)show the comparisons between various order approximationof the optimal HAM and the numerical solutions for the case
0 2 4 6 8 10
02
04
06
08
1
Nr = 01Numerical
Numerical
NumericalNr = 03Nr = 05
Nt = 01Nb = 01Q = 01Le = 10K = 10n = 10b = 00
120578
120601(120578
)Figure 13 Effect of buoyancy ratio 119873119903 on the dimensionlessconcentration profile
0 2 4 6 8 10
02
04
06
08
1
12
NumericalNumerical
Numerical
f998400 (120578
)
120578
Nb = 01
Nb = 03
Nb = 05
Nt = 01Nr = 01Q = 01Le = 10K = 10n = 10b = 00
Figure 14 Effect of Brownian motion 119873119887 on the dimensionlessvelocity profile
119873119887 = 01 119873119905 = 01 119873119903 = 01 119876 = 01 Le = 10 119870 = 10119899 = 10 119887 = 00 For better presentation these results arepresented in Tables 3 4 and 5 Also Tables 6 7 and 8 presentthe comparisons between various order approximation of theoptimal HAM and the numerical solutions for the case119873119887 =03 119873119905 = 005 119873119903 = 03 119876 = 01 Le = 20 119870 = 05
Mathematical Problems in Engineering 13
0 2 4 6 8 10
02
04
06
08
1
0
120578
NumericalNumerical
Numerical
Nb = 01
Nb = 03
Nb = 05
Nt = 01Nr = 01Q = 01Le = 10K = 10n = 10b = 00
120579(120578
)
Figure 15 Effect of Brownian motion 119873119887 on the dimensionlesstemperature profile
02
04
06
08
1
00
2 4 6 8 10120578
NumericalNumerical
Numerical
Nb = 01
Nb = 03
Nb = 05
Nt = 01Nr = 01Q = 01Le = 10K = 10n = 10b = 00
120601(120578
)
Figure 16 Effect of Brownian motion 119873119887 on the dimensionlessconcentration profile
119899 = 20 119887 = 01 It is observed that the results of 12th-order approximation of the optimal HAM are very close tothe numerical solutions which confirm the validity of thesemethods
In the following figures effects of various physicalparameters on the dimensionless velocity temperature and
Table 3 Comparison of values of 1198911015840(120578) obtained by various ordersof the HAM with numerical solution when 119873119887 = 01119873119905 =
temperature profiles will be investigated These results havebeen obtained by 12th order of the HAM and have beenvalidated by numerical results Figures 11ndash13 respectivelyrepresent that the comparison of solutions of 1198911015840(120578) 120579(120578)and 120601(120578) for fix values 119873119887 = 01 119873119905 = 01 119876 = 01Le = 10 119870 = 10 119899 = 10 119887 = 00 and different values ofbuoyancy ratio119873119903 In Figure 11 it is clear that velocity of thefluid tremendously decreases in the near of horizontal platewith an increase in the buoyancy ratio It is observed that thetemperature increases slightly but concentration of the fluiddoes not vary sensibly (Figures 12 and 13) The effects of theBrownian motion 119873119887 are depicted in Figures 14 15 and 16when119873119903 = 01119873119905 = 01119876 = 01 Le = 10119870 = 10 119899 = 10
14 Mathematical Problems in Engineering
Table 6 Comparison of values of 1198911015840(120578) obtained by various ordersof the HAM with numerical solution when 119873119887 = 03119873119905 =
119887 = 00 If 119873119887 increases 1198911015840(120578) and 120579(120578) increase lightly and120601(120578) decreases The dimensionless velocity dimensionlesstemperature and dimensionless concentration profiles fordifferent values of thermophoresis 119873119905 with constant values119873119903 = 01 119873119887 = 01 119876 = 01 Le = 10 119870 = 10119899 = 10 119887 = 00 are presented in Figures 17 18 and19 It is observed that the velocity and temperature of thefluid are not impressible from119873119905 but concentration increasesslightly In Figures 20 21 and 22 respectively comparison ofsolutions of 1198911015840(120578) 120579(120578) and 120601(120578) for 119873119903 = 01 119873119887 = 01119873119905 = 01 Le = 10 119870 = 10 119899 = 10 119887 = 00 anddifferent values of heat generationabsorption parameter 119876are shown It is observed that in presence of heat source1198911015840(120578) 120579(120578) both increase while heat sink causes that 1198911015840(120578)
120579(120578) both decrease extremely These effects are more visible
Table 9 Values of reduced Nusselt number and Sherwood numberobtained by HAM for different values of the parameters Le 119870 119899 119887and 119876 when119873119887 = 119873119903 = 119873119905 = 01
Figure 17 Effect of thermophoresis 119873119905 on the dimensionlessvelocity profile
for temperature distribution For 120601(120578) the effect of changing119876 is seen to be almost insignificant This is in agreementwith the physical factThe dimensionless profiles for differentvalues of Lewis number Le with constant parameters 119873119903 =01 119873119887 = 01 119873119905 = 01 119876 = 01 119870 = 10 119899 = 10119887 = 00 are shown in Figures 23 24 and 25 It is clear that forthese constant parameters variation of Lewis number has anyconsiderable effect on the velocity and temperature profilesbut concentration profile decreases with the increase of Lewisnumber What is similar to these results can be detected inFigures 26 27 and 28 for chemical reaction parameter 119870Figures 29 30 and 31 respectively present the comparisonof solutions of 1198911015840(120578) 120579(120578) and 120601(120578) for 119873119903 = 05 119873119887 = 05119873119905 = 01 119876 = 001 Le = 10 119870 = 10 119887 = 00 and differentvalues of order of chemical reaction 119899 It is observed that
Mathematical Problems in Engineering 15
0 2 4 6 80
1
120578
Nt = 00
Nt = 005
Nt = 10NumericalNumerical
Numerical
08
06
04
02
10
Q = 01Le = 10K = 10n = 10b = 00
Nb = 01
120579(120578
)
Nr = 01
Figure 18 Effect of thermophoresis 119873119905 on the dimensionlesstemperature profile
0 2 4 6 80
1
120601(120578
)
120578
Nt = 00
Nt = 005
Nt = 10NumericalNumerical
Numerical
08
06
04
02
10
Q = 01Le = 10K = 10n = 10b = 00
Nb = 01Nr = 01
Figure 19 Effect of thermophoresis 119873119905 on the dimensionlessconcentration profile
for higher order of chemical reaction velocity decreases andtemperature and concentration profiles increase extremelyUltimately Figures 32 33 and 34 depict the effect of thermalslip parameter 119887 on velocity temperature and concentrationfunctions when 119873119903 = 01 119873119887 = 01 119873119905 = 01 119876 = 01Le = 20 119870 = 15 119899 = 10 It is seen that the velocityand temperature of the fluid decrease with the increase of the
0 2 4 6 80
1
12
10
02
04
06
08
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numericalf998400 (120578
)
120578
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 20 Effect of generationabsorption heat parameter119876 on thedimensionless velocity profile
0 2 4 6 8 100
02
04
06
08
1
120579(120578
)
120578
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numerical
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 21 Effect of generationabsorption heat parameter 119876 on thedimensionless temperature profile
thermal slip parameter in the near of horizontal plate whilethe concentration of the fluid does not vary patently
Also for investigation of the parameters of physicalinterest Table 9 is presented In this table numerical valuesof reduced Nusselt number and Sherwood number obtainedby HAM for different values of the parameters 119890 119870 119899 119887 and119876 can be compared when119873119887 = 119873119903 = 119873119905 = 01
16 Mathematical Problems in Engineering
10
02
04
06
08
0 2 4 6 80
1
120578
120601(120578
)
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numerical
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 22 Effect of generationabsorption heat parameter119876 on thedimensionless concentration profile
0 2 4 6 80
1
12
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 23 Effect of Lewis number Le on the dimensionless velocityprofile
9 Conclusions
In this paper we studied the steady laminar incompressiblefree convective flow of a nanofluid past a chemically reactingupward facing horizontal plate in porous medium takinginto account heat generation and the thermal slip boundarycondition
0 2 4 6 80
1
120579(120578
)
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 24 Effect of Lewis number Le on the dimensionlesstemperature profile
0 2 4 6 80
1
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 25 Effect of Lewis number Le on the dimensionlessconcentration profile
The governing partial differential equations have beentransformed by similarity transformations into a system ofordinary differential equations which are solved by OHAMand numerical method (fourth-order Runge-Kutta schemewith the shooting method) Dimensionless velocity temper-ature and concentration functions are presented for various
Mathematical Problems in Engineering 17
0 2 4 6 80
1
12
K = 00
K = 05
K = 10NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 26 Effect of chemical reaction parameter 119870 on the dimen-sionless velocity profile
0 2 4 6 80
1
K = 00
K = 05
K = 10NumericalNumerical
Numerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 27 Effect of chemical reaction parameter 119870 on the dimen-sionless temperature profile
values of parameters for the problem for example buoy-ancy ratio 119873119903 Brownian motion 119873119887 thermophoresis 119873119905heat generationabsorption parameter 119876 Lewis number Lechemical reaction parameter119870 order of chemical reaction 119899and thermal slip parameter 119887 From the present investigationthe following may be concluded
(i) The velocity 1198911015840(120578) increases with increasing 119873119887 anddecreases with increasing119873119903 119899 and 119887
0 2 4 6 80
1
K = 00
K = 05
K = 10NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 28 Effect of chemical reaction parameter 119870 on the dimen-sionless concentration profile
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 29 Effect of order of chemical reaction 119899 on the dimension-less velocity profile
(ii) The temperature 120579(120578) increases with 119873119903 119873119887 119899 anddecreases with 119887
(iii) Parameters of119873119905 Le and119870 have no effect on velocityand temperature profiles
(iv) The concentration distribution 120601(120578) increases withincreasing N119905 119899 and decreases with 119873119887 Le and 119870but119873119903 119876 and 119887 are not affected by it
18 Mathematical Problems in Engineering
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 30 Effect of order of chemical reaction 119899 on the dimension-less temperature profile
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 31 Effect of order of chemical reaction 119899 on the dimension-less concentration profile
(v) In presence of heat source (119876 gt 0) 1198911015840(120578) 120579(120578) bothincrease while heat sink (119876 lt 0) causes that 1198911015840(120578)120579(120578) both decrease extremely These effects are morevisible for temperature distribution
In addition numerical results for the reducedNusselt andSherwood numbers are tabulated It is concluded that thereduced Nusselt number increases with the Lewis numberand reaction parameter whist it decreases with the order of
0 2 4 6 8
1
12
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 32 Effect of order of thermal slip parameter 119887 on thedimensionless velocity profile
0 2 4 6 80
1
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 33 Effect of order of thermal slip parameter 119887 on thedimensionless temperature profile
chemical reaction thermal slip and generation parametersIt is further concluded that the reduced Sherwood numberenhances with the Lewis number generation and reactionparameters whist it suppresses with thermal slip and orderof chemical reaction
The convergence of the solution series and the power ofHAM in controlling and adjusting the convergence regionand rate of solution series were discussed The proper range
Mathematical Problems in Engineering 19
0 2 4 6 80
1
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 34 Effect of order of thermal slip parameter 119887 on thedimensionless concentration profile
of auxiliary parameter ℏ to ensure the convergence of thesolution series was obtained through the so called ℏ-curveThe HAM provides us with a convenient way to control theconvergence of approximation series which is a fundamentalqualitative difference in analysis between the HAM andother methods The numerical results of the above problemsdisplay a fast convergence with minimal calculations Thisshows that the HAM is a very efficient method Finally theagreement between analytical and numerical results of thepresent study with previous published results is excellent
Nomenclature
119887 Thermal slip parameter119862 Dimensional concentration119862119891 Skin friction factor1198631 Thermal slip factor119863119861 Brownian diffusion coefficient119863119879 Thermophoretic diffusion coefficient119891 Dimensionless velocity functions119892 Gravitation acceleration119870 Chemical reaction parameter119896 Permeability of the porous media1198960 The constant reaction rate119896119898 Effective thermal conductivity of the porous
medium119896(119909) Variable reaction rateL Linear operator of the HAM119871 Length of horizontal plateLe Lewis numberN Nonlinear operator of the HAM
119899 Order of chemical reaction119873119887 Brownian motion119873119903 Buoyancy ratio119873119905 ThermophoresisNu Nusselt number119875 PressurePr Prandtl number119876 Heat generationabsorption parameter1198760 Heat generationabsorption constant119902 Internal heat generation rate
Ra Rayleigh numberSh Sherwood number119879 Temperature119905 Time119906 Velocity in 119909-direction119880119903 Reference velocity Velocity vectorV Velocity in 119910-direction119909 Distance along the surface119910 Distance normal to the surface
Greek Letters
120572119898 Thermal diffusivity120573 Volumetric expansion coefficient of nanofluid120576 Porosity of the porous media120601 Dimensionless concentration120578 Similarity variable120583 Dynamic viscosity120579 Dimensionless temperature120588119891 Density of the base fluid120588119901 Density of the nanoparticles(120588119862)119891 Effective heat capacity of the fluid(120588119862)119875 Effective heat capacity of the nanoparticle material120595 Stream function
Subscript Superscript
infin Conditions far away from the surface1015840 Differentiation with respect to 120578
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
M M Rashidi wishes to thank the Centre for DifferentialEquations Continuum Mechanics and Applications Schoolof Computational and Applied Mathematics University ofthe Witwatersrand for the hospitality during his visit EMomoniat thanks theNational Research Foundation of SouthAfrica for their support
20 Mathematical Problems in Engineering
References
[1] E Abu-Nada F O Hakan and I Pop ldquoBuoyancy inducedflow in a nanofluid filled enclosure partially exposed to forcedconvectionrdquo Superlattices and Microstructures vol 51 no 3 pp381ndash395 2012
[2] S Kakac and A Pramuanjaroenkij ldquoReview of convective heattransfer enhancement with nanofluidsrdquo International Journal ofHeat and Mass Transfer vol 52 no 13-14 pp 3187ndash3196 2009
[3] J A Eastman S U S Choi S Li W Yu and L J ThompsonldquoAnomalously increased effective thermal conductivities ofethylene glycol-based nanofluids containing copper nanoparti-clesrdquo Applied Physics Letters vol 78 no 6 pp 718ndash720 2001
[4] Y Xuan and Q Li ldquoHeat transfer enhancement of nanofluidsrdquoInternational Journal of Heat and Fluid Flow vol 21 no 1 pp58ndash64 2000
[5] S K Das N Putra P Thiesen and W Roetzel ldquoTemperaturedependence of thermal conductivity enhancement for nanoflu-idsrdquo Journal of Heat Transfer vol 125 no 4 pp 567ndash574 2003
[6] D H Kumar H E Patel V R R Kumar T SundararajanT Pradeep and S K Das ldquoModel for heat conduction innanofluidsrdquo Physical Review Letters vol 93 no 14 Article ID144301 2004
[7] H E Patel T Sundararajan T Pradeep A Dasgupta NDasgupta and S K Das ldquoA micro-convection model forthermal conductivity of nanofluidsrdquo Pramana vol 65 no 5 pp863ndash869 2005
[8] A V Kuznetsov ldquoNon-oscillatory and oscillatory nanofluidbio-thermal convection in a horizontal layer of finite depthrdquoEuropean Journal of Mechanics vol 30 no 2 pp 156ndash165 2011
[9] M Napoli J C T Eijkel and S Pennathur ldquoNanofluidictechnology for biomolecule applications a critical reviewrdquo Labon a Chip vol 10 no 8 pp 957ndash985 2010
[10] J C Crepeau and R Clarksean ldquoSimilarity solutions of naturalconvection with internal heat generationrdquo Journal of HeatTransfer vol 119 no 1 pp 183ndash185 1997
[11] A Z Sahin ldquoTransient heat conduction in semi-infinite solidwith spatially decaying exponential heat generationrdquo Interna-tional Communications in Heat andMass Transfer vol 19 no 3pp 349ndash358 1992
[12] O D Makind and A Aziz ldquoMixed convection from a con-vectively heated vertical plate to a fluid with internal heatgenerationrdquo Journal of Heat Transfer vol 133 Article ID 1225016 pages 2011
[13] J Buongiorno ldquoConvective transport in nanofluidsrdquo Journal ofHeat Transfer vol 128 no 3 pp 240ndash250 2006
[14] AV Kuznetsov andDANield ldquoNatural convective boundary-layer flow of a nanofluid past a vertical platerdquo InternationalJournal of Thermal Sciences vol 49 no 2 pp 243ndash247 2010
[15] L B Godson D Raja L D Mohan and S WongwisescldquoEnhancement of heat transfer using nanofluids-an overviewrdquoRenewable and Sustainable Energy Reviews vol 14 no 2 pp629ndash641 2010
[16] D A Nield and A BejanConvection in PorousMedia SpringerNew York NY USA 3rd edition 2006
[17] P Vadasz Emerging Topics in Heat and Mass Transfer in PorousMedia Springer New York NY USA 2008
[18] K Vafai Porous Media Applications in Biological Systems andBiotechnology CRC Press New York NY USA 2010
[19] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application to
heat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977
[20] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for natural convective boundary-layer flow in a porousmedium saturated by a nanofluidrdquo International Journal of Heatand Mass Transfer vol 52 no 25-26 pp 5792ndash5795 2009
[21] P Cheng and I D Chang ldquoBuoyancy induced flows in asaturated porous medium adjacent to impermeable horizontalsurfacesrdquo International Journal of Heat and Mass Transfer vol19 no 11 pp 1267ndash1272 1976
[22] I D Chang and P Cheng ldquoMatched asymptotic expansionsfor free convection about an impermeable horizontal surfacein a porous mediumrdquo International Journal of Heat and MassTransfer vol 26 no 2 pp 163ndash174 1983
[23] D S Shiunlin and B Gebhart ldquoBuoyancy-induced flow adja-cent to a horizontal surface submerged in porous mediumsaturated with cold waterrdquo International Journal of Heat andMass Transfer vol 29 no 4 pp 611ndash623 1986
[24] J H Merkin and G Zhang ldquoOn the similarity solutions for freeconvection in a saturated porousmedium adjacent to imperme-able horizontal surfacesrdquoWarme- und Stoffubertragung vol 25no 3 pp 179ndash184 1990
[25] M A Chaudhary J H Merkin and I Pop ldquoNatural convectionfrom a horizontal permeable surface in a porous mediummdashnumerical and asymptotic solutionsrdquo Transport in PorousMedia vol 22 no 3 pp 327ndash344 1996
[26] R S R Gorla and A Chamkha ldquoNatural convective boundarylayer flow over a nonisothermal vertical plate embedded ina porous medium saturated with a nanofluidrdquo Nanoscale andMicroscale Thermophysical Engineering vol 15 no 2 pp 81ndash942011
[27] W A Khan and I Pop ldquoFree convection boundary layer flowpast a horizontal flat plate embedded in a porous medium filledwith a nanofluidrdquo Journal of Heat Transfer vol 133 no 9 ArticleID 094501 2011
[28] A Aziz W A Khan and I Pop ldquoFree convection boundarylayer flow past a horizontal flat plate embedded in porousmedium filled by nanofluid containing gyrotactic microorgan-ismsrdquo International Journal of Thermal Sciences vol 56 pp 48ndash57 2012
[29] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer New York NYUSA 2009
[30] A A Avramenko S G Kobzar I V Shevchuk A V Kuznetsovand L T Iwanisov ldquoSymmetry of turbulent boundary-layerflows investigation of different eddy viscosity modelsrdquo ActaMechanica vol 151 no 1-2 pp 1ndash14 2001
[31] A V Kuznetsov A A Avramenko and P Geng ldquoAnalyticalinvestigation of a falling plume caused by bioconvection ofoxytactic bacteria in a fluid saturated porous mediumrdquo Inter-national Journal of Engineering Science vol 42 no 5-6 pp 557ndash569 2004
[32] M Jalil S Asghar and M Mushtaq ldquoLie group analysis ofmixed convection flow with mass transfer over a stretchingsurface with suction or injectionrdquo Mathematical Problems inEngineering vol 2010 Article ID 264901 14 pages 2010
[33] M A A Hamad M J Uddin and A I M Ismail ldquoRadiationeffects on heat and mass transfer in MHD stagnation-pointflow over a permeable flat plate with thermal convectivesurface boundary condition temperature dependent viscosityand thermal conductivityrdquoNuclear Engineering and Design vol242 pp 194ndash200 2012
Mathematical Problems in Engineering 21
[34] A Aziz M J Uddin M A A Hamad and A I MdIsmail ldquoMHD flow over an inclined radiating plate with thetemperature-dependent thermal conductivity variable reactiveindex and heat generationrdquoHeat Transfer vol 41 no 3 pp 241ndash259 2012
[35] A G Hansen Similarity Analysis of Boundary Layer Problemsin Engineering Prentice Hall Englewood Cliffs NJ USA 1964
[36] W FAmesNonlinear Partial Differential Equations in Engineer-ing Academic Press New York NY USA 1972
[37] R Seshadri and T Y Na Group Invariance in EngineeringBoundary Value Problems Springer New York NY USA 1985
[38] D Shang Theory of Heat Transfer with Forced Convection FilmFlows vol 3 of Heat and Mass Transfer 2010
[40] E Erfani M M Rashidi and A B Parsa ldquoThe modifieddifferential transform method for solving off-centered stag-nation flow toward a rotating discrdquo International Journal ofComputational Methods vol 7 no 4 pp 655ndash670 2010
[41] M M Rashidi N Laraqi and A Basiri Parsa ldquoAnalyticalmodeling of heat convection in magnetized micropolar fluid byusing modified differential transform methodrdquo Heat Transfervol 40 no 3 pp 187ndash204 2011
[42] M M Rashidi and M Keimanesh ldquoUsing differential trans-form method and pade approximant for solving mhd flow ina laminar liquid film from a horizontal stretching surfacerdquoMathematical Problems in Engineering vol 2010 Article ID491319 14 pages 2010
[43] M M Rashidi and G Domairry ldquoNew analytical solution ofthe three-dimensional NavierStokes equationsrdquoModern PhysicsLetters B vol 23 no 26 pp 3147ndash3155 2009
[44] M M Rashidi M Keimanesh O A Beg and T K HungldquoMagnetohydrodynamic biorheological transport phenomenain a porous medium a simulation of magnetic blood flowcontrol and filtrationrdquo International Journal for NumericalMethods in Biomedical Engineering vol 27 no 6 pp 805ndash8212011
[45] S J Liao The proposed homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992
[46] M M Rashidi G Domairry and S Dinarvand ldquoApproximatesolutions for the Burger and regularized long wave equationsby means of the homotopy analysis methodrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 14 no 3 pp708ndash717 2009
[47] Z Ziabakhsh and G Domairry ldquoAnalytic solution of naturalconvection flow of a non-Newtonian fluid between two verticalflat plates using homotopy analysis methodrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 14 no 5pp 1868ndash1880 2009
[48] M M Rashidi S A Mohimanian Pour T Hayat and SObaidat ldquoAnalytic approximate solutions for steady flow over arotating disk in porousmediumwith heat transfer by homotopyanalysis methodrdquo Computers and Fluids vol 54 no 1 pp 1ndash92012
[49] M Sajid T Hayat and S Asghar ldquoComparison between theHAM and HPM solutions of thin film flows of non-Newtonianfluids on a moving beltrdquo Nonlinear Dynamics vol 50 no 1-2pp 27ndash35 2007
[50] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for the double-diffusive natural convective boundary
layer flow in a porous medium saturated by a nanofluidrdquoInternational Journal of Heat and Mass Transfer vol 54 no 1-3pp 374ndash378 2011
[51] K Vajravelu and A Hadjinicolaou ldquoConvective heat transferin an electrically conducting fluid at a stretching surfacewith uniform free streamrdquo International Journal of EngineeringScience vol 35 pp 1237ndash1244 1997
[52] S Mukhopadhyay and G C Layek ldquoEffects of variable fluidviscosity on flow past a heated stretching sheet embedded ina porous medium in presence of heat sourcesinkrdquo Meccanicavol 47 no 4 pp 863ndash876 2012
[53] S J Liao ldquoAn optimal homotopy-analysis approach for stronglynonlinear differential equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 8 pp 2003ndash20162010
[54] A Aziz ldquoA similarity solution for laminar thermal boundarylayer over a flat plate with a convective surface boundary con-ditionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1064ndash1068 2009
By using the symbolic software MATHEMATICA tosolve the system of linear equations (49) with the boundaryconditions (46) it can be obtained finally
1198651 (120578) = minus
4
3 (1 + 119887)
ℏ +
4119890minus120578
3 (1 + 119887)
ℏ +
4119873119903
3 (1 + 119887)
ℏ
+
4119887119873119903
3 (1 + 119887)
ℏ minus
4119873119903119890minus120578
3 (1 + 119887)
ℏ minus
4119887119873119903119890minus120578
3 (1 + 119887)
ℏ
+
2119890minus120578120578
3 (1 + 119887)
ℏ minus
2119873119903119890minus120578120578
3 (1 + 119887)
ℏ minus
2119887119873119903119890minus120578120578
3 (1 + 119887)
ℏ
Θ1 (120578) =119890minus2120578
2 (1 + 119887)
ℏ minus
119890minus120578
2(1 + 119887)2ℏ minus
119887119890minus120578
2(1 + 119887)2ℏ
minus
119873119887119890minus3120578
6(1 + 119887)2ℏ +
119887119873119887119890minus3120578
6(1 + 119887)2ℏ minus
119873119887119890minus120578
6(1 + 119887)2ℏ
minus
119887119873119887119890minus120578
6(1 + 119887)2ℏ +
119873119905119890minus3120578
6(1 + 119887)2ℏ minus
119873119905119890minus120578
6(1 + 119887)2ℏ
+
119876119890minus2120578
2 (1 + 119887)
ℏ minus
119876119890minus120578
2(1 + 119887)2ℏ minus
119887119876119890minus120578
2(1 + 119887)2ℏ
Φ1 (120578) =119890minus2120578
2 (1 + 119887)
ℏ +
119887119890minus2120578
2 (1 + 119887)
ℏ minus
119890minus120578
2 (1 + 119887)
ℏ
minus
119887119890minus120578
2 (1 + 119887)
ℏ minus
119870119890minus2120578
2 (1 + 119887)
ℏ minus
119887119870eminus2120578
2 (1 + 119887)
ℏ
+
119870119890minus120578
2 (1 + 119887)
ℏ +
119887119870119890minus120578
2 (1 + 119887)
ℏ +
119873119905
119873119887
119890minus2120578
2 (1 + 119887)
ℏ
minus
119873119905
119873119887
119890minus120578
2 (1 + 119887)
ℏ
(56)
The higher orders solutions of 119865(120578) Θ(120578) and Φ(120578) weretoo long to be mentioned here therefore they are showngraphically
51 Convergence of the HAM Solution As mentioned by Liao[45] HAM provides us with great freedom in choosing thesolution of a nonlinear problem by different base functionsThis has a great effect on the convergence region because theconvergence region and rate of a series are chiefly determinedby the base functions used to express the solution We usedseveral terms in evaluating the approximate solution 119865app asympsum119899119894=0 119865119894Θapp asymp sum
119899119894=0Θ119894Φapp asymp sum
119899119894=0Φ119894 note that the solution
series contains the auxiliary parameter ℏ which provides uswith a simple way to adjust and control the convergence ofthe solution series Generally by means of the so-called ℏ-curve that is a curve of a versus ℏ As pointed by Liao [45]the valid region of ℏ is a horizontal line segment Figures2 3 and 4 show the ℏ-curve with the various order of theHAM for 11989110158401015840(0) 1205791015840(0) 1206011015840(0) respectively when 119873119887 = 01
Figure 6 The residual error of 120579(120578) for (34) using 12th order ofapproximations
119873119905 = 01 119873119903 = 01 119876 = 01 Le = 10 119870 = 10 119899 = 10119887 = 00 It can be seen that when the order of series is 12 thesegment of the horizontal line is more than the other ordersFor example it can be found that for 11986510158401015840(0) the acceptablerange ofℏ is betweenminus05 andminus20 for 12th order of theHAMbut for 8th order of the HAM the acceptable range of ℏ isbetween minus10 and minus20 or minus10 and minus15 for 4th order of theHAM so horizontal line segment of 12th order of the HAM ismore than others Therefore it is straightforward to choosean appropriate range for ℏ which ensures the convergence
10 Mathematical Problems in Engineering
minus00002
120578
= minus10
= minus111 (optimal value)= minus12
Nt = 01Nr = 01
Q = 01Le = 10K = 10n = 10b = 00
0006
0004
0002
0
0 2 4 6 8 10
Nb = 01
Res (120601
(120578))
ℏ
ℏ
ℏ
Figure 7 The residual error of 120601(120578) for (35) using 12th order ofapproximations
of the solution series To choose optimal value of auxiliaryparameter the averaged residual errors (see [53] for moredetails) are defined as
119864119865119898 =1
119870
119870
sum
119894=0
[N119865(119898
sum
119894=0
119865119894 (119894Δ119909))]
2
119864Θ119898 =1
119870
119870
sum
119894=0
[NΘ(119898
sum
119894=0
Θ119894 (119894Δ119909))]
2
119864Φ119898 =1
119870
119870
sum
119894=0
[NΦ(119898
sum
119894=0
Φ119894 (119894Δ119909))]
2
(57)
where Δ119909 = 10119870 and 119870 = 20 For given order ofapproximation 119898 and the optimal values ℏ are given bythe minimum of 119864119898 corresponding to nonlinear algebraicequations
119889119864119865119898
119889ℏ
= 0
119889119864Θ119898
119889ℏ
= 0
119889119864Φ119898
119889ℏ
= 0 (58)
It is noted that the optimal value of ℏ is replaced intothe equations Table 1 shows optimal values obtained for theauxiliary parameter ℏ for various quantities of119873119905 119876 and 119899when119873119887 = 01119873119903 = 01 Le = 10 119870 = 10 119887 = 00 To see
the accuracy of the solutions the residual errors are definedfor the system as (for119898 order approximation)
Res119865119898
=
1198892119865119898
1198891205782minus
2
3
120578(
119889Θ119898
119889120578
minus 119873119903
119889Φ119898
119889120578
) (59)
ResΘ119898
=
1198892Θ119898
1198891205782+
1
3
119865119898
119889Θ119898
d120578+ 119873119887
119889Θ119898
119889120578
119889Φ119898
119889120578
+ 119873119905(
119889Θ119898
119889120578
)
2
+ 119876Θ119898
(60)
ResΦ119898
=
1198892Φ119898
1198891205782+
Le3
119865119898
119889Φ119898
119889120578
+
119873119905
119873119887
1198892Θ119898
1198891205782
minus 119870(
119889Φ119898
119889120578
)
119899
(61)
Figures 5 6 and 7 show the residual errors for 12th-orderdeformation solutions when 119873119887 = 01 119873119905 = 01 119873119903 = 01119876 = 01 Le = 10 119870 = 10 119899 = 10 119887 = 00 For example ℏ =minus145 has the minimum range of residual curve in Figure 5and so on
Graphical representation of results is very useful todemonstrate the efficiency and accuracy of the HAM for theabove problem
6 Comparisons and Verification
It is worth citing that for isothermal plate (119887 = 0) and in theabsence of internal heat generationabsorption (119876 = 0) andchemical reaction (119870 = 0) our problem reduces to Gorla andChamkha [26] and Khan and Pop [27] To verify the accuracyof our results the present results are compared in Table 2with Gorla and Chamkha [26] and are found to be in goodagreement
7 Physical Quantities
Theparameters of physical interest of the present problem arethe local skin friction factor 119862119891119909 the local Nusselt numberNu119909 and the local Sherwood number Sh119909 respectivelyPhysically 119862119891119909 indicates wall shear stress Nu119909 indicates therate of wall heat transfer whilst Sh119909 indicates the rate ofwall nanoparticle volume fractionThe following relations areused to find these quantities
119862119891119909 =2120583
1205881198802119903
(
120597119906
120597119910
)
119910=0
Nu119909 =minus119909
119879119891 minus 119879infin
(
120597119879
120597119910
)
119910=0
Sh119909 =minus119909
119862119908 minus 119862infin
(
120597119862
120597119910
)
119910=0
(62)
Mathematical Problems in Engineering 11
Table 2 Comparison of present solution with Gorla and Chamkha[26] for different values of buoyancy and nanofluid parameters
Present results (HAM) Gorla and Chamkha [26]minus1205791015840(0) minus120601
1015840(0) minus120579
1015840(0) minus120601
1015840(0)
Nr 119873119887 = 03119873119905 = 01 Le = 10 119887 = 119870 = 119876 = 0
01 032578 148242 326119864 minus 01 148416402 032385 146704 325119864 minus 01 146816103 032188 145125 322119864 minus 01 145266404 031985 143503 321119864 minus 01 143639205 031777 141833 319119864 minus 01 1419499Nt 119873119887 = 03119873119903 = 05 Le = 10 119887 = 119870 = 119876 = 0
01 031777 141833 319119864 minus 01 141949902 030486 141491 305119864 minus 01 141653603 02927 141561 293119864 minus 01 141686604 028125 141991 282119864 minus 01 142158205 027046 142737 271119864 minus 01 1429226Nb 119873119905 = 01119873119903 = 05 Le = 10 119887 = 119870 = 119876 = 0
01 03672 132611 368119864 minus 01 132745402 034271 139216 343119864 minus 01 139361503 031777 141833 319119864 minus 01 141949904 029399 143428 294119864 minus 01 143546405 027161 144598 272119864 minus 01 144772
Figure 10 Comparison of dimensionless concentration profile 120601(120578)obtained by various orders of the HAM
Here Ra119909 = 119892119870120573(1minus119862infin)Δ119879119909(120572119898]) is the local Rayleighnumber Pr = ]120572119898 is the Prandtl number and 119880119903 = (1 minus
119862infin)119892119870120573Δ119879120572119898 is the reference velocity
8 Results and Discussion
Graphical representation of results is very useful to discussthe physical features presented by the solutions This sectiondescribes the influence of some interesting parameters on
12 Mathematical Problems in Engineering
0 2 4 6 8 10
02
04
06
08
1
Nr = 01Numerical
Numerical
NumericalNr = 03Nr = 05
f998400 (120578)
120578
Nt = 01Nb = 01Q = 01Le = 10K = 10n = 10b = 00
Figure 11 Effect of buoyancy ratio119873119903 on the dimensionless velocityprofile
0 2 4 6 8 10
02
04
06
08
1
Nt = 01Nb = 01Q = 01Le = 10K = 10n = 10b = 00
Nr = 01Numerical
Numerical
NumericalNr = 03Nr = 05
120578
120579(120578
)
Figure 12 Effect of buoyancy ratio 119873119903 on the dimensionlesstemperature profile
the velocity and temperature fields Equations (33) to (35)with boundary conditions in (36) were solved analytically byHAM and numerically using Runge-Kutta-Fehlberg fourth-fifth order proposed by Aziz [54]
Figures 8 9 and 10 respectively for 1198911015840(120578) 120579(120578) and 120601(120578)show the comparisons between various order approximationof the optimal HAM and the numerical solutions for the case
0 2 4 6 8 10
02
04
06
08
1
Nr = 01Numerical
Numerical
NumericalNr = 03Nr = 05
Nt = 01Nb = 01Q = 01Le = 10K = 10n = 10b = 00
120578
120601(120578
)Figure 13 Effect of buoyancy ratio 119873119903 on the dimensionlessconcentration profile
0 2 4 6 8 10
02
04
06
08
1
12
NumericalNumerical
Numerical
f998400 (120578
)
120578
Nb = 01
Nb = 03
Nb = 05
Nt = 01Nr = 01Q = 01Le = 10K = 10n = 10b = 00
Figure 14 Effect of Brownian motion 119873119887 on the dimensionlessvelocity profile
119873119887 = 01 119873119905 = 01 119873119903 = 01 119876 = 01 Le = 10 119870 = 10119899 = 10 119887 = 00 For better presentation these results arepresented in Tables 3 4 and 5 Also Tables 6 7 and 8 presentthe comparisons between various order approximation of theoptimal HAM and the numerical solutions for the case119873119887 =03 119873119905 = 005 119873119903 = 03 119876 = 01 Le = 20 119870 = 05
Mathematical Problems in Engineering 13
0 2 4 6 8 10
02
04
06
08
1
0
120578
NumericalNumerical
Numerical
Nb = 01
Nb = 03
Nb = 05
Nt = 01Nr = 01Q = 01Le = 10K = 10n = 10b = 00
120579(120578
)
Figure 15 Effect of Brownian motion 119873119887 on the dimensionlesstemperature profile
02
04
06
08
1
00
2 4 6 8 10120578
NumericalNumerical
Numerical
Nb = 01
Nb = 03
Nb = 05
Nt = 01Nr = 01Q = 01Le = 10K = 10n = 10b = 00
120601(120578
)
Figure 16 Effect of Brownian motion 119873119887 on the dimensionlessconcentration profile
119899 = 20 119887 = 01 It is observed that the results of 12th-order approximation of the optimal HAM are very close tothe numerical solutions which confirm the validity of thesemethods
In the following figures effects of various physicalparameters on the dimensionless velocity temperature and
Table 3 Comparison of values of 1198911015840(120578) obtained by various ordersof the HAM with numerical solution when 119873119887 = 01119873119905 =
temperature profiles will be investigated These results havebeen obtained by 12th order of the HAM and have beenvalidated by numerical results Figures 11ndash13 respectivelyrepresent that the comparison of solutions of 1198911015840(120578) 120579(120578)and 120601(120578) for fix values 119873119887 = 01 119873119905 = 01 119876 = 01Le = 10 119870 = 10 119899 = 10 119887 = 00 and different values ofbuoyancy ratio119873119903 In Figure 11 it is clear that velocity of thefluid tremendously decreases in the near of horizontal platewith an increase in the buoyancy ratio It is observed that thetemperature increases slightly but concentration of the fluiddoes not vary sensibly (Figures 12 and 13) The effects of theBrownian motion 119873119887 are depicted in Figures 14 15 and 16when119873119903 = 01119873119905 = 01119876 = 01 Le = 10119870 = 10 119899 = 10
14 Mathematical Problems in Engineering
Table 6 Comparison of values of 1198911015840(120578) obtained by various ordersof the HAM with numerical solution when 119873119887 = 03119873119905 =
119887 = 00 If 119873119887 increases 1198911015840(120578) and 120579(120578) increase lightly and120601(120578) decreases The dimensionless velocity dimensionlesstemperature and dimensionless concentration profiles fordifferent values of thermophoresis 119873119905 with constant values119873119903 = 01 119873119887 = 01 119876 = 01 Le = 10 119870 = 10119899 = 10 119887 = 00 are presented in Figures 17 18 and19 It is observed that the velocity and temperature of thefluid are not impressible from119873119905 but concentration increasesslightly In Figures 20 21 and 22 respectively comparison ofsolutions of 1198911015840(120578) 120579(120578) and 120601(120578) for 119873119903 = 01 119873119887 = 01119873119905 = 01 Le = 10 119870 = 10 119899 = 10 119887 = 00 anddifferent values of heat generationabsorption parameter 119876are shown It is observed that in presence of heat source1198911015840(120578) 120579(120578) both increase while heat sink causes that 1198911015840(120578)
120579(120578) both decrease extremely These effects are more visible
Table 9 Values of reduced Nusselt number and Sherwood numberobtained by HAM for different values of the parameters Le 119870 119899 119887and 119876 when119873119887 = 119873119903 = 119873119905 = 01
Figure 17 Effect of thermophoresis 119873119905 on the dimensionlessvelocity profile
for temperature distribution For 120601(120578) the effect of changing119876 is seen to be almost insignificant This is in agreementwith the physical factThe dimensionless profiles for differentvalues of Lewis number Le with constant parameters 119873119903 =01 119873119887 = 01 119873119905 = 01 119876 = 01 119870 = 10 119899 = 10119887 = 00 are shown in Figures 23 24 and 25 It is clear that forthese constant parameters variation of Lewis number has anyconsiderable effect on the velocity and temperature profilesbut concentration profile decreases with the increase of Lewisnumber What is similar to these results can be detected inFigures 26 27 and 28 for chemical reaction parameter 119870Figures 29 30 and 31 respectively present the comparisonof solutions of 1198911015840(120578) 120579(120578) and 120601(120578) for 119873119903 = 05 119873119887 = 05119873119905 = 01 119876 = 001 Le = 10 119870 = 10 119887 = 00 and differentvalues of order of chemical reaction 119899 It is observed that
Mathematical Problems in Engineering 15
0 2 4 6 80
1
120578
Nt = 00
Nt = 005
Nt = 10NumericalNumerical
Numerical
08
06
04
02
10
Q = 01Le = 10K = 10n = 10b = 00
Nb = 01
120579(120578
)
Nr = 01
Figure 18 Effect of thermophoresis 119873119905 on the dimensionlesstemperature profile
0 2 4 6 80
1
120601(120578
)
120578
Nt = 00
Nt = 005
Nt = 10NumericalNumerical
Numerical
08
06
04
02
10
Q = 01Le = 10K = 10n = 10b = 00
Nb = 01Nr = 01
Figure 19 Effect of thermophoresis 119873119905 on the dimensionlessconcentration profile
for higher order of chemical reaction velocity decreases andtemperature and concentration profiles increase extremelyUltimately Figures 32 33 and 34 depict the effect of thermalslip parameter 119887 on velocity temperature and concentrationfunctions when 119873119903 = 01 119873119887 = 01 119873119905 = 01 119876 = 01Le = 20 119870 = 15 119899 = 10 It is seen that the velocityand temperature of the fluid decrease with the increase of the
0 2 4 6 80
1
12
10
02
04
06
08
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numericalf998400 (120578
)
120578
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 20 Effect of generationabsorption heat parameter119876 on thedimensionless velocity profile
0 2 4 6 8 100
02
04
06
08
1
120579(120578
)
120578
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numerical
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 21 Effect of generationabsorption heat parameter 119876 on thedimensionless temperature profile
thermal slip parameter in the near of horizontal plate whilethe concentration of the fluid does not vary patently
Also for investigation of the parameters of physicalinterest Table 9 is presented In this table numerical valuesof reduced Nusselt number and Sherwood number obtainedby HAM for different values of the parameters 119890 119870 119899 119887 and119876 can be compared when119873119887 = 119873119903 = 119873119905 = 01
16 Mathematical Problems in Engineering
10
02
04
06
08
0 2 4 6 80
1
120578
120601(120578
)
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numerical
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 22 Effect of generationabsorption heat parameter119876 on thedimensionless concentration profile
0 2 4 6 80
1
12
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 23 Effect of Lewis number Le on the dimensionless velocityprofile
9 Conclusions
In this paper we studied the steady laminar incompressiblefree convective flow of a nanofluid past a chemically reactingupward facing horizontal plate in porous medium takinginto account heat generation and the thermal slip boundarycondition
0 2 4 6 80
1
120579(120578
)
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 24 Effect of Lewis number Le on the dimensionlesstemperature profile
0 2 4 6 80
1
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 25 Effect of Lewis number Le on the dimensionlessconcentration profile
The governing partial differential equations have beentransformed by similarity transformations into a system ofordinary differential equations which are solved by OHAMand numerical method (fourth-order Runge-Kutta schemewith the shooting method) Dimensionless velocity temper-ature and concentration functions are presented for various
Mathematical Problems in Engineering 17
0 2 4 6 80
1
12
K = 00
K = 05
K = 10NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 26 Effect of chemical reaction parameter 119870 on the dimen-sionless velocity profile
0 2 4 6 80
1
K = 00
K = 05
K = 10NumericalNumerical
Numerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 27 Effect of chemical reaction parameter 119870 on the dimen-sionless temperature profile
values of parameters for the problem for example buoy-ancy ratio 119873119903 Brownian motion 119873119887 thermophoresis 119873119905heat generationabsorption parameter 119876 Lewis number Lechemical reaction parameter119870 order of chemical reaction 119899and thermal slip parameter 119887 From the present investigationthe following may be concluded
(i) The velocity 1198911015840(120578) increases with increasing 119873119887 anddecreases with increasing119873119903 119899 and 119887
0 2 4 6 80
1
K = 00
K = 05
K = 10NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 28 Effect of chemical reaction parameter 119870 on the dimen-sionless concentration profile
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 29 Effect of order of chemical reaction 119899 on the dimension-less velocity profile
(ii) The temperature 120579(120578) increases with 119873119903 119873119887 119899 anddecreases with 119887
(iii) Parameters of119873119905 Le and119870 have no effect on velocityand temperature profiles
(iv) The concentration distribution 120601(120578) increases withincreasing N119905 119899 and decreases with 119873119887 Le and 119870but119873119903 119876 and 119887 are not affected by it
18 Mathematical Problems in Engineering
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 30 Effect of order of chemical reaction 119899 on the dimension-less temperature profile
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 31 Effect of order of chemical reaction 119899 on the dimension-less concentration profile
(v) In presence of heat source (119876 gt 0) 1198911015840(120578) 120579(120578) bothincrease while heat sink (119876 lt 0) causes that 1198911015840(120578)120579(120578) both decrease extremely These effects are morevisible for temperature distribution
In addition numerical results for the reducedNusselt andSherwood numbers are tabulated It is concluded that thereduced Nusselt number increases with the Lewis numberand reaction parameter whist it decreases with the order of
0 2 4 6 8
1
12
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 32 Effect of order of thermal slip parameter 119887 on thedimensionless velocity profile
0 2 4 6 80
1
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 33 Effect of order of thermal slip parameter 119887 on thedimensionless temperature profile
chemical reaction thermal slip and generation parametersIt is further concluded that the reduced Sherwood numberenhances with the Lewis number generation and reactionparameters whist it suppresses with thermal slip and orderof chemical reaction
The convergence of the solution series and the power ofHAM in controlling and adjusting the convergence regionand rate of solution series were discussed The proper range
Mathematical Problems in Engineering 19
0 2 4 6 80
1
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 34 Effect of order of thermal slip parameter 119887 on thedimensionless concentration profile
of auxiliary parameter ℏ to ensure the convergence of thesolution series was obtained through the so called ℏ-curveThe HAM provides us with a convenient way to control theconvergence of approximation series which is a fundamentalqualitative difference in analysis between the HAM andother methods The numerical results of the above problemsdisplay a fast convergence with minimal calculations Thisshows that the HAM is a very efficient method Finally theagreement between analytical and numerical results of thepresent study with previous published results is excellent
Nomenclature
119887 Thermal slip parameter119862 Dimensional concentration119862119891 Skin friction factor1198631 Thermal slip factor119863119861 Brownian diffusion coefficient119863119879 Thermophoretic diffusion coefficient119891 Dimensionless velocity functions119892 Gravitation acceleration119870 Chemical reaction parameter119896 Permeability of the porous media1198960 The constant reaction rate119896119898 Effective thermal conductivity of the porous
medium119896(119909) Variable reaction rateL Linear operator of the HAM119871 Length of horizontal plateLe Lewis numberN Nonlinear operator of the HAM
119899 Order of chemical reaction119873119887 Brownian motion119873119903 Buoyancy ratio119873119905 ThermophoresisNu Nusselt number119875 PressurePr Prandtl number119876 Heat generationabsorption parameter1198760 Heat generationabsorption constant119902 Internal heat generation rate
Ra Rayleigh numberSh Sherwood number119879 Temperature119905 Time119906 Velocity in 119909-direction119880119903 Reference velocity Velocity vectorV Velocity in 119910-direction119909 Distance along the surface119910 Distance normal to the surface
Greek Letters
120572119898 Thermal diffusivity120573 Volumetric expansion coefficient of nanofluid120576 Porosity of the porous media120601 Dimensionless concentration120578 Similarity variable120583 Dynamic viscosity120579 Dimensionless temperature120588119891 Density of the base fluid120588119901 Density of the nanoparticles(120588119862)119891 Effective heat capacity of the fluid(120588119862)119875 Effective heat capacity of the nanoparticle material120595 Stream function
Subscript Superscript
infin Conditions far away from the surface1015840 Differentiation with respect to 120578
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
M M Rashidi wishes to thank the Centre for DifferentialEquations Continuum Mechanics and Applications Schoolof Computational and Applied Mathematics University ofthe Witwatersrand for the hospitality during his visit EMomoniat thanks theNational Research Foundation of SouthAfrica for their support
20 Mathematical Problems in Engineering
References
[1] E Abu-Nada F O Hakan and I Pop ldquoBuoyancy inducedflow in a nanofluid filled enclosure partially exposed to forcedconvectionrdquo Superlattices and Microstructures vol 51 no 3 pp381ndash395 2012
[2] S Kakac and A Pramuanjaroenkij ldquoReview of convective heattransfer enhancement with nanofluidsrdquo International Journal ofHeat and Mass Transfer vol 52 no 13-14 pp 3187ndash3196 2009
[3] J A Eastman S U S Choi S Li W Yu and L J ThompsonldquoAnomalously increased effective thermal conductivities ofethylene glycol-based nanofluids containing copper nanoparti-clesrdquo Applied Physics Letters vol 78 no 6 pp 718ndash720 2001
[4] Y Xuan and Q Li ldquoHeat transfer enhancement of nanofluidsrdquoInternational Journal of Heat and Fluid Flow vol 21 no 1 pp58ndash64 2000
[5] S K Das N Putra P Thiesen and W Roetzel ldquoTemperaturedependence of thermal conductivity enhancement for nanoflu-idsrdquo Journal of Heat Transfer vol 125 no 4 pp 567ndash574 2003
[6] D H Kumar H E Patel V R R Kumar T SundararajanT Pradeep and S K Das ldquoModel for heat conduction innanofluidsrdquo Physical Review Letters vol 93 no 14 Article ID144301 2004
[7] H E Patel T Sundararajan T Pradeep A Dasgupta NDasgupta and S K Das ldquoA micro-convection model forthermal conductivity of nanofluidsrdquo Pramana vol 65 no 5 pp863ndash869 2005
[8] A V Kuznetsov ldquoNon-oscillatory and oscillatory nanofluidbio-thermal convection in a horizontal layer of finite depthrdquoEuropean Journal of Mechanics vol 30 no 2 pp 156ndash165 2011
[9] M Napoli J C T Eijkel and S Pennathur ldquoNanofluidictechnology for biomolecule applications a critical reviewrdquo Labon a Chip vol 10 no 8 pp 957ndash985 2010
[10] J C Crepeau and R Clarksean ldquoSimilarity solutions of naturalconvection with internal heat generationrdquo Journal of HeatTransfer vol 119 no 1 pp 183ndash185 1997
[11] A Z Sahin ldquoTransient heat conduction in semi-infinite solidwith spatially decaying exponential heat generationrdquo Interna-tional Communications in Heat andMass Transfer vol 19 no 3pp 349ndash358 1992
[12] O D Makind and A Aziz ldquoMixed convection from a con-vectively heated vertical plate to a fluid with internal heatgenerationrdquo Journal of Heat Transfer vol 133 Article ID 1225016 pages 2011
[13] J Buongiorno ldquoConvective transport in nanofluidsrdquo Journal ofHeat Transfer vol 128 no 3 pp 240ndash250 2006
[14] AV Kuznetsov andDANield ldquoNatural convective boundary-layer flow of a nanofluid past a vertical platerdquo InternationalJournal of Thermal Sciences vol 49 no 2 pp 243ndash247 2010
[15] L B Godson D Raja L D Mohan and S WongwisescldquoEnhancement of heat transfer using nanofluids-an overviewrdquoRenewable and Sustainable Energy Reviews vol 14 no 2 pp629ndash641 2010
[16] D A Nield and A BejanConvection in PorousMedia SpringerNew York NY USA 3rd edition 2006
[17] P Vadasz Emerging Topics in Heat and Mass Transfer in PorousMedia Springer New York NY USA 2008
[18] K Vafai Porous Media Applications in Biological Systems andBiotechnology CRC Press New York NY USA 2010
[19] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application to
heat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977
[20] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for natural convective boundary-layer flow in a porousmedium saturated by a nanofluidrdquo International Journal of Heatand Mass Transfer vol 52 no 25-26 pp 5792ndash5795 2009
[21] P Cheng and I D Chang ldquoBuoyancy induced flows in asaturated porous medium adjacent to impermeable horizontalsurfacesrdquo International Journal of Heat and Mass Transfer vol19 no 11 pp 1267ndash1272 1976
[22] I D Chang and P Cheng ldquoMatched asymptotic expansionsfor free convection about an impermeable horizontal surfacein a porous mediumrdquo International Journal of Heat and MassTransfer vol 26 no 2 pp 163ndash174 1983
[23] D S Shiunlin and B Gebhart ldquoBuoyancy-induced flow adja-cent to a horizontal surface submerged in porous mediumsaturated with cold waterrdquo International Journal of Heat andMass Transfer vol 29 no 4 pp 611ndash623 1986
[24] J H Merkin and G Zhang ldquoOn the similarity solutions for freeconvection in a saturated porousmedium adjacent to imperme-able horizontal surfacesrdquoWarme- und Stoffubertragung vol 25no 3 pp 179ndash184 1990
[25] M A Chaudhary J H Merkin and I Pop ldquoNatural convectionfrom a horizontal permeable surface in a porous mediummdashnumerical and asymptotic solutionsrdquo Transport in PorousMedia vol 22 no 3 pp 327ndash344 1996
[26] R S R Gorla and A Chamkha ldquoNatural convective boundarylayer flow over a nonisothermal vertical plate embedded ina porous medium saturated with a nanofluidrdquo Nanoscale andMicroscale Thermophysical Engineering vol 15 no 2 pp 81ndash942011
[27] W A Khan and I Pop ldquoFree convection boundary layer flowpast a horizontal flat plate embedded in a porous medium filledwith a nanofluidrdquo Journal of Heat Transfer vol 133 no 9 ArticleID 094501 2011
[28] A Aziz W A Khan and I Pop ldquoFree convection boundarylayer flow past a horizontal flat plate embedded in porousmedium filled by nanofluid containing gyrotactic microorgan-ismsrdquo International Journal of Thermal Sciences vol 56 pp 48ndash57 2012
[29] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer New York NYUSA 2009
[30] A A Avramenko S G Kobzar I V Shevchuk A V Kuznetsovand L T Iwanisov ldquoSymmetry of turbulent boundary-layerflows investigation of different eddy viscosity modelsrdquo ActaMechanica vol 151 no 1-2 pp 1ndash14 2001
[31] A V Kuznetsov A A Avramenko and P Geng ldquoAnalyticalinvestigation of a falling plume caused by bioconvection ofoxytactic bacteria in a fluid saturated porous mediumrdquo Inter-national Journal of Engineering Science vol 42 no 5-6 pp 557ndash569 2004
[32] M Jalil S Asghar and M Mushtaq ldquoLie group analysis ofmixed convection flow with mass transfer over a stretchingsurface with suction or injectionrdquo Mathematical Problems inEngineering vol 2010 Article ID 264901 14 pages 2010
[33] M A A Hamad M J Uddin and A I M Ismail ldquoRadiationeffects on heat and mass transfer in MHD stagnation-pointflow over a permeable flat plate with thermal convectivesurface boundary condition temperature dependent viscosityand thermal conductivityrdquoNuclear Engineering and Design vol242 pp 194ndash200 2012
Mathematical Problems in Engineering 21
[34] A Aziz M J Uddin M A A Hamad and A I MdIsmail ldquoMHD flow over an inclined radiating plate with thetemperature-dependent thermal conductivity variable reactiveindex and heat generationrdquoHeat Transfer vol 41 no 3 pp 241ndash259 2012
[35] A G Hansen Similarity Analysis of Boundary Layer Problemsin Engineering Prentice Hall Englewood Cliffs NJ USA 1964
[36] W FAmesNonlinear Partial Differential Equations in Engineer-ing Academic Press New York NY USA 1972
[37] R Seshadri and T Y Na Group Invariance in EngineeringBoundary Value Problems Springer New York NY USA 1985
[38] D Shang Theory of Heat Transfer with Forced Convection FilmFlows vol 3 of Heat and Mass Transfer 2010
[40] E Erfani M M Rashidi and A B Parsa ldquoThe modifieddifferential transform method for solving off-centered stag-nation flow toward a rotating discrdquo International Journal ofComputational Methods vol 7 no 4 pp 655ndash670 2010
[41] M M Rashidi N Laraqi and A Basiri Parsa ldquoAnalyticalmodeling of heat convection in magnetized micropolar fluid byusing modified differential transform methodrdquo Heat Transfervol 40 no 3 pp 187ndash204 2011
[42] M M Rashidi and M Keimanesh ldquoUsing differential trans-form method and pade approximant for solving mhd flow ina laminar liquid film from a horizontal stretching surfacerdquoMathematical Problems in Engineering vol 2010 Article ID491319 14 pages 2010
[43] M M Rashidi and G Domairry ldquoNew analytical solution ofthe three-dimensional NavierStokes equationsrdquoModern PhysicsLetters B vol 23 no 26 pp 3147ndash3155 2009
[44] M M Rashidi M Keimanesh O A Beg and T K HungldquoMagnetohydrodynamic biorheological transport phenomenain a porous medium a simulation of magnetic blood flowcontrol and filtrationrdquo International Journal for NumericalMethods in Biomedical Engineering vol 27 no 6 pp 805ndash8212011
[45] S J Liao The proposed homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992
[46] M M Rashidi G Domairry and S Dinarvand ldquoApproximatesolutions for the Burger and regularized long wave equationsby means of the homotopy analysis methodrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 14 no 3 pp708ndash717 2009
[47] Z Ziabakhsh and G Domairry ldquoAnalytic solution of naturalconvection flow of a non-Newtonian fluid between two verticalflat plates using homotopy analysis methodrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 14 no 5pp 1868ndash1880 2009
[48] M M Rashidi S A Mohimanian Pour T Hayat and SObaidat ldquoAnalytic approximate solutions for steady flow over arotating disk in porousmediumwith heat transfer by homotopyanalysis methodrdquo Computers and Fluids vol 54 no 1 pp 1ndash92012
[49] M Sajid T Hayat and S Asghar ldquoComparison between theHAM and HPM solutions of thin film flows of non-Newtonianfluids on a moving beltrdquo Nonlinear Dynamics vol 50 no 1-2pp 27ndash35 2007
[50] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for the double-diffusive natural convective boundary
layer flow in a porous medium saturated by a nanofluidrdquoInternational Journal of Heat and Mass Transfer vol 54 no 1-3pp 374ndash378 2011
[51] K Vajravelu and A Hadjinicolaou ldquoConvective heat transferin an electrically conducting fluid at a stretching surfacewith uniform free streamrdquo International Journal of EngineeringScience vol 35 pp 1237ndash1244 1997
[52] S Mukhopadhyay and G C Layek ldquoEffects of variable fluidviscosity on flow past a heated stretching sheet embedded ina porous medium in presence of heat sourcesinkrdquo Meccanicavol 47 no 4 pp 863ndash876 2012
[53] S J Liao ldquoAn optimal homotopy-analysis approach for stronglynonlinear differential equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 8 pp 2003ndash20162010
[54] A Aziz ldquoA similarity solution for laminar thermal boundarylayer over a flat plate with a convective surface boundary con-ditionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1064ndash1068 2009
Figure 7 The residual error of 120601(120578) for (35) using 12th order ofapproximations
of the solution series To choose optimal value of auxiliaryparameter the averaged residual errors (see [53] for moredetails) are defined as
119864119865119898 =1
119870
119870
sum
119894=0
[N119865(119898
sum
119894=0
119865119894 (119894Δ119909))]
2
119864Θ119898 =1
119870
119870
sum
119894=0
[NΘ(119898
sum
119894=0
Θ119894 (119894Δ119909))]
2
119864Φ119898 =1
119870
119870
sum
119894=0
[NΦ(119898
sum
119894=0
Φ119894 (119894Δ119909))]
2
(57)
where Δ119909 = 10119870 and 119870 = 20 For given order ofapproximation 119898 and the optimal values ℏ are given bythe minimum of 119864119898 corresponding to nonlinear algebraicequations
119889119864119865119898
119889ℏ
= 0
119889119864Θ119898
119889ℏ
= 0
119889119864Φ119898
119889ℏ
= 0 (58)
It is noted that the optimal value of ℏ is replaced intothe equations Table 1 shows optimal values obtained for theauxiliary parameter ℏ for various quantities of119873119905 119876 and 119899when119873119887 = 01119873119903 = 01 Le = 10 119870 = 10 119887 = 00 To see
the accuracy of the solutions the residual errors are definedfor the system as (for119898 order approximation)
Res119865119898
=
1198892119865119898
1198891205782minus
2
3
120578(
119889Θ119898
119889120578
minus 119873119903
119889Φ119898
119889120578
) (59)
ResΘ119898
=
1198892Θ119898
1198891205782+
1
3
119865119898
119889Θ119898
d120578+ 119873119887
119889Θ119898
119889120578
119889Φ119898
119889120578
+ 119873119905(
119889Θ119898
119889120578
)
2
+ 119876Θ119898
(60)
ResΦ119898
=
1198892Φ119898
1198891205782+
Le3
119865119898
119889Φ119898
119889120578
+
119873119905
119873119887
1198892Θ119898
1198891205782
minus 119870(
119889Φ119898
119889120578
)
119899
(61)
Figures 5 6 and 7 show the residual errors for 12th-orderdeformation solutions when 119873119887 = 01 119873119905 = 01 119873119903 = 01119876 = 01 Le = 10 119870 = 10 119899 = 10 119887 = 00 For example ℏ =minus145 has the minimum range of residual curve in Figure 5and so on
Graphical representation of results is very useful todemonstrate the efficiency and accuracy of the HAM for theabove problem
6 Comparisons and Verification
It is worth citing that for isothermal plate (119887 = 0) and in theabsence of internal heat generationabsorption (119876 = 0) andchemical reaction (119870 = 0) our problem reduces to Gorla andChamkha [26] and Khan and Pop [27] To verify the accuracyof our results the present results are compared in Table 2with Gorla and Chamkha [26] and are found to be in goodagreement
7 Physical Quantities
Theparameters of physical interest of the present problem arethe local skin friction factor 119862119891119909 the local Nusselt numberNu119909 and the local Sherwood number Sh119909 respectivelyPhysically 119862119891119909 indicates wall shear stress Nu119909 indicates therate of wall heat transfer whilst Sh119909 indicates the rate ofwall nanoparticle volume fractionThe following relations areused to find these quantities
119862119891119909 =2120583
1205881198802119903
(
120597119906
120597119910
)
119910=0
Nu119909 =minus119909
119879119891 minus 119879infin
(
120597119879
120597119910
)
119910=0
Sh119909 =minus119909
119862119908 minus 119862infin
(
120597119862
120597119910
)
119910=0
(62)
Mathematical Problems in Engineering 11
Table 2 Comparison of present solution with Gorla and Chamkha[26] for different values of buoyancy and nanofluid parameters
Present results (HAM) Gorla and Chamkha [26]minus1205791015840(0) minus120601
1015840(0) minus120579
1015840(0) minus120601
1015840(0)
Nr 119873119887 = 03119873119905 = 01 Le = 10 119887 = 119870 = 119876 = 0
01 032578 148242 326119864 minus 01 148416402 032385 146704 325119864 minus 01 146816103 032188 145125 322119864 minus 01 145266404 031985 143503 321119864 minus 01 143639205 031777 141833 319119864 minus 01 1419499Nt 119873119887 = 03119873119903 = 05 Le = 10 119887 = 119870 = 119876 = 0
01 031777 141833 319119864 minus 01 141949902 030486 141491 305119864 minus 01 141653603 02927 141561 293119864 minus 01 141686604 028125 141991 282119864 minus 01 142158205 027046 142737 271119864 minus 01 1429226Nb 119873119905 = 01119873119903 = 05 Le = 10 119887 = 119870 = 119876 = 0
01 03672 132611 368119864 minus 01 132745402 034271 139216 343119864 minus 01 139361503 031777 141833 319119864 minus 01 141949904 029399 143428 294119864 minus 01 143546405 027161 144598 272119864 minus 01 144772
Figure 10 Comparison of dimensionless concentration profile 120601(120578)obtained by various orders of the HAM
Here Ra119909 = 119892119870120573(1minus119862infin)Δ119879119909(120572119898]) is the local Rayleighnumber Pr = ]120572119898 is the Prandtl number and 119880119903 = (1 minus
119862infin)119892119870120573Δ119879120572119898 is the reference velocity
8 Results and Discussion
Graphical representation of results is very useful to discussthe physical features presented by the solutions This sectiondescribes the influence of some interesting parameters on
12 Mathematical Problems in Engineering
0 2 4 6 8 10
02
04
06
08
1
Nr = 01Numerical
Numerical
NumericalNr = 03Nr = 05
f998400 (120578)
120578
Nt = 01Nb = 01Q = 01Le = 10K = 10n = 10b = 00
Figure 11 Effect of buoyancy ratio119873119903 on the dimensionless velocityprofile
0 2 4 6 8 10
02
04
06
08
1
Nt = 01Nb = 01Q = 01Le = 10K = 10n = 10b = 00
Nr = 01Numerical
Numerical
NumericalNr = 03Nr = 05
120578
120579(120578
)
Figure 12 Effect of buoyancy ratio 119873119903 on the dimensionlesstemperature profile
the velocity and temperature fields Equations (33) to (35)with boundary conditions in (36) were solved analytically byHAM and numerically using Runge-Kutta-Fehlberg fourth-fifth order proposed by Aziz [54]
Figures 8 9 and 10 respectively for 1198911015840(120578) 120579(120578) and 120601(120578)show the comparisons between various order approximationof the optimal HAM and the numerical solutions for the case
0 2 4 6 8 10
02
04
06
08
1
Nr = 01Numerical
Numerical
NumericalNr = 03Nr = 05
Nt = 01Nb = 01Q = 01Le = 10K = 10n = 10b = 00
120578
120601(120578
)Figure 13 Effect of buoyancy ratio 119873119903 on the dimensionlessconcentration profile
0 2 4 6 8 10
02
04
06
08
1
12
NumericalNumerical
Numerical
f998400 (120578
)
120578
Nb = 01
Nb = 03
Nb = 05
Nt = 01Nr = 01Q = 01Le = 10K = 10n = 10b = 00
Figure 14 Effect of Brownian motion 119873119887 on the dimensionlessvelocity profile
119873119887 = 01 119873119905 = 01 119873119903 = 01 119876 = 01 Le = 10 119870 = 10119899 = 10 119887 = 00 For better presentation these results arepresented in Tables 3 4 and 5 Also Tables 6 7 and 8 presentthe comparisons between various order approximation of theoptimal HAM and the numerical solutions for the case119873119887 =03 119873119905 = 005 119873119903 = 03 119876 = 01 Le = 20 119870 = 05
Mathematical Problems in Engineering 13
0 2 4 6 8 10
02
04
06
08
1
0
120578
NumericalNumerical
Numerical
Nb = 01
Nb = 03
Nb = 05
Nt = 01Nr = 01Q = 01Le = 10K = 10n = 10b = 00
120579(120578
)
Figure 15 Effect of Brownian motion 119873119887 on the dimensionlesstemperature profile
02
04
06
08
1
00
2 4 6 8 10120578
NumericalNumerical
Numerical
Nb = 01
Nb = 03
Nb = 05
Nt = 01Nr = 01Q = 01Le = 10K = 10n = 10b = 00
120601(120578
)
Figure 16 Effect of Brownian motion 119873119887 on the dimensionlessconcentration profile
119899 = 20 119887 = 01 It is observed that the results of 12th-order approximation of the optimal HAM are very close tothe numerical solutions which confirm the validity of thesemethods
In the following figures effects of various physicalparameters on the dimensionless velocity temperature and
Table 3 Comparison of values of 1198911015840(120578) obtained by various ordersof the HAM with numerical solution when 119873119887 = 01119873119905 =
temperature profiles will be investigated These results havebeen obtained by 12th order of the HAM and have beenvalidated by numerical results Figures 11ndash13 respectivelyrepresent that the comparison of solutions of 1198911015840(120578) 120579(120578)and 120601(120578) for fix values 119873119887 = 01 119873119905 = 01 119876 = 01Le = 10 119870 = 10 119899 = 10 119887 = 00 and different values ofbuoyancy ratio119873119903 In Figure 11 it is clear that velocity of thefluid tremendously decreases in the near of horizontal platewith an increase in the buoyancy ratio It is observed that thetemperature increases slightly but concentration of the fluiddoes not vary sensibly (Figures 12 and 13) The effects of theBrownian motion 119873119887 are depicted in Figures 14 15 and 16when119873119903 = 01119873119905 = 01119876 = 01 Le = 10119870 = 10 119899 = 10
14 Mathematical Problems in Engineering
Table 6 Comparison of values of 1198911015840(120578) obtained by various ordersof the HAM with numerical solution when 119873119887 = 03119873119905 =
119887 = 00 If 119873119887 increases 1198911015840(120578) and 120579(120578) increase lightly and120601(120578) decreases The dimensionless velocity dimensionlesstemperature and dimensionless concentration profiles fordifferent values of thermophoresis 119873119905 with constant values119873119903 = 01 119873119887 = 01 119876 = 01 Le = 10 119870 = 10119899 = 10 119887 = 00 are presented in Figures 17 18 and19 It is observed that the velocity and temperature of thefluid are not impressible from119873119905 but concentration increasesslightly In Figures 20 21 and 22 respectively comparison ofsolutions of 1198911015840(120578) 120579(120578) and 120601(120578) for 119873119903 = 01 119873119887 = 01119873119905 = 01 Le = 10 119870 = 10 119899 = 10 119887 = 00 anddifferent values of heat generationabsorption parameter 119876are shown It is observed that in presence of heat source1198911015840(120578) 120579(120578) both increase while heat sink causes that 1198911015840(120578)
120579(120578) both decrease extremely These effects are more visible
Table 9 Values of reduced Nusselt number and Sherwood numberobtained by HAM for different values of the parameters Le 119870 119899 119887and 119876 when119873119887 = 119873119903 = 119873119905 = 01
Figure 17 Effect of thermophoresis 119873119905 on the dimensionlessvelocity profile
for temperature distribution For 120601(120578) the effect of changing119876 is seen to be almost insignificant This is in agreementwith the physical factThe dimensionless profiles for differentvalues of Lewis number Le with constant parameters 119873119903 =01 119873119887 = 01 119873119905 = 01 119876 = 01 119870 = 10 119899 = 10119887 = 00 are shown in Figures 23 24 and 25 It is clear that forthese constant parameters variation of Lewis number has anyconsiderable effect on the velocity and temperature profilesbut concentration profile decreases with the increase of Lewisnumber What is similar to these results can be detected inFigures 26 27 and 28 for chemical reaction parameter 119870Figures 29 30 and 31 respectively present the comparisonof solutions of 1198911015840(120578) 120579(120578) and 120601(120578) for 119873119903 = 05 119873119887 = 05119873119905 = 01 119876 = 001 Le = 10 119870 = 10 119887 = 00 and differentvalues of order of chemical reaction 119899 It is observed that
Mathematical Problems in Engineering 15
0 2 4 6 80
1
120578
Nt = 00
Nt = 005
Nt = 10NumericalNumerical
Numerical
08
06
04
02
10
Q = 01Le = 10K = 10n = 10b = 00
Nb = 01
120579(120578
)
Nr = 01
Figure 18 Effect of thermophoresis 119873119905 on the dimensionlesstemperature profile
0 2 4 6 80
1
120601(120578
)
120578
Nt = 00
Nt = 005
Nt = 10NumericalNumerical
Numerical
08
06
04
02
10
Q = 01Le = 10K = 10n = 10b = 00
Nb = 01Nr = 01
Figure 19 Effect of thermophoresis 119873119905 on the dimensionlessconcentration profile
for higher order of chemical reaction velocity decreases andtemperature and concentration profiles increase extremelyUltimately Figures 32 33 and 34 depict the effect of thermalslip parameter 119887 on velocity temperature and concentrationfunctions when 119873119903 = 01 119873119887 = 01 119873119905 = 01 119876 = 01Le = 20 119870 = 15 119899 = 10 It is seen that the velocityand temperature of the fluid decrease with the increase of the
0 2 4 6 80
1
12
10
02
04
06
08
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numericalf998400 (120578
)
120578
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 20 Effect of generationabsorption heat parameter119876 on thedimensionless velocity profile
0 2 4 6 8 100
02
04
06
08
1
120579(120578
)
120578
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numerical
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 21 Effect of generationabsorption heat parameter 119876 on thedimensionless temperature profile
thermal slip parameter in the near of horizontal plate whilethe concentration of the fluid does not vary patently
Also for investigation of the parameters of physicalinterest Table 9 is presented In this table numerical valuesof reduced Nusselt number and Sherwood number obtainedby HAM for different values of the parameters 119890 119870 119899 119887 and119876 can be compared when119873119887 = 119873119903 = 119873119905 = 01
16 Mathematical Problems in Engineering
10
02
04
06
08
0 2 4 6 80
1
120578
120601(120578
)
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numerical
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 22 Effect of generationabsorption heat parameter119876 on thedimensionless concentration profile
0 2 4 6 80
1
12
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 23 Effect of Lewis number Le on the dimensionless velocityprofile
9 Conclusions
In this paper we studied the steady laminar incompressiblefree convective flow of a nanofluid past a chemically reactingupward facing horizontal plate in porous medium takinginto account heat generation and the thermal slip boundarycondition
0 2 4 6 80
1
120579(120578
)
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 24 Effect of Lewis number Le on the dimensionlesstemperature profile
0 2 4 6 80
1
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 25 Effect of Lewis number Le on the dimensionlessconcentration profile
The governing partial differential equations have beentransformed by similarity transformations into a system ofordinary differential equations which are solved by OHAMand numerical method (fourth-order Runge-Kutta schemewith the shooting method) Dimensionless velocity temper-ature and concentration functions are presented for various
Mathematical Problems in Engineering 17
0 2 4 6 80
1
12
K = 00
K = 05
K = 10NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 26 Effect of chemical reaction parameter 119870 on the dimen-sionless velocity profile
0 2 4 6 80
1
K = 00
K = 05
K = 10NumericalNumerical
Numerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 27 Effect of chemical reaction parameter 119870 on the dimen-sionless temperature profile
values of parameters for the problem for example buoy-ancy ratio 119873119903 Brownian motion 119873119887 thermophoresis 119873119905heat generationabsorption parameter 119876 Lewis number Lechemical reaction parameter119870 order of chemical reaction 119899and thermal slip parameter 119887 From the present investigationthe following may be concluded
(i) The velocity 1198911015840(120578) increases with increasing 119873119887 anddecreases with increasing119873119903 119899 and 119887
0 2 4 6 80
1
K = 00
K = 05
K = 10NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 28 Effect of chemical reaction parameter 119870 on the dimen-sionless concentration profile
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 29 Effect of order of chemical reaction 119899 on the dimension-less velocity profile
(ii) The temperature 120579(120578) increases with 119873119903 119873119887 119899 anddecreases with 119887
(iii) Parameters of119873119905 Le and119870 have no effect on velocityand temperature profiles
(iv) The concentration distribution 120601(120578) increases withincreasing N119905 119899 and decreases with 119873119887 Le and 119870but119873119903 119876 and 119887 are not affected by it
18 Mathematical Problems in Engineering
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 30 Effect of order of chemical reaction 119899 on the dimension-less temperature profile
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 31 Effect of order of chemical reaction 119899 on the dimension-less concentration profile
(v) In presence of heat source (119876 gt 0) 1198911015840(120578) 120579(120578) bothincrease while heat sink (119876 lt 0) causes that 1198911015840(120578)120579(120578) both decrease extremely These effects are morevisible for temperature distribution
In addition numerical results for the reducedNusselt andSherwood numbers are tabulated It is concluded that thereduced Nusselt number increases with the Lewis numberand reaction parameter whist it decreases with the order of
0 2 4 6 8
1
12
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 32 Effect of order of thermal slip parameter 119887 on thedimensionless velocity profile
0 2 4 6 80
1
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 33 Effect of order of thermal slip parameter 119887 on thedimensionless temperature profile
chemical reaction thermal slip and generation parametersIt is further concluded that the reduced Sherwood numberenhances with the Lewis number generation and reactionparameters whist it suppresses with thermal slip and orderof chemical reaction
The convergence of the solution series and the power ofHAM in controlling and adjusting the convergence regionand rate of solution series were discussed The proper range
Mathematical Problems in Engineering 19
0 2 4 6 80
1
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 34 Effect of order of thermal slip parameter 119887 on thedimensionless concentration profile
of auxiliary parameter ℏ to ensure the convergence of thesolution series was obtained through the so called ℏ-curveThe HAM provides us with a convenient way to control theconvergence of approximation series which is a fundamentalqualitative difference in analysis between the HAM andother methods The numerical results of the above problemsdisplay a fast convergence with minimal calculations Thisshows that the HAM is a very efficient method Finally theagreement between analytical and numerical results of thepresent study with previous published results is excellent
Nomenclature
119887 Thermal slip parameter119862 Dimensional concentration119862119891 Skin friction factor1198631 Thermal slip factor119863119861 Brownian diffusion coefficient119863119879 Thermophoretic diffusion coefficient119891 Dimensionless velocity functions119892 Gravitation acceleration119870 Chemical reaction parameter119896 Permeability of the porous media1198960 The constant reaction rate119896119898 Effective thermal conductivity of the porous
medium119896(119909) Variable reaction rateL Linear operator of the HAM119871 Length of horizontal plateLe Lewis numberN Nonlinear operator of the HAM
119899 Order of chemical reaction119873119887 Brownian motion119873119903 Buoyancy ratio119873119905 ThermophoresisNu Nusselt number119875 PressurePr Prandtl number119876 Heat generationabsorption parameter1198760 Heat generationabsorption constant119902 Internal heat generation rate
Ra Rayleigh numberSh Sherwood number119879 Temperature119905 Time119906 Velocity in 119909-direction119880119903 Reference velocity Velocity vectorV Velocity in 119910-direction119909 Distance along the surface119910 Distance normal to the surface
Greek Letters
120572119898 Thermal diffusivity120573 Volumetric expansion coefficient of nanofluid120576 Porosity of the porous media120601 Dimensionless concentration120578 Similarity variable120583 Dynamic viscosity120579 Dimensionless temperature120588119891 Density of the base fluid120588119901 Density of the nanoparticles(120588119862)119891 Effective heat capacity of the fluid(120588119862)119875 Effective heat capacity of the nanoparticle material120595 Stream function
Subscript Superscript
infin Conditions far away from the surface1015840 Differentiation with respect to 120578
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
M M Rashidi wishes to thank the Centre for DifferentialEquations Continuum Mechanics and Applications Schoolof Computational and Applied Mathematics University ofthe Witwatersrand for the hospitality during his visit EMomoniat thanks theNational Research Foundation of SouthAfrica for their support
20 Mathematical Problems in Engineering
References
[1] E Abu-Nada F O Hakan and I Pop ldquoBuoyancy inducedflow in a nanofluid filled enclosure partially exposed to forcedconvectionrdquo Superlattices and Microstructures vol 51 no 3 pp381ndash395 2012
[2] S Kakac and A Pramuanjaroenkij ldquoReview of convective heattransfer enhancement with nanofluidsrdquo International Journal ofHeat and Mass Transfer vol 52 no 13-14 pp 3187ndash3196 2009
[3] J A Eastman S U S Choi S Li W Yu and L J ThompsonldquoAnomalously increased effective thermal conductivities ofethylene glycol-based nanofluids containing copper nanoparti-clesrdquo Applied Physics Letters vol 78 no 6 pp 718ndash720 2001
[4] Y Xuan and Q Li ldquoHeat transfer enhancement of nanofluidsrdquoInternational Journal of Heat and Fluid Flow vol 21 no 1 pp58ndash64 2000
[5] S K Das N Putra P Thiesen and W Roetzel ldquoTemperaturedependence of thermal conductivity enhancement for nanoflu-idsrdquo Journal of Heat Transfer vol 125 no 4 pp 567ndash574 2003
[6] D H Kumar H E Patel V R R Kumar T SundararajanT Pradeep and S K Das ldquoModel for heat conduction innanofluidsrdquo Physical Review Letters vol 93 no 14 Article ID144301 2004
[7] H E Patel T Sundararajan T Pradeep A Dasgupta NDasgupta and S K Das ldquoA micro-convection model forthermal conductivity of nanofluidsrdquo Pramana vol 65 no 5 pp863ndash869 2005
[8] A V Kuznetsov ldquoNon-oscillatory and oscillatory nanofluidbio-thermal convection in a horizontal layer of finite depthrdquoEuropean Journal of Mechanics vol 30 no 2 pp 156ndash165 2011
[9] M Napoli J C T Eijkel and S Pennathur ldquoNanofluidictechnology for biomolecule applications a critical reviewrdquo Labon a Chip vol 10 no 8 pp 957ndash985 2010
[10] J C Crepeau and R Clarksean ldquoSimilarity solutions of naturalconvection with internal heat generationrdquo Journal of HeatTransfer vol 119 no 1 pp 183ndash185 1997
[11] A Z Sahin ldquoTransient heat conduction in semi-infinite solidwith spatially decaying exponential heat generationrdquo Interna-tional Communications in Heat andMass Transfer vol 19 no 3pp 349ndash358 1992
[12] O D Makind and A Aziz ldquoMixed convection from a con-vectively heated vertical plate to a fluid with internal heatgenerationrdquo Journal of Heat Transfer vol 133 Article ID 1225016 pages 2011
[13] J Buongiorno ldquoConvective transport in nanofluidsrdquo Journal ofHeat Transfer vol 128 no 3 pp 240ndash250 2006
[14] AV Kuznetsov andDANield ldquoNatural convective boundary-layer flow of a nanofluid past a vertical platerdquo InternationalJournal of Thermal Sciences vol 49 no 2 pp 243ndash247 2010
[15] L B Godson D Raja L D Mohan and S WongwisescldquoEnhancement of heat transfer using nanofluids-an overviewrdquoRenewable and Sustainable Energy Reviews vol 14 no 2 pp629ndash641 2010
[16] D A Nield and A BejanConvection in PorousMedia SpringerNew York NY USA 3rd edition 2006
[17] P Vadasz Emerging Topics in Heat and Mass Transfer in PorousMedia Springer New York NY USA 2008
[18] K Vafai Porous Media Applications in Biological Systems andBiotechnology CRC Press New York NY USA 2010
[19] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application to
heat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977
[20] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for natural convective boundary-layer flow in a porousmedium saturated by a nanofluidrdquo International Journal of Heatand Mass Transfer vol 52 no 25-26 pp 5792ndash5795 2009
[21] P Cheng and I D Chang ldquoBuoyancy induced flows in asaturated porous medium adjacent to impermeable horizontalsurfacesrdquo International Journal of Heat and Mass Transfer vol19 no 11 pp 1267ndash1272 1976
[22] I D Chang and P Cheng ldquoMatched asymptotic expansionsfor free convection about an impermeable horizontal surfacein a porous mediumrdquo International Journal of Heat and MassTransfer vol 26 no 2 pp 163ndash174 1983
[23] D S Shiunlin and B Gebhart ldquoBuoyancy-induced flow adja-cent to a horizontal surface submerged in porous mediumsaturated with cold waterrdquo International Journal of Heat andMass Transfer vol 29 no 4 pp 611ndash623 1986
[24] J H Merkin and G Zhang ldquoOn the similarity solutions for freeconvection in a saturated porousmedium adjacent to imperme-able horizontal surfacesrdquoWarme- und Stoffubertragung vol 25no 3 pp 179ndash184 1990
[25] M A Chaudhary J H Merkin and I Pop ldquoNatural convectionfrom a horizontal permeable surface in a porous mediummdashnumerical and asymptotic solutionsrdquo Transport in PorousMedia vol 22 no 3 pp 327ndash344 1996
[26] R S R Gorla and A Chamkha ldquoNatural convective boundarylayer flow over a nonisothermal vertical plate embedded ina porous medium saturated with a nanofluidrdquo Nanoscale andMicroscale Thermophysical Engineering vol 15 no 2 pp 81ndash942011
[27] W A Khan and I Pop ldquoFree convection boundary layer flowpast a horizontal flat plate embedded in a porous medium filledwith a nanofluidrdquo Journal of Heat Transfer vol 133 no 9 ArticleID 094501 2011
[28] A Aziz W A Khan and I Pop ldquoFree convection boundarylayer flow past a horizontal flat plate embedded in porousmedium filled by nanofluid containing gyrotactic microorgan-ismsrdquo International Journal of Thermal Sciences vol 56 pp 48ndash57 2012
[29] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer New York NYUSA 2009
[30] A A Avramenko S G Kobzar I V Shevchuk A V Kuznetsovand L T Iwanisov ldquoSymmetry of turbulent boundary-layerflows investigation of different eddy viscosity modelsrdquo ActaMechanica vol 151 no 1-2 pp 1ndash14 2001
[31] A V Kuznetsov A A Avramenko and P Geng ldquoAnalyticalinvestigation of a falling plume caused by bioconvection ofoxytactic bacteria in a fluid saturated porous mediumrdquo Inter-national Journal of Engineering Science vol 42 no 5-6 pp 557ndash569 2004
[32] M Jalil S Asghar and M Mushtaq ldquoLie group analysis ofmixed convection flow with mass transfer over a stretchingsurface with suction or injectionrdquo Mathematical Problems inEngineering vol 2010 Article ID 264901 14 pages 2010
[33] M A A Hamad M J Uddin and A I M Ismail ldquoRadiationeffects on heat and mass transfer in MHD stagnation-pointflow over a permeable flat plate with thermal convectivesurface boundary condition temperature dependent viscosityand thermal conductivityrdquoNuclear Engineering and Design vol242 pp 194ndash200 2012
Mathematical Problems in Engineering 21
[34] A Aziz M J Uddin M A A Hamad and A I MdIsmail ldquoMHD flow over an inclined radiating plate with thetemperature-dependent thermal conductivity variable reactiveindex and heat generationrdquoHeat Transfer vol 41 no 3 pp 241ndash259 2012
[35] A G Hansen Similarity Analysis of Boundary Layer Problemsin Engineering Prentice Hall Englewood Cliffs NJ USA 1964
[36] W FAmesNonlinear Partial Differential Equations in Engineer-ing Academic Press New York NY USA 1972
[37] R Seshadri and T Y Na Group Invariance in EngineeringBoundary Value Problems Springer New York NY USA 1985
[38] D Shang Theory of Heat Transfer with Forced Convection FilmFlows vol 3 of Heat and Mass Transfer 2010
[40] E Erfani M M Rashidi and A B Parsa ldquoThe modifieddifferential transform method for solving off-centered stag-nation flow toward a rotating discrdquo International Journal ofComputational Methods vol 7 no 4 pp 655ndash670 2010
[41] M M Rashidi N Laraqi and A Basiri Parsa ldquoAnalyticalmodeling of heat convection in magnetized micropolar fluid byusing modified differential transform methodrdquo Heat Transfervol 40 no 3 pp 187ndash204 2011
[42] M M Rashidi and M Keimanesh ldquoUsing differential trans-form method and pade approximant for solving mhd flow ina laminar liquid film from a horizontal stretching surfacerdquoMathematical Problems in Engineering vol 2010 Article ID491319 14 pages 2010
[43] M M Rashidi and G Domairry ldquoNew analytical solution ofthe three-dimensional NavierStokes equationsrdquoModern PhysicsLetters B vol 23 no 26 pp 3147ndash3155 2009
[44] M M Rashidi M Keimanesh O A Beg and T K HungldquoMagnetohydrodynamic biorheological transport phenomenain a porous medium a simulation of magnetic blood flowcontrol and filtrationrdquo International Journal for NumericalMethods in Biomedical Engineering vol 27 no 6 pp 805ndash8212011
[45] S J Liao The proposed homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992
[46] M M Rashidi G Domairry and S Dinarvand ldquoApproximatesolutions for the Burger and regularized long wave equationsby means of the homotopy analysis methodrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 14 no 3 pp708ndash717 2009
[47] Z Ziabakhsh and G Domairry ldquoAnalytic solution of naturalconvection flow of a non-Newtonian fluid between two verticalflat plates using homotopy analysis methodrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 14 no 5pp 1868ndash1880 2009
[48] M M Rashidi S A Mohimanian Pour T Hayat and SObaidat ldquoAnalytic approximate solutions for steady flow over arotating disk in porousmediumwith heat transfer by homotopyanalysis methodrdquo Computers and Fluids vol 54 no 1 pp 1ndash92012
[49] M Sajid T Hayat and S Asghar ldquoComparison between theHAM and HPM solutions of thin film flows of non-Newtonianfluids on a moving beltrdquo Nonlinear Dynamics vol 50 no 1-2pp 27ndash35 2007
[50] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for the double-diffusive natural convective boundary
layer flow in a porous medium saturated by a nanofluidrdquoInternational Journal of Heat and Mass Transfer vol 54 no 1-3pp 374ndash378 2011
[51] K Vajravelu and A Hadjinicolaou ldquoConvective heat transferin an electrically conducting fluid at a stretching surfacewith uniform free streamrdquo International Journal of EngineeringScience vol 35 pp 1237ndash1244 1997
[52] S Mukhopadhyay and G C Layek ldquoEffects of variable fluidviscosity on flow past a heated stretching sheet embedded ina porous medium in presence of heat sourcesinkrdquo Meccanicavol 47 no 4 pp 863ndash876 2012
[53] S J Liao ldquoAn optimal homotopy-analysis approach for stronglynonlinear differential equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 8 pp 2003ndash20162010
[54] A Aziz ldquoA similarity solution for laminar thermal boundarylayer over a flat plate with a convective surface boundary con-ditionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1064ndash1068 2009
Figure 10 Comparison of dimensionless concentration profile 120601(120578)obtained by various orders of the HAM
Here Ra119909 = 119892119870120573(1minus119862infin)Δ119879119909(120572119898]) is the local Rayleighnumber Pr = ]120572119898 is the Prandtl number and 119880119903 = (1 minus
119862infin)119892119870120573Δ119879120572119898 is the reference velocity
8 Results and Discussion
Graphical representation of results is very useful to discussthe physical features presented by the solutions This sectiondescribes the influence of some interesting parameters on
12 Mathematical Problems in Engineering
0 2 4 6 8 10
02
04
06
08
1
Nr = 01Numerical
Numerical
NumericalNr = 03Nr = 05
f998400 (120578)
120578
Nt = 01Nb = 01Q = 01Le = 10K = 10n = 10b = 00
Figure 11 Effect of buoyancy ratio119873119903 on the dimensionless velocityprofile
0 2 4 6 8 10
02
04
06
08
1
Nt = 01Nb = 01Q = 01Le = 10K = 10n = 10b = 00
Nr = 01Numerical
Numerical
NumericalNr = 03Nr = 05
120578
120579(120578
)
Figure 12 Effect of buoyancy ratio 119873119903 on the dimensionlesstemperature profile
the velocity and temperature fields Equations (33) to (35)with boundary conditions in (36) were solved analytically byHAM and numerically using Runge-Kutta-Fehlberg fourth-fifth order proposed by Aziz [54]
Figures 8 9 and 10 respectively for 1198911015840(120578) 120579(120578) and 120601(120578)show the comparisons between various order approximationof the optimal HAM and the numerical solutions for the case
0 2 4 6 8 10
02
04
06
08
1
Nr = 01Numerical
Numerical
NumericalNr = 03Nr = 05
Nt = 01Nb = 01Q = 01Le = 10K = 10n = 10b = 00
120578
120601(120578
)Figure 13 Effect of buoyancy ratio 119873119903 on the dimensionlessconcentration profile
0 2 4 6 8 10
02
04
06
08
1
12
NumericalNumerical
Numerical
f998400 (120578
)
120578
Nb = 01
Nb = 03
Nb = 05
Nt = 01Nr = 01Q = 01Le = 10K = 10n = 10b = 00
Figure 14 Effect of Brownian motion 119873119887 on the dimensionlessvelocity profile
119873119887 = 01 119873119905 = 01 119873119903 = 01 119876 = 01 Le = 10 119870 = 10119899 = 10 119887 = 00 For better presentation these results arepresented in Tables 3 4 and 5 Also Tables 6 7 and 8 presentthe comparisons between various order approximation of theoptimal HAM and the numerical solutions for the case119873119887 =03 119873119905 = 005 119873119903 = 03 119876 = 01 Le = 20 119870 = 05
Mathematical Problems in Engineering 13
0 2 4 6 8 10
02
04
06
08
1
0
120578
NumericalNumerical
Numerical
Nb = 01
Nb = 03
Nb = 05
Nt = 01Nr = 01Q = 01Le = 10K = 10n = 10b = 00
120579(120578
)
Figure 15 Effect of Brownian motion 119873119887 on the dimensionlesstemperature profile
02
04
06
08
1
00
2 4 6 8 10120578
NumericalNumerical
Numerical
Nb = 01
Nb = 03
Nb = 05
Nt = 01Nr = 01Q = 01Le = 10K = 10n = 10b = 00
120601(120578
)
Figure 16 Effect of Brownian motion 119873119887 on the dimensionlessconcentration profile
119899 = 20 119887 = 01 It is observed that the results of 12th-order approximation of the optimal HAM are very close tothe numerical solutions which confirm the validity of thesemethods
In the following figures effects of various physicalparameters on the dimensionless velocity temperature and
Table 3 Comparison of values of 1198911015840(120578) obtained by various ordersof the HAM with numerical solution when 119873119887 = 01119873119905 =
temperature profiles will be investigated These results havebeen obtained by 12th order of the HAM and have beenvalidated by numerical results Figures 11ndash13 respectivelyrepresent that the comparison of solutions of 1198911015840(120578) 120579(120578)and 120601(120578) for fix values 119873119887 = 01 119873119905 = 01 119876 = 01Le = 10 119870 = 10 119899 = 10 119887 = 00 and different values ofbuoyancy ratio119873119903 In Figure 11 it is clear that velocity of thefluid tremendously decreases in the near of horizontal platewith an increase in the buoyancy ratio It is observed that thetemperature increases slightly but concentration of the fluiddoes not vary sensibly (Figures 12 and 13) The effects of theBrownian motion 119873119887 are depicted in Figures 14 15 and 16when119873119903 = 01119873119905 = 01119876 = 01 Le = 10119870 = 10 119899 = 10
14 Mathematical Problems in Engineering
Table 6 Comparison of values of 1198911015840(120578) obtained by various ordersof the HAM with numerical solution when 119873119887 = 03119873119905 =
119887 = 00 If 119873119887 increases 1198911015840(120578) and 120579(120578) increase lightly and120601(120578) decreases The dimensionless velocity dimensionlesstemperature and dimensionless concentration profiles fordifferent values of thermophoresis 119873119905 with constant values119873119903 = 01 119873119887 = 01 119876 = 01 Le = 10 119870 = 10119899 = 10 119887 = 00 are presented in Figures 17 18 and19 It is observed that the velocity and temperature of thefluid are not impressible from119873119905 but concentration increasesslightly In Figures 20 21 and 22 respectively comparison ofsolutions of 1198911015840(120578) 120579(120578) and 120601(120578) for 119873119903 = 01 119873119887 = 01119873119905 = 01 Le = 10 119870 = 10 119899 = 10 119887 = 00 anddifferent values of heat generationabsorption parameter 119876are shown It is observed that in presence of heat source1198911015840(120578) 120579(120578) both increase while heat sink causes that 1198911015840(120578)
120579(120578) both decrease extremely These effects are more visible
Table 9 Values of reduced Nusselt number and Sherwood numberobtained by HAM for different values of the parameters Le 119870 119899 119887and 119876 when119873119887 = 119873119903 = 119873119905 = 01
Figure 17 Effect of thermophoresis 119873119905 on the dimensionlessvelocity profile
for temperature distribution For 120601(120578) the effect of changing119876 is seen to be almost insignificant This is in agreementwith the physical factThe dimensionless profiles for differentvalues of Lewis number Le with constant parameters 119873119903 =01 119873119887 = 01 119873119905 = 01 119876 = 01 119870 = 10 119899 = 10119887 = 00 are shown in Figures 23 24 and 25 It is clear that forthese constant parameters variation of Lewis number has anyconsiderable effect on the velocity and temperature profilesbut concentration profile decreases with the increase of Lewisnumber What is similar to these results can be detected inFigures 26 27 and 28 for chemical reaction parameter 119870Figures 29 30 and 31 respectively present the comparisonof solutions of 1198911015840(120578) 120579(120578) and 120601(120578) for 119873119903 = 05 119873119887 = 05119873119905 = 01 119876 = 001 Le = 10 119870 = 10 119887 = 00 and differentvalues of order of chemical reaction 119899 It is observed that
Mathematical Problems in Engineering 15
0 2 4 6 80
1
120578
Nt = 00
Nt = 005
Nt = 10NumericalNumerical
Numerical
08
06
04
02
10
Q = 01Le = 10K = 10n = 10b = 00
Nb = 01
120579(120578
)
Nr = 01
Figure 18 Effect of thermophoresis 119873119905 on the dimensionlesstemperature profile
0 2 4 6 80
1
120601(120578
)
120578
Nt = 00
Nt = 005
Nt = 10NumericalNumerical
Numerical
08
06
04
02
10
Q = 01Le = 10K = 10n = 10b = 00
Nb = 01Nr = 01
Figure 19 Effect of thermophoresis 119873119905 on the dimensionlessconcentration profile
for higher order of chemical reaction velocity decreases andtemperature and concentration profiles increase extremelyUltimately Figures 32 33 and 34 depict the effect of thermalslip parameter 119887 on velocity temperature and concentrationfunctions when 119873119903 = 01 119873119887 = 01 119873119905 = 01 119876 = 01Le = 20 119870 = 15 119899 = 10 It is seen that the velocityand temperature of the fluid decrease with the increase of the
0 2 4 6 80
1
12
10
02
04
06
08
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numericalf998400 (120578
)
120578
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 20 Effect of generationabsorption heat parameter119876 on thedimensionless velocity profile
0 2 4 6 8 100
02
04
06
08
1
120579(120578
)
120578
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numerical
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 21 Effect of generationabsorption heat parameter 119876 on thedimensionless temperature profile
thermal slip parameter in the near of horizontal plate whilethe concentration of the fluid does not vary patently
Also for investigation of the parameters of physicalinterest Table 9 is presented In this table numerical valuesof reduced Nusselt number and Sherwood number obtainedby HAM for different values of the parameters 119890 119870 119899 119887 and119876 can be compared when119873119887 = 119873119903 = 119873119905 = 01
16 Mathematical Problems in Engineering
10
02
04
06
08
0 2 4 6 80
1
120578
120601(120578
)
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numerical
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 22 Effect of generationabsorption heat parameter119876 on thedimensionless concentration profile
0 2 4 6 80
1
12
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 23 Effect of Lewis number Le on the dimensionless velocityprofile
9 Conclusions
In this paper we studied the steady laminar incompressiblefree convective flow of a nanofluid past a chemically reactingupward facing horizontal plate in porous medium takinginto account heat generation and the thermal slip boundarycondition
0 2 4 6 80
1
120579(120578
)
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 24 Effect of Lewis number Le on the dimensionlesstemperature profile
0 2 4 6 80
1
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 25 Effect of Lewis number Le on the dimensionlessconcentration profile
The governing partial differential equations have beentransformed by similarity transformations into a system ofordinary differential equations which are solved by OHAMand numerical method (fourth-order Runge-Kutta schemewith the shooting method) Dimensionless velocity temper-ature and concentration functions are presented for various
Mathematical Problems in Engineering 17
0 2 4 6 80
1
12
K = 00
K = 05
K = 10NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 26 Effect of chemical reaction parameter 119870 on the dimen-sionless velocity profile
0 2 4 6 80
1
K = 00
K = 05
K = 10NumericalNumerical
Numerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 27 Effect of chemical reaction parameter 119870 on the dimen-sionless temperature profile
values of parameters for the problem for example buoy-ancy ratio 119873119903 Brownian motion 119873119887 thermophoresis 119873119905heat generationabsorption parameter 119876 Lewis number Lechemical reaction parameter119870 order of chemical reaction 119899and thermal slip parameter 119887 From the present investigationthe following may be concluded
(i) The velocity 1198911015840(120578) increases with increasing 119873119887 anddecreases with increasing119873119903 119899 and 119887
0 2 4 6 80
1
K = 00
K = 05
K = 10NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 28 Effect of chemical reaction parameter 119870 on the dimen-sionless concentration profile
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 29 Effect of order of chemical reaction 119899 on the dimension-less velocity profile
(ii) The temperature 120579(120578) increases with 119873119903 119873119887 119899 anddecreases with 119887
(iii) Parameters of119873119905 Le and119870 have no effect on velocityand temperature profiles
(iv) The concentration distribution 120601(120578) increases withincreasing N119905 119899 and decreases with 119873119887 Le and 119870but119873119903 119876 and 119887 are not affected by it
18 Mathematical Problems in Engineering
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 30 Effect of order of chemical reaction 119899 on the dimension-less temperature profile
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 31 Effect of order of chemical reaction 119899 on the dimension-less concentration profile
(v) In presence of heat source (119876 gt 0) 1198911015840(120578) 120579(120578) bothincrease while heat sink (119876 lt 0) causes that 1198911015840(120578)120579(120578) both decrease extremely These effects are morevisible for temperature distribution
In addition numerical results for the reducedNusselt andSherwood numbers are tabulated It is concluded that thereduced Nusselt number increases with the Lewis numberand reaction parameter whist it decreases with the order of
0 2 4 6 8
1
12
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 32 Effect of order of thermal slip parameter 119887 on thedimensionless velocity profile
0 2 4 6 80
1
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 33 Effect of order of thermal slip parameter 119887 on thedimensionless temperature profile
chemical reaction thermal slip and generation parametersIt is further concluded that the reduced Sherwood numberenhances with the Lewis number generation and reactionparameters whist it suppresses with thermal slip and orderof chemical reaction
The convergence of the solution series and the power ofHAM in controlling and adjusting the convergence regionand rate of solution series were discussed The proper range
Mathematical Problems in Engineering 19
0 2 4 6 80
1
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 34 Effect of order of thermal slip parameter 119887 on thedimensionless concentration profile
of auxiliary parameter ℏ to ensure the convergence of thesolution series was obtained through the so called ℏ-curveThe HAM provides us with a convenient way to control theconvergence of approximation series which is a fundamentalqualitative difference in analysis between the HAM andother methods The numerical results of the above problemsdisplay a fast convergence with minimal calculations Thisshows that the HAM is a very efficient method Finally theagreement between analytical and numerical results of thepresent study with previous published results is excellent
Nomenclature
119887 Thermal slip parameter119862 Dimensional concentration119862119891 Skin friction factor1198631 Thermal slip factor119863119861 Brownian diffusion coefficient119863119879 Thermophoretic diffusion coefficient119891 Dimensionless velocity functions119892 Gravitation acceleration119870 Chemical reaction parameter119896 Permeability of the porous media1198960 The constant reaction rate119896119898 Effective thermal conductivity of the porous
medium119896(119909) Variable reaction rateL Linear operator of the HAM119871 Length of horizontal plateLe Lewis numberN Nonlinear operator of the HAM
119899 Order of chemical reaction119873119887 Brownian motion119873119903 Buoyancy ratio119873119905 ThermophoresisNu Nusselt number119875 PressurePr Prandtl number119876 Heat generationabsorption parameter1198760 Heat generationabsorption constant119902 Internal heat generation rate
Ra Rayleigh numberSh Sherwood number119879 Temperature119905 Time119906 Velocity in 119909-direction119880119903 Reference velocity Velocity vectorV Velocity in 119910-direction119909 Distance along the surface119910 Distance normal to the surface
Greek Letters
120572119898 Thermal diffusivity120573 Volumetric expansion coefficient of nanofluid120576 Porosity of the porous media120601 Dimensionless concentration120578 Similarity variable120583 Dynamic viscosity120579 Dimensionless temperature120588119891 Density of the base fluid120588119901 Density of the nanoparticles(120588119862)119891 Effective heat capacity of the fluid(120588119862)119875 Effective heat capacity of the nanoparticle material120595 Stream function
Subscript Superscript
infin Conditions far away from the surface1015840 Differentiation with respect to 120578
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
M M Rashidi wishes to thank the Centre for DifferentialEquations Continuum Mechanics and Applications Schoolof Computational and Applied Mathematics University ofthe Witwatersrand for the hospitality during his visit EMomoniat thanks theNational Research Foundation of SouthAfrica for their support
20 Mathematical Problems in Engineering
References
[1] E Abu-Nada F O Hakan and I Pop ldquoBuoyancy inducedflow in a nanofluid filled enclosure partially exposed to forcedconvectionrdquo Superlattices and Microstructures vol 51 no 3 pp381ndash395 2012
[2] S Kakac and A Pramuanjaroenkij ldquoReview of convective heattransfer enhancement with nanofluidsrdquo International Journal ofHeat and Mass Transfer vol 52 no 13-14 pp 3187ndash3196 2009
[3] J A Eastman S U S Choi S Li W Yu and L J ThompsonldquoAnomalously increased effective thermal conductivities ofethylene glycol-based nanofluids containing copper nanoparti-clesrdquo Applied Physics Letters vol 78 no 6 pp 718ndash720 2001
[4] Y Xuan and Q Li ldquoHeat transfer enhancement of nanofluidsrdquoInternational Journal of Heat and Fluid Flow vol 21 no 1 pp58ndash64 2000
[5] S K Das N Putra P Thiesen and W Roetzel ldquoTemperaturedependence of thermal conductivity enhancement for nanoflu-idsrdquo Journal of Heat Transfer vol 125 no 4 pp 567ndash574 2003
[6] D H Kumar H E Patel V R R Kumar T SundararajanT Pradeep and S K Das ldquoModel for heat conduction innanofluidsrdquo Physical Review Letters vol 93 no 14 Article ID144301 2004
[7] H E Patel T Sundararajan T Pradeep A Dasgupta NDasgupta and S K Das ldquoA micro-convection model forthermal conductivity of nanofluidsrdquo Pramana vol 65 no 5 pp863ndash869 2005
[8] A V Kuznetsov ldquoNon-oscillatory and oscillatory nanofluidbio-thermal convection in a horizontal layer of finite depthrdquoEuropean Journal of Mechanics vol 30 no 2 pp 156ndash165 2011
[9] M Napoli J C T Eijkel and S Pennathur ldquoNanofluidictechnology for biomolecule applications a critical reviewrdquo Labon a Chip vol 10 no 8 pp 957ndash985 2010
[10] J C Crepeau and R Clarksean ldquoSimilarity solutions of naturalconvection with internal heat generationrdquo Journal of HeatTransfer vol 119 no 1 pp 183ndash185 1997
[11] A Z Sahin ldquoTransient heat conduction in semi-infinite solidwith spatially decaying exponential heat generationrdquo Interna-tional Communications in Heat andMass Transfer vol 19 no 3pp 349ndash358 1992
[12] O D Makind and A Aziz ldquoMixed convection from a con-vectively heated vertical plate to a fluid with internal heatgenerationrdquo Journal of Heat Transfer vol 133 Article ID 1225016 pages 2011
[13] J Buongiorno ldquoConvective transport in nanofluidsrdquo Journal ofHeat Transfer vol 128 no 3 pp 240ndash250 2006
[14] AV Kuznetsov andDANield ldquoNatural convective boundary-layer flow of a nanofluid past a vertical platerdquo InternationalJournal of Thermal Sciences vol 49 no 2 pp 243ndash247 2010
[15] L B Godson D Raja L D Mohan and S WongwisescldquoEnhancement of heat transfer using nanofluids-an overviewrdquoRenewable and Sustainable Energy Reviews vol 14 no 2 pp629ndash641 2010
[16] D A Nield and A BejanConvection in PorousMedia SpringerNew York NY USA 3rd edition 2006
[17] P Vadasz Emerging Topics in Heat and Mass Transfer in PorousMedia Springer New York NY USA 2008
[18] K Vafai Porous Media Applications in Biological Systems andBiotechnology CRC Press New York NY USA 2010
[19] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application to
heat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977
[20] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for natural convective boundary-layer flow in a porousmedium saturated by a nanofluidrdquo International Journal of Heatand Mass Transfer vol 52 no 25-26 pp 5792ndash5795 2009
[21] P Cheng and I D Chang ldquoBuoyancy induced flows in asaturated porous medium adjacent to impermeable horizontalsurfacesrdquo International Journal of Heat and Mass Transfer vol19 no 11 pp 1267ndash1272 1976
[22] I D Chang and P Cheng ldquoMatched asymptotic expansionsfor free convection about an impermeable horizontal surfacein a porous mediumrdquo International Journal of Heat and MassTransfer vol 26 no 2 pp 163ndash174 1983
[23] D S Shiunlin and B Gebhart ldquoBuoyancy-induced flow adja-cent to a horizontal surface submerged in porous mediumsaturated with cold waterrdquo International Journal of Heat andMass Transfer vol 29 no 4 pp 611ndash623 1986
[24] J H Merkin and G Zhang ldquoOn the similarity solutions for freeconvection in a saturated porousmedium adjacent to imperme-able horizontal surfacesrdquoWarme- und Stoffubertragung vol 25no 3 pp 179ndash184 1990
[25] M A Chaudhary J H Merkin and I Pop ldquoNatural convectionfrom a horizontal permeable surface in a porous mediummdashnumerical and asymptotic solutionsrdquo Transport in PorousMedia vol 22 no 3 pp 327ndash344 1996
[26] R S R Gorla and A Chamkha ldquoNatural convective boundarylayer flow over a nonisothermal vertical plate embedded ina porous medium saturated with a nanofluidrdquo Nanoscale andMicroscale Thermophysical Engineering vol 15 no 2 pp 81ndash942011
[27] W A Khan and I Pop ldquoFree convection boundary layer flowpast a horizontal flat plate embedded in a porous medium filledwith a nanofluidrdquo Journal of Heat Transfer vol 133 no 9 ArticleID 094501 2011
[28] A Aziz W A Khan and I Pop ldquoFree convection boundarylayer flow past a horizontal flat plate embedded in porousmedium filled by nanofluid containing gyrotactic microorgan-ismsrdquo International Journal of Thermal Sciences vol 56 pp 48ndash57 2012
[29] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer New York NYUSA 2009
[30] A A Avramenko S G Kobzar I V Shevchuk A V Kuznetsovand L T Iwanisov ldquoSymmetry of turbulent boundary-layerflows investigation of different eddy viscosity modelsrdquo ActaMechanica vol 151 no 1-2 pp 1ndash14 2001
[31] A V Kuznetsov A A Avramenko and P Geng ldquoAnalyticalinvestigation of a falling plume caused by bioconvection ofoxytactic bacteria in a fluid saturated porous mediumrdquo Inter-national Journal of Engineering Science vol 42 no 5-6 pp 557ndash569 2004
[32] M Jalil S Asghar and M Mushtaq ldquoLie group analysis ofmixed convection flow with mass transfer over a stretchingsurface with suction or injectionrdquo Mathematical Problems inEngineering vol 2010 Article ID 264901 14 pages 2010
[33] M A A Hamad M J Uddin and A I M Ismail ldquoRadiationeffects on heat and mass transfer in MHD stagnation-pointflow over a permeable flat plate with thermal convectivesurface boundary condition temperature dependent viscosityand thermal conductivityrdquoNuclear Engineering and Design vol242 pp 194ndash200 2012
Mathematical Problems in Engineering 21
[34] A Aziz M J Uddin M A A Hamad and A I MdIsmail ldquoMHD flow over an inclined radiating plate with thetemperature-dependent thermal conductivity variable reactiveindex and heat generationrdquoHeat Transfer vol 41 no 3 pp 241ndash259 2012
[35] A G Hansen Similarity Analysis of Boundary Layer Problemsin Engineering Prentice Hall Englewood Cliffs NJ USA 1964
[36] W FAmesNonlinear Partial Differential Equations in Engineer-ing Academic Press New York NY USA 1972
[37] R Seshadri and T Y Na Group Invariance in EngineeringBoundary Value Problems Springer New York NY USA 1985
[38] D Shang Theory of Heat Transfer with Forced Convection FilmFlows vol 3 of Heat and Mass Transfer 2010
[40] E Erfani M M Rashidi and A B Parsa ldquoThe modifieddifferential transform method for solving off-centered stag-nation flow toward a rotating discrdquo International Journal ofComputational Methods vol 7 no 4 pp 655ndash670 2010
[41] M M Rashidi N Laraqi and A Basiri Parsa ldquoAnalyticalmodeling of heat convection in magnetized micropolar fluid byusing modified differential transform methodrdquo Heat Transfervol 40 no 3 pp 187ndash204 2011
[42] M M Rashidi and M Keimanesh ldquoUsing differential trans-form method and pade approximant for solving mhd flow ina laminar liquid film from a horizontal stretching surfacerdquoMathematical Problems in Engineering vol 2010 Article ID491319 14 pages 2010
[43] M M Rashidi and G Domairry ldquoNew analytical solution ofthe three-dimensional NavierStokes equationsrdquoModern PhysicsLetters B vol 23 no 26 pp 3147ndash3155 2009
[44] M M Rashidi M Keimanesh O A Beg and T K HungldquoMagnetohydrodynamic biorheological transport phenomenain a porous medium a simulation of magnetic blood flowcontrol and filtrationrdquo International Journal for NumericalMethods in Biomedical Engineering vol 27 no 6 pp 805ndash8212011
[45] S J Liao The proposed homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992
[46] M M Rashidi G Domairry and S Dinarvand ldquoApproximatesolutions for the Burger and regularized long wave equationsby means of the homotopy analysis methodrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 14 no 3 pp708ndash717 2009
[47] Z Ziabakhsh and G Domairry ldquoAnalytic solution of naturalconvection flow of a non-Newtonian fluid between two verticalflat plates using homotopy analysis methodrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 14 no 5pp 1868ndash1880 2009
[48] M M Rashidi S A Mohimanian Pour T Hayat and SObaidat ldquoAnalytic approximate solutions for steady flow over arotating disk in porousmediumwith heat transfer by homotopyanalysis methodrdquo Computers and Fluids vol 54 no 1 pp 1ndash92012
[49] M Sajid T Hayat and S Asghar ldquoComparison between theHAM and HPM solutions of thin film flows of non-Newtonianfluids on a moving beltrdquo Nonlinear Dynamics vol 50 no 1-2pp 27ndash35 2007
[50] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for the double-diffusive natural convective boundary
layer flow in a porous medium saturated by a nanofluidrdquoInternational Journal of Heat and Mass Transfer vol 54 no 1-3pp 374ndash378 2011
[51] K Vajravelu and A Hadjinicolaou ldquoConvective heat transferin an electrically conducting fluid at a stretching surfacewith uniform free streamrdquo International Journal of EngineeringScience vol 35 pp 1237ndash1244 1997
[52] S Mukhopadhyay and G C Layek ldquoEffects of variable fluidviscosity on flow past a heated stretching sheet embedded ina porous medium in presence of heat sourcesinkrdquo Meccanicavol 47 no 4 pp 863ndash876 2012
[53] S J Liao ldquoAn optimal homotopy-analysis approach for stronglynonlinear differential equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 8 pp 2003ndash20162010
[54] A Aziz ldquoA similarity solution for laminar thermal boundarylayer over a flat plate with a convective surface boundary con-ditionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1064ndash1068 2009
Figure 11 Effect of buoyancy ratio119873119903 on the dimensionless velocityprofile
0 2 4 6 8 10
02
04
06
08
1
Nt = 01Nb = 01Q = 01Le = 10K = 10n = 10b = 00
Nr = 01Numerical
Numerical
NumericalNr = 03Nr = 05
120578
120579(120578
)
Figure 12 Effect of buoyancy ratio 119873119903 on the dimensionlesstemperature profile
the velocity and temperature fields Equations (33) to (35)with boundary conditions in (36) were solved analytically byHAM and numerically using Runge-Kutta-Fehlberg fourth-fifth order proposed by Aziz [54]
Figures 8 9 and 10 respectively for 1198911015840(120578) 120579(120578) and 120601(120578)show the comparisons between various order approximationof the optimal HAM and the numerical solutions for the case
0 2 4 6 8 10
02
04
06
08
1
Nr = 01Numerical
Numerical
NumericalNr = 03Nr = 05
Nt = 01Nb = 01Q = 01Le = 10K = 10n = 10b = 00
120578
120601(120578
)Figure 13 Effect of buoyancy ratio 119873119903 on the dimensionlessconcentration profile
0 2 4 6 8 10
02
04
06
08
1
12
NumericalNumerical
Numerical
f998400 (120578
)
120578
Nb = 01
Nb = 03
Nb = 05
Nt = 01Nr = 01Q = 01Le = 10K = 10n = 10b = 00
Figure 14 Effect of Brownian motion 119873119887 on the dimensionlessvelocity profile
119873119887 = 01 119873119905 = 01 119873119903 = 01 119876 = 01 Le = 10 119870 = 10119899 = 10 119887 = 00 For better presentation these results arepresented in Tables 3 4 and 5 Also Tables 6 7 and 8 presentthe comparisons between various order approximation of theoptimal HAM and the numerical solutions for the case119873119887 =03 119873119905 = 005 119873119903 = 03 119876 = 01 Le = 20 119870 = 05
Mathematical Problems in Engineering 13
0 2 4 6 8 10
02
04
06
08
1
0
120578
NumericalNumerical
Numerical
Nb = 01
Nb = 03
Nb = 05
Nt = 01Nr = 01Q = 01Le = 10K = 10n = 10b = 00
120579(120578
)
Figure 15 Effect of Brownian motion 119873119887 on the dimensionlesstemperature profile
02
04
06
08
1
00
2 4 6 8 10120578
NumericalNumerical
Numerical
Nb = 01
Nb = 03
Nb = 05
Nt = 01Nr = 01Q = 01Le = 10K = 10n = 10b = 00
120601(120578
)
Figure 16 Effect of Brownian motion 119873119887 on the dimensionlessconcentration profile
119899 = 20 119887 = 01 It is observed that the results of 12th-order approximation of the optimal HAM are very close tothe numerical solutions which confirm the validity of thesemethods
In the following figures effects of various physicalparameters on the dimensionless velocity temperature and
Table 3 Comparison of values of 1198911015840(120578) obtained by various ordersof the HAM with numerical solution when 119873119887 = 01119873119905 =
temperature profiles will be investigated These results havebeen obtained by 12th order of the HAM and have beenvalidated by numerical results Figures 11ndash13 respectivelyrepresent that the comparison of solutions of 1198911015840(120578) 120579(120578)and 120601(120578) for fix values 119873119887 = 01 119873119905 = 01 119876 = 01Le = 10 119870 = 10 119899 = 10 119887 = 00 and different values ofbuoyancy ratio119873119903 In Figure 11 it is clear that velocity of thefluid tremendously decreases in the near of horizontal platewith an increase in the buoyancy ratio It is observed that thetemperature increases slightly but concentration of the fluiddoes not vary sensibly (Figures 12 and 13) The effects of theBrownian motion 119873119887 are depicted in Figures 14 15 and 16when119873119903 = 01119873119905 = 01119876 = 01 Le = 10119870 = 10 119899 = 10
14 Mathematical Problems in Engineering
Table 6 Comparison of values of 1198911015840(120578) obtained by various ordersof the HAM with numerical solution when 119873119887 = 03119873119905 =
119887 = 00 If 119873119887 increases 1198911015840(120578) and 120579(120578) increase lightly and120601(120578) decreases The dimensionless velocity dimensionlesstemperature and dimensionless concentration profiles fordifferent values of thermophoresis 119873119905 with constant values119873119903 = 01 119873119887 = 01 119876 = 01 Le = 10 119870 = 10119899 = 10 119887 = 00 are presented in Figures 17 18 and19 It is observed that the velocity and temperature of thefluid are not impressible from119873119905 but concentration increasesslightly In Figures 20 21 and 22 respectively comparison ofsolutions of 1198911015840(120578) 120579(120578) and 120601(120578) for 119873119903 = 01 119873119887 = 01119873119905 = 01 Le = 10 119870 = 10 119899 = 10 119887 = 00 anddifferent values of heat generationabsorption parameter 119876are shown It is observed that in presence of heat source1198911015840(120578) 120579(120578) both increase while heat sink causes that 1198911015840(120578)
120579(120578) both decrease extremely These effects are more visible
Table 9 Values of reduced Nusselt number and Sherwood numberobtained by HAM for different values of the parameters Le 119870 119899 119887and 119876 when119873119887 = 119873119903 = 119873119905 = 01
Figure 17 Effect of thermophoresis 119873119905 on the dimensionlessvelocity profile
for temperature distribution For 120601(120578) the effect of changing119876 is seen to be almost insignificant This is in agreementwith the physical factThe dimensionless profiles for differentvalues of Lewis number Le with constant parameters 119873119903 =01 119873119887 = 01 119873119905 = 01 119876 = 01 119870 = 10 119899 = 10119887 = 00 are shown in Figures 23 24 and 25 It is clear that forthese constant parameters variation of Lewis number has anyconsiderable effect on the velocity and temperature profilesbut concentration profile decreases with the increase of Lewisnumber What is similar to these results can be detected inFigures 26 27 and 28 for chemical reaction parameter 119870Figures 29 30 and 31 respectively present the comparisonof solutions of 1198911015840(120578) 120579(120578) and 120601(120578) for 119873119903 = 05 119873119887 = 05119873119905 = 01 119876 = 001 Le = 10 119870 = 10 119887 = 00 and differentvalues of order of chemical reaction 119899 It is observed that
Mathematical Problems in Engineering 15
0 2 4 6 80
1
120578
Nt = 00
Nt = 005
Nt = 10NumericalNumerical
Numerical
08
06
04
02
10
Q = 01Le = 10K = 10n = 10b = 00
Nb = 01
120579(120578
)
Nr = 01
Figure 18 Effect of thermophoresis 119873119905 on the dimensionlesstemperature profile
0 2 4 6 80
1
120601(120578
)
120578
Nt = 00
Nt = 005
Nt = 10NumericalNumerical
Numerical
08
06
04
02
10
Q = 01Le = 10K = 10n = 10b = 00
Nb = 01Nr = 01
Figure 19 Effect of thermophoresis 119873119905 on the dimensionlessconcentration profile
for higher order of chemical reaction velocity decreases andtemperature and concentration profiles increase extremelyUltimately Figures 32 33 and 34 depict the effect of thermalslip parameter 119887 on velocity temperature and concentrationfunctions when 119873119903 = 01 119873119887 = 01 119873119905 = 01 119876 = 01Le = 20 119870 = 15 119899 = 10 It is seen that the velocityand temperature of the fluid decrease with the increase of the
0 2 4 6 80
1
12
10
02
04
06
08
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numericalf998400 (120578
)
120578
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 20 Effect of generationabsorption heat parameter119876 on thedimensionless velocity profile
0 2 4 6 8 100
02
04
06
08
1
120579(120578
)
120578
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numerical
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 21 Effect of generationabsorption heat parameter 119876 on thedimensionless temperature profile
thermal slip parameter in the near of horizontal plate whilethe concentration of the fluid does not vary patently
Also for investigation of the parameters of physicalinterest Table 9 is presented In this table numerical valuesof reduced Nusselt number and Sherwood number obtainedby HAM for different values of the parameters 119890 119870 119899 119887 and119876 can be compared when119873119887 = 119873119903 = 119873119905 = 01
16 Mathematical Problems in Engineering
10
02
04
06
08
0 2 4 6 80
1
120578
120601(120578
)
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numerical
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 22 Effect of generationabsorption heat parameter119876 on thedimensionless concentration profile
0 2 4 6 80
1
12
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 23 Effect of Lewis number Le on the dimensionless velocityprofile
9 Conclusions
In this paper we studied the steady laminar incompressiblefree convective flow of a nanofluid past a chemically reactingupward facing horizontal plate in porous medium takinginto account heat generation and the thermal slip boundarycondition
0 2 4 6 80
1
120579(120578
)
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 24 Effect of Lewis number Le on the dimensionlesstemperature profile
0 2 4 6 80
1
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 25 Effect of Lewis number Le on the dimensionlessconcentration profile
The governing partial differential equations have beentransformed by similarity transformations into a system ofordinary differential equations which are solved by OHAMand numerical method (fourth-order Runge-Kutta schemewith the shooting method) Dimensionless velocity temper-ature and concentration functions are presented for various
Mathematical Problems in Engineering 17
0 2 4 6 80
1
12
K = 00
K = 05
K = 10NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 26 Effect of chemical reaction parameter 119870 on the dimen-sionless velocity profile
0 2 4 6 80
1
K = 00
K = 05
K = 10NumericalNumerical
Numerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 27 Effect of chemical reaction parameter 119870 on the dimen-sionless temperature profile
values of parameters for the problem for example buoy-ancy ratio 119873119903 Brownian motion 119873119887 thermophoresis 119873119905heat generationabsorption parameter 119876 Lewis number Lechemical reaction parameter119870 order of chemical reaction 119899and thermal slip parameter 119887 From the present investigationthe following may be concluded
(i) The velocity 1198911015840(120578) increases with increasing 119873119887 anddecreases with increasing119873119903 119899 and 119887
0 2 4 6 80
1
K = 00
K = 05
K = 10NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 28 Effect of chemical reaction parameter 119870 on the dimen-sionless concentration profile
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 29 Effect of order of chemical reaction 119899 on the dimension-less velocity profile
(ii) The temperature 120579(120578) increases with 119873119903 119873119887 119899 anddecreases with 119887
(iii) Parameters of119873119905 Le and119870 have no effect on velocityand temperature profiles
(iv) The concentration distribution 120601(120578) increases withincreasing N119905 119899 and decreases with 119873119887 Le and 119870but119873119903 119876 and 119887 are not affected by it
18 Mathematical Problems in Engineering
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 30 Effect of order of chemical reaction 119899 on the dimension-less temperature profile
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 31 Effect of order of chemical reaction 119899 on the dimension-less concentration profile
(v) In presence of heat source (119876 gt 0) 1198911015840(120578) 120579(120578) bothincrease while heat sink (119876 lt 0) causes that 1198911015840(120578)120579(120578) both decrease extremely These effects are morevisible for temperature distribution
In addition numerical results for the reducedNusselt andSherwood numbers are tabulated It is concluded that thereduced Nusselt number increases with the Lewis numberand reaction parameter whist it decreases with the order of
0 2 4 6 8
1
12
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 32 Effect of order of thermal slip parameter 119887 on thedimensionless velocity profile
0 2 4 6 80
1
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 33 Effect of order of thermal slip parameter 119887 on thedimensionless temperature profile
chemical reaction thermal slip and generation parametersIt is further concluded that the reduced Sherwood numberenhances with the Lewis number generation and reactionparameters whist it suppresses with thermal slip and orderof chemical reaction
The convergence of the solution series and the power ofHAM in controlling and adjusting the convergence regionand rate of solution series were discussed The proper range
Mathematical Problems in Engineering 19
0 2 4 6 80
1
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 34 Effect of order of thermal slip parameter 119887 on thedimensionless concentration profile
of auxiliary parameter ℏ to ensure the convergence of thesolution series was obtained through the so called ℏ-curveThe HAM provides us with a convenient way to control theconvergence of approximation series which is a fundamentalqualitative difference in analysis between the HAM andother methods The numerical results of the above problemsdisplay a fast convergence with minimal calculations Thisshows that the HAM is a very efficient method Finally theagreement between analytical and numerical results of thepresent study with previous published results is excellent
Nomenclature
119887 Thermal slip parameter119862 Dimensional concentration119862119891 Skin friction factor1198631 Thermal slip factor119863119861 Brownian diffusion coefficient119863119879 Thermophoretic diffusion coefficient119891 Dimensionless velocity functions119892 Gravitation acceleration119870 Chemical reaction parameter119896 Permeability of the porous media1198960 The constant reaction rate119896119898 Effective thermal conductivity of the porous
medium119896(119909) Variable reaction rateL Linear operator of the HAM119871 Length of horizontal plateLe Lewis numberN Nonlinear operator of the HAM
119899 Order of chemical reaction119873119887 Brownian motion119873119903 Buoyancy ratio119873119905 ThermophoresisNu Nusselt number119875 PressurePr Prandtl number119876 Heat generationabsorption parameter1198760 Heat generationabsorption constant119902 Internal heat generation rate
Ra Rayleigh numberSh Sherwood number119879 Temperature119905 Time119906 Velocity in 119909-direction119880119903 Reference velocity Velocity vectorV Velocity in 119910-direction119909 Distance along the surface119910 Distance normal to the surface
Greek Letters
120572119898 Thermal diffusivity120573 Volumetric expansion coefficient of nanofluid120576 Porosity of the porous media120601 Dimensionless concentration120578 Similarity variable120583 Dynamic viscosity120579 Dimensionless temperature120588119891 Density of the base fluid120588119901 Density of the nanoparticles(120588119862)119891 Effective heat capacity of the fluid(120588119862)119875 Effective heat capacity of the nanoparticle material120595 Stream function
Subscript Superscript
infin Conditions far away from the surface1015840 Differentiation with respect to 120578
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
M M Rashidi wishes to thank the Centre for DifferentialEquations Continuum Mechanics and Applications Schoolof Computational and Applied Mathematics University ofthe Witwatersrand for the hospitality during his visit EMomoniat thanks theNational Research Foundation of SouthAfrica for their support
20 Mathematical Problems in Engineering
References
[1] E Abu-Nada F O Hakan and I Pop ldquoBuoyancy inducedflow in a nanofluid filled enclosure partially exposed to forcedconvectionrdquo Superlattices and Microstructures vol 51 no 3 pp381ndash395 2012
[2] S Kakac and A Pramuanjaroenkij ldquoReview of convective heattransfer enhancement with nanofluidsrdquo International Journal ofHeat and Mass Transfer vol 52 no 13-14 pp 3187ndash3196 2009
[3] J A Eastman S U S Choi S Li W Yu and L J ThompsonldquoAnomalously increased effective thermal conductivities ofethylene glycol-based nanofluids containing copper nanoparti-clesrdquo Applied Physics Letters vol 78 no 6 pp 718ndash720 2001
[4] Y Xuan and Q Li ldquoHeat transfer enhancement of nanofluidsrdquoInternational Journal of Heat and Fluid Flow vol 21 no 1 pp58ndash64 2000
[5] S K Das N Putra P Thiesen and W Roetzel ldquoTemperaturedependence of thermal conductivity enhancement for nanoflu-idsrdquo Journal of Heat Transfer vol 125 no 4 pp 567ndash574 2003
[6] D H Kumar H E Patel V R R Kumar T SundararajanT Pradeep and S K Das ldquoModel for heat conduction innanofluidsrdquo Physical Review Letters vol 93 no 14 Article ID144301 2004
[7] H E Patel T Sundararajan T Pradeep A Dasgupta NDasgupta and S K Das ldquoA micro-convection model forthermal conductivity of nanofluidsrdquo Pramana vol 65 no 5 pp863ndash869 2005
[8] A V Kuznetsov ldquoNon-oscillatory and oscillatory nanofluidbio-thermal convection in a horizontal layer of finite depthrdquoEuropean Journal of Mechanics vol 30 no 2 pp 156ndash165 2011
[9] M Napoli J C T Eijkel and S Pennathur ldquoNanofluidictechnology for biomolecule applications a critical reviewrdquo Labon a Chip vol 10 no 8 pp 957ndash985 2010
[10] J C Crepeau and R Clarksean ldquoSimilarity solutions of naturalconvection with internal heat generationrdquo Journal of HeatTransfer vol 119 no 1 pp 183ndash185 1997
[11] A Z Sahin ldquoTransient heat conduction in semi-infinite solidwith spatially decaying exponential heat generationrdquo Interna-tional Communications in Heat andMass Transfer vol 19 no 3pp 349ndash358 1992
[12] O D Makind and A Aziz ldquoMixed convection from a con-vectively heated vertical plate to a fluid with internal heatgenerationrdquo Journal of Heat Transfer vol 133 Article ID 1225016 pages 2011
[13] J Buongiorno ldquoConvective transport in nanofluidsrdquo Journal ofHeat Transfer vol 128 no 3 pp 240ndash250 2006
[14] AV Kuznetsov andDANield ldquoNatural convective boundary-layer flow of a nanofluid past a vertical platerdquo InternationalJournal of Thermal Sciences vol 49 no 2 pp 243ndash247 2010
[15] L B Godson D Raja L D Mohan and S WongwisescldquoEnhancement of heat transfer using nanofluids-an overviewrdquoRenewable and Sustainable Energy Reviews vol 14 no 2 pp629ndash641 2010
[16] D A Nield and A BejanConvection in PorousMedia SpringerNew York NY USA 3rd edition 2006
[17] P Vadasz Emerging Topics in Heat and Mass Transfer in PorousMedia Springer New York NY USA 2008
[18] K Vafai Porous Media Applications in Biological Systems andBiotechnology CRC Press New York NY USA 2010
[19] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application to
heat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977
[20] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for natural convective boundary-layer flow in a porousmedium saturated by a nanofluidrdquo International Journal of Heatand Mass Transfer vol 52 no 25-26 pp 5792ndash5795 2009
[21] P Cheng and I D Chang ldquoBuoyancy induced flows in asaturated porous medium adjacent to impermeable horizontalsurfacesrdquo International Journal of Heat and Mass Transfer vol19 no 11 pp 1267ndash1272 1976
[22] I D Chang and P Cheng ldquoMatched asymptotic expansionsfor free convection about an impermeable horizontal surfacein a porous mediumrdquo International Journal of Heat and MassTransfer vol 26 no 2 pp 163ndash174 1983
[23] D S Shiunlin and B Gebhart ldquoBuoyancy-induced flow adja-cent to a horizontal surface submerged in porous mediumsaturated with cold waterrdquo International Journal of Heat andMass Transfer vol 29 no 4 pp 611ndash623 1986
[24] J H Merkin and G Zhang ldquoOn the similarity solutions for freeconvection in a saturated porousmedium adjacent to imperme-able horizontal surfacesrdquoWarme- und Stoffubertragung vol 25no 3 pp 179ndash184 1990
[25] M A Chaudhary J H Merkin and I Pop ldquoNatural convectionfrom a horizontal permeable surface in a porous mediummdashnumerical and asymptotic solutionsrdquo Transport in PorousMedia vol 22 no 3 pp 327ndash344 1996
[26] R S R Gorla and A Chamkha ldquoNatural convective boundarylayer flow over a nonisothermal vertical plate embedded ina porous medium saturated with a nanofluidrdquo Nanoscale andMicroscale Thermophysical Engineering vol 15 no 2 pp 81ndash942011
[27] W A Khan and I Pop ldquoFree convection boundary layer flowpast a horizontal flat plate embedded in a porous medium filledwith a nanofluidrdquo Journal of Heat Transfer vol 133 no 9 ArticleID 094501 2011
[28] A Aziz W A Khan and I Pop ldquoFree convection boundarylayer flow past a horizontal flat plate embedded in porousmedium filled by nanofluid containing gyrotactic microorgan-ismsrdquo International Journal of Thermal Sciences vol 56 pp 48ndash57 2012
[29] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer New York NYUSA 2009
[30] A A Avramenko S G Kobzar I V Shevchuk A V Kuznetsovand L T Iwanisov ldquoSymmetry of turbulent boundary-layerflows investigation of different eddy viscosity modelsrdquo ActaMechanica vol 151 no 1-2 pp 1ndash14 2001
[31] A V Kuznetsov A A Avramenko and P Geng ldquoAnalyticalinvestigation of a falling plume caused by bioconvection ofoxytactic bacteria in a fluid saturated porous mediumrdquo Inter-national Journal of Engineering Science vol 42 no 5-6 pp 557ndash569 2004
[32] M Jalil S Asghar and M Mushtaq ldquoLie group analysis ofmixed convection flow with mass transfer over a stretchingsurface with suction or injectionrdquo Mathematical Problems inEngineering vol 2010 Article ID 264901 14 pages 2010
[33] M A A Hamad M J Uddin and A I M Ismail ldquoRadiationeffects on heat and mass transfer in MHD stagnation-pointflow over a permeable flat plate with thermal convectivesurface boundary condition temperature dependent viscosityand thermal conductivityrdquoNuclear Engineering and Design vol242 pp 194ndash200 2012
Mathematical Problems in Engineering 21
[34] A Aziz M J Uddin M A A Hamad and A I MdIsmail ldquoMHD flow over an inclined radiating plate with thetemperature-dependent thermal conductivity variable reactiveindex and heat generationrdquoHeat Transfer vol 41 no 3 pp 241ndash259 2012
[35] A G Hansen Similarity Analysis of Boundary Layer Problemsin Engineering Prentice Hall Englewood Cliffs NJ USA 1964
[36] W FAmesNonlinear Partial Differential Equations in Engineer-ing Academic Press New York NY USA 1972
[37] R Seshadri and T Y Na Group Invariance in EngineeringBoundary Value Problems Springer New York NY USA 1985
[38] D Shang Theory of Heat Transfer with Forced Convection FilmFlows vol 3 of Heat and Mass Transfer 2010
[40] E Erfani M M Rashidi and A B Parsa ldquoThe modifieddifferential transform method for solving off-centered stag-nation flow toward a rotating discrdquo International Journal ofComputational Methods vol 7 no 4 pp 655ndash670 2010
[41] M M Rashidi N Laraqi and A Basiri Parsa ldquoAnalyticalmodeling of heat convection in magnetized micropolar fluid byusing modified differential transform methodrdquo Heat Transfervol 40 no 3 pp 187ndash204 2011
[42] M M Rashidi and M Keimanesh ldquoUsing differential trans-form method and pade approximant for solving mhd flow ina laminar liquid film from a horizontal stretching surfacerdquoMathematical Problems in Engineering vol 2010 Article ID491319 14 pages 2010
[43] M M Rashidi and G Domairry ldquoNew analytical solution ofthe three-dimensional NavierStokes equationsrdquoModern PhysicsLetters B vol 23 no 26 pp 3147ndash3155 2009
[44] M M Rashidi M Keimanesh O A Beg and T K HungldquoMagnetohydrodynamic biorheological transport phenomenain a porous medium a simulation of magnetic blood flowcontrol and filtrationrdquo International Journal for NumericalMethods in Biomedical Engineering vol 27 no 6 pp 805ndash8212011
[45] S J Liao The proposed homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992
[46] M M Rashidi G Domairry and S Dinarvand ldquoApproximatesolutions for the Burger and regularized long wave equationsby means of the homotopy analysis methodrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 14 no 3 pp708ndash717 2009
[47] Z Ziabakhsh and G Domairry ldquoAnalytic solution of naturalconvection flow of a non-Newtonian fluid between two verticalflat plates using homotopy analysis methodrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 14 no 5pp 1868ndash1880 2009
[48] M M Rashidi S A Mohimanian Pour T Hayat and SObaidat ldquoAnalytic approximate solutions for steady flow over arotating disk in porousmediumwith heat transfer by homotopyanalysis methodrdquo Computers and Fluids vol 54 no 1 pp 1ndash92012
[49] M Sajid T Hayat and S Asghar ldquoComparison between theHAM and HPM solutions of thin film flows of non-Newtonianfluids on a moving beltrdquo Nonlinear Dynamics vol 50 no 1-2pp 27ndash35 2007
[50] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for the double-diffusive natural convective boundary
layer flow in a porous medium saturated by a nanofluidrdquoInternational Journal of Heat and Mass Transfer vol 54 no 1-3pp 374ndash378 2011
[51] K Vajravelu and A Hadjinicolaou ldquoConvective heat transferin an electrically conducting fluid at a stretching surfacewith uniform free streamrdquo International Journal of EngineeringScience vol 35 pp 1237ndash1244 1997
[52] S Mukhopadhyay and G C Layek ldquoEffects of variable fluidviscosity on flow past a heated stretching sheet embedded ina porous medium in presence of heat sourcesinkrdquo Meccanicavol 47 no 4 pp 863ndash876 2012
[53] S J Liao ldquoAn optimal homotopy-analysis approach for stronglynonlinear differential equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 8 pp 2003ndash20162010
[54] A Aziz ldquoA similarity solution for laminar thermal boundarylayer over a flat plate with a convective surface boundary con-ditionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1064ndash1068 2009
Figure 15 Effect of Brownian motion 119873119887 on the dimensionlesstemperature profile
02
04
06
08
1
00
2 4 6 8 10120578
NumericalNumerical
Numerical
Nb = 01
Nb = 03
Nb = 05
Nt = 01Nr = 01Q = 01Le = 10K = 10n = 10b = 00
120601(120578
)
Figure 16 Effect of Brownian motion 119873119887 on the dimensionlessconcentration profile
119899 = 20 119887 = 01 It is observed that the results of 12th-order approximation of the optimal HAM are very close tothe numerical solutions which confirm the validity of thesemethods
In the following figures effects of various physicalparameters on the dimensionless velocity temperature and
Table 3 Comparison of values of 1198911015840(120578) obtained by various ordersof the HAM with numerical solution when 119873119887 = 01119873119905 =
temperature profiles will be investigated These results havebeen obtained by 12th order of the HAM and have beenvalidated by numerical results Figures 11ndash13 respectivelyrepresent that the comparison of solutions of 1198911015840(120578) 120579(120578)and 120601(120578) for fix values 119873119887 = 01 119873119905 = 01 119876 = 01Le = 10 119870 = 10 119899 = 10 119887 = 00 and different values ofbuoyancy ratio119873119903 In Figure 11 it is clear that velocity of thefluid tremendously decreases in the near of horizontal platewith an increase in the buoyancy ratio It is observed that thetemperature increases slightly but concentration of the fluiddoes not vary sensibly (Figures 12 and 13) The effects of theBrownian motion 119873119887 are depicted in Figures 14 15 and 16when119873119903 = 01119873119905 = 01119876 = 01 Le = 10119870 = 10 119899 = 10
14 Mathematical Problems in Engineering
Table 6 Comparison of values of 1198911015840(120578) obtained by various ordersof the HAM with numerical solution when 119873119887 = 03119873119905 =
119887 = 00 If 119873119887 increases 1198911015840(120578) and 120579(120578) increase lightly and120601(120578) decreases The dimensionless velocity dimensionlesstemperature and dimensionless concentration profiles fordifferent values of thermophoresis 119873119905 with constant values119873119903 = 01 119873119887 = 01 119876 = 01 Le = 10 119870 = 10119899 = 10 119887 = 00 are presented in Figures 17 18 and19 It is observed that the velocity and temperature of thefluid are not impressible from119873119905 but concentration increasesslightly In Figures 20 21 and 22 respectively comparison ofsolutions of 1198911015840(120578) 120579(120578) and 120601(120578) for 119873119903 = 01 119873119887 = 01119873119905 = 01 Le = 10 119870 = 10 119899 = 10 119887 = 00 anddifferent values of heat generationabsorption parameter 119876are shown It is observed that in presence of heat source1198911015840(120578) 120579(120578) both increase while heat sink causes that 1198911015840(120578)
120579(120578) both decrease extremely These effects are more visible
Table 9 Values of reduced Nusselt number and Sherwood numberobtained by HAM for different values of the parameters Le 119870 119899 119887and 119876 when119873119887 = 119873119903 = 119873119905 = 01
Figure 17 Effect of thermophoresis 119873119905 on the dimensionlessvelocity profile
for temperature distribution For 120601(120578) the effect of changing119876 is seen to be almost insignificant This is in agreementwith the physical factThe dimensionless profiles for differentvalues of Lewis number Le with constant parameters 119873119903 =01 119873119887 = 01 119873119905 = 01 119876 = 01 119870 = 10 119899 = 10119887 = 00 are shown in Figures 23 24 and 25 It is clear that forthese constant parameters variation of Lewis number has anyconsiderable effect on the velocity and temperature profilesbut concentration profile decreases with the increase of Lewisnumber What is similar to these results can be detected inFigures 26 27 and 28 for chemical reaction parameter 119870Figures 29 30 and 31 respectively present the comparisonof solutions of 1198911015840(120578) 120579(120578) and 120601(120578) for 119873119903 = 05 119873119887 = 05119873119905 = 01 119876 = 001 Le = 10 119870 = 10 119887 = 00 and differentvalues of order of chemical reaction 119899 It is observed that
Mathematical Problems in Engineering 15
0 2 4 6 80
1
120578
Nt = 00
Nt = 005
Nt = 10NumericalNumerical
Numerical
08
06
04
02
10
Q = 01Le = 10K = 10n = 10b = 00
Nb = 01
120579(120578
)
Nr = 01
Figure 18 Effect of thermophoresis 119873119905 on the dimensionlesstemperature profile
0 2 4 6 80
1
120601(120578
)
120578
Nt = 00
Nt = 005
Nt = 10NumericalNumerical
Numerical
08
06
04
02
10
Q = 01Le = 10K = 10n = 10b = 00
Nb = 01Nr = 01
Figure 19 Effect of thermophoresis 119873119905 on the dimensionlessconcentration profile
for higher order of chemical reaction velocity decreases andtemperature and concentration profiles increase extremelyUltimately Figures 32 33 and 34 depict the effect of thermalslip parameter 119887 on velocity temperature and concentrationfunctions when 119873119903 = 01 119873119887 = 01 119873119905 = 01 119876 = 01Le = 20 119870 = 15 119899 = 10 It is seen that the velocityand temperature of the fluid decrease with the increase of the
0 2 4 6 80
1
12
10
02
04
06
08
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numericalf998400 (120578
)
120578
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 20 Effect of generationabsorption heat parameter119876 on thedimensionless velocity profile
0 2 4 6 8 100
02
04
06
08
1
120579(120578
)
120578
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numerical
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 21 Effect of generationabsorption heat parameter 119876 on thedimensionless temperature profile
thermal slip parameter in the near of horizontal plate whilethe concentration of the fluid does not vary patently
Also for investigation of the parameters of physicalinterest Table 9 is presented In this table numerical valuesof reduced Nusselt number and Sherwood number obtainedby HAM for different values of the parameters 119890 119870 119899 119887 and119876 can be compared when119873119887 = 119873119903 = 119873119905 = 01
16 Mathematical Problems in Engineering
10
02
04
06
08
0 2 4 6 80
1
120578
120601(120578
)
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numerical
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 22 Effect of generationabsorption heat parameter119876 on thedimensionless concentration profile
0 2 4 6 80
1
12
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 23 Effect of Lewis number Le on the dimensionless velocityprofile
9 Conclusions
In this paper we studied the steady laminar incompressiblefree convective flow of a nanofluid past a chemically reactingupward facing horizontal plate in porous medium takinginto account heat generation and the thermal slip boundarycondition
0 2 4 6 80
1
120579(120578
)
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 24 Effect of Lewis number Le on the dimensionlesstemperature profile
0 2 4 6 80
1
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 25 Effect of Lewis number Le on the dimensionlessconcentration profile
The governing partial differential equations have beentransformed by similarity transformations into a system ofordinary differential equations which are solved by OHAMand numerical method (fourth-order Runge-Kutta schemewith the shooting method) Dimensionless velocity temper-ature and concentration functions are presented for various
Mathematical Problems in Engineering 17
0 2 4 6 80
1
12
K = 00
K = 05
K = 10NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 26 Effect of chemical reaction parameter 119870 on the dimen-sionless velocity profile
0 2 4 6 80
1
K = 00
K = 05
K = 10NumericalNumerical
Numerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 27 Effect of chemical reaction parameter 119870 on the dimen-sionless temperature profile
values of parameters for the problem for example buoy-ancy ratio 119873119903 Brownian motion 119873119887 thermophoresis 119873119905heat generationabsorption parameter 119876 Lewis number Lechemical reaction parameter119870 order of chemical reaction 119899and thermal slip parameter 119887 From the present investigationthe following may be concluded
(i) The velocity 1198911015840(120578) increases with increasing 119873119887 anddecreases with increasing119873119903 119899 and 119887
0 2 4 6 80
1
K = 00
K = 05
K = 10NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 28 Effect of chemical reaction parameter 119870 on the dimen-sionless concentration profile
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 29 Effect of order of chemical reaction 119899 on the dimension-less velocity profile
(ii) The temperature 120579(120578) increases with 119873119903 119873119887 119899 anddecreases with 119887
(iii) Parameters of119873119905 Le and119870 have no effect on velocityand temperature profiles
(iv) The concentration distribution 120601(120578) increases withincreasing N119905 119899 and decreases with 119873119887 Le and 119870but119873119903 119876 and 119887 are not affected by it
18 Mathematical Problems in Engineering
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 30 Effect of order of chemical reaction 119899 on the dimension-less temperature profile
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 31 Effect of order of chemical reaction 119899 on the dimension-less concentration profile
(v) In presence of heat source (119876 gt 0) 1198911015840(120578) 120579(120578) bothincrease while heat sink (119876 lt 0) causes that 1198911015840(120578)120579(120578) both decrease extremely These effects are morevisible for temperature distribution
In addition numerical results for the reducedNusselt andSherwood numbers are tabulated It is concluded that thereduced Nusselt number increases with the Lewis numberand reaction parameter whist it decreases with the order of
0 2 4 6 8
1
12
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 32 Effect of order of thermal slip parameter 119887 on thedimensionless velocity profile
0 2 4 6 80
1
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 33 Effect of order of thermal slip parameter 119887 on thedimensionless temperature profile
chemical reaction thermal slip and generation parametersIt is further concluded that the reduced Sherwood numberenhances with the Lewis number generation and reactionparameters whist it suppresses with thermal slip and orderof chemical reaction
The convergence of the solution series and the power ofHAM in controlling and adjusting the convergence regionand rate of solution series were discussed The proper range
Mathematical Problems in Engineering 19
0 2 4 6 80
1
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 34 Effect of order of thermal slip parameter 119887 on thedimensionless concentration profile
of auxiliary parameter ℏ to ensure the convergence of thesolution series was obtained through the so called ℏ-curveThe HAM provides us with a convenient way to control theconvergence of approximation series which is a fundamentalqualitative difference in analysis between the HAM andother methods The numerical results of the above problemsdisplay a fast convergence with minimal calculations Thisshows that the HAM is a very efficient method Finally theagreement between analytical and numerical results of thepresent study with previous published results is excellent
Nomenclature
119887 Thermal slip parameter119862 Dimensional concentration119862119891 Skin friction factor1198631 Thermal slip factor119863119861 Brownian diffusion coefficient119863119879 Thermophoretic diffusion coefficient119891 Dimensionless velocity functions119892 Gravitation acceleration119870 Chemical reaction parameter119896 Permeability of the porous media1198960 The constant reaction rate119896119898 Effective thermal conductivity of the porous
medium119896(119909) Variable reaction rateL Linear operator of the HAM119871 Length of horizontal plateLe Lewis numberN Nonlinear operator of the HAM
119899 Order of chemical reaction119873119887 Brownian motion119873119903 Buoyancy ratio119873119905 ThermophoresisNu Nusselt number119875 PressurePr Prandtl number119876 Heat generationabsorption parameter1198760 Heat generationabsorption constant119902 Internal heat generation rate
Ra Rayleigh numberSh Sherwood number119879 Temperature119905 Time119906 Velocity in 119909-direction119880119903 Reference velocity Velocity vectorV Velocity in 119910-direction119909 Distance along the surface119910 Distance normal to the surface
Greek Letters
120572119898 Thermal diffusivity120573 Volumetric expansion coefficient of nanofluid120576 Porosity of the porous media120601 Dimensionless concentration120578 Similarity variable120583 Dynamic viscosity120579 Dimensionless temperature120588119891 Density of the base fluid120588119901 Density of the nanoparticles(120588119862)119891 Effective heat capacity of the fluid(120588119862)119875 Effective heat capacity of the nanoparticle material120595 Stream function
Subscript Superscript
infin Conditions far away from the surface1015840 Differentiation with respect to 120578
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
M M Rashidi wishes to thank the Centre for DifferentialEquations Continuum Mechanics and Applications Schoolof Computational and Applied Mathematics University ofthe Witwatersrand for the hospitality during his visit EMomoniat thanks theNational Research Foundation of SouthAfrica for their support
20 Mathematical Problems in Engineering
References
[1] E Abu-Nada F O Hakan and I Pop ldquoBuoyancy inducedflow in a nanofluid filled enclosure partially exposed to forcedconvectionrdquo Superlattices and Microstructures vol 51 no 3 pp381ndash395 2012
[2] S Kakac and A Pramuanjaroenkij ldquoReview of convective heattransfer enhancement with nanofluidsrdquo International Journal ofHeat and Mass Transfer vol 52 no 13-14 pp 3187ndash3196 2009
[3] J A Eastman S U S Choi S Li W Yu and L J ThompsonldquoAnomalously increased effective thermal conductivities ofethylene glycol-based nanofluids containing copper nanoparti-clesrdquo Applied Physics Letters vol 78 no 6 pp 718ndash720 2001
[4] Y Xuan and Q Li ldquoHeat transfer enhancement of nanofluidsrdquoInternational Journal of Heat and Fluid Flow vol 21 no 1 pp58ndash64 2000
[5] S K Das N Putra P Thiesen and W Roetzel ldquoTemperaturedependence of thermal conductivity enhancement for nanoflu-idsrdquo Journal of Heat Transfer vol 125 no 4 pp 567ndash574 2003
[6] D H Kumar H E Patel V R R Kumar T SundararajanT Pradeep and S K Das ldquoModel for heat conduction innanofluidsrdquo Physical Review Letters vol 93 no 14 Article ID144301 2004
[7] H E Patel T Sundararajan T Pradeep A Dasgupta NDasgupta and S K Das ldquoA micro-convection model forthermal conductivity of nanofluidsrdquo Pramana vol 65 no 5 pp863ndash869 2005
[8] A V Kuznetsov ldquoNon-oscillatory and oscillatory nanofluidbio-thermal convection in a horizontal layer of finite depthrdquoEuropean Journal of Mechanics vol 30 no 2 pp 156ndash165 2011
[9] M Napoli J C T Eijkel and S Pennathur ldquoNanofluidictechnology for biomolecule applications a critical reviewrdquo Labon a Chip vol 10 no 8 pp 957ndash985 2010
[10] J C Crepeau and R Clarksean ldquoSimilarity solutions of naturalconvection with internal heat generationrdquo Journal of HeatTransfer vol 119 no 1 pp 183ndash185 1997
[11] A Z Sahin ldquoTransient heat conduction in semi-infinite solidwith spatially decaying exponential heat generationrdquo Interna-tional Communications in Heat andMass Transfer vol 19 no 3pp 349ndash358 1992
[12] O D Makind and A Aziz ldquoMixed convection from a con-vectively heated vertical plate to a fluid with internal heatgenerationrdquo Journal of Heat Transfer vol 133 Article ID 1225016 pages 2011
[13] J Buongiorno ldquoConvective transport in nanofluidsrdquo Journal ofHeat Transfer vol 128 no 3 pp 240ndash250 2006
[14] AV Kuznetsov andDANield ldquoNatural convective boundary-layer flow of a nanofluid past a vertical platerdquo InternationalJournal of Thermal Sciences vol 49 no 2 pp 243ndash247 2010
[15] L B Godson D Raja L D Mohan and S WongwisescldquoEnhancement of heat transfer using nanofluids-an overviewrdquoRenewable and Sustainable Energy Reviews vol 14 no 2 pp629ndash641 2010
[16] D A Nield and A BejanConvection in PorousMedia SpringerNew York NY USA 3rd edition 2006
[17] P Vadasz Emerging Topics in Heat and Mass Transfer in PorousMedia Springer New York NY USA 2008
[18] K Vafai Porous Media Applications in Biological Systems andBiotechnology CRC Press New York NY USA 2010
[19] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application to
heat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977
[20] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for natural convective boundary-layer flow in a porousmedium saturated by a nanofluidrdquo International Journal of Heatand Mass Transfer vol 52 no 25-26 pp 5792ndash5795 2009
[21] P Cheng and I D Chang ldquoBuoyancy induced flows in asaturated porous medium adjacent to impermeable horizontalsurfacesrdquo International Journal of Heat and Mass Transfer vol19 no 11 pp 1267ndash1272 1976
[22] I D Chang and P Cheng ldquoMatched asymptotic expansionsfor free convection about an impermeable horizontal surfacein a porous mediumrdquo International Journal of Heat and MassTransfer vol 26 no 2 pp 163ndash174 1983
[23] D S Shiunlin and B Gebhart ldquoBuoyancy-induced flow adja-cent to a horizontal surface submerged in porous mediumsaturated with cold waterrdquo International Journal of Heat andMass Transfer vol 29 no 4 pp 611ndash623 1986
[24] J H Merkin and G Zhang ldquoOn the similarity solutions for freeconvection in a saturated porousmedium adjacent to imperme-able horizontal surfacesrdquoWarme- und Stoffubertragung vol 25no 3 pp 179ndash184 1990
[25] M A Chaudhary J H Merkin and I Pop ldquoNatural convectionfrom a horizontal permeable surface in a porous mediummdashnumerical and asymptotic solutionsrdquo Transport in PorousMedia vol 22 no 3 pp 327ndash344 1996
[26] R S R Gorla and A Chamkha ldquoNatural convective boundarylayer flow over a nonisothermal vertical plate embedded ina porous medium saturated with a nanofluidrdquo Nanoscale andMicroscale Thermophysical Engineering vol 15 no 2 pp 81ndash942011
[27] W A Khan and I Pop ldquoFree convection boundary layer flowpast a horizontal flat plate embedded in a porous medium filledwith a nanofluidrdquo Journal of Heat Transfer vol 133 no 9 ArticleID 094501 2011
[28] A Aziz W A Khan and I Pop ldquoFree convection boundarylayer flow past a horizontal flat plate embedded in porousmedium filled by nanofluid containing gyrotactic microorgan-ismsrdquo International Journal of Thermal Sciences vol 56 pp 48ndash57 2012
[29] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer New York NYUSA 2009
[30] A A Avramenko S G Kobzar I V Shevchuk A V Kuznetsovand L T Iwanisov ldquoSymmetry of turbulent boundary-layerflows investigation of different eddy viscosity modelsrdquo ActaMechanica vol 151 no 1-2 pp 1ndash14 2001
[31] A V Kuznetsov A A Avramenko and P Geng ldquoAnalyticalinvestigation of a falling plume caused by bioconvection ofoxytactic bacteria in a fluid saturated porous mediumrdquo Inter-national Journal of Engineering Science vol 42 no 5-6 pp 557ndash569 2004
[32] M Jalil S Asghar and M Mushtaq ldquoLie group analysis ofmixed convection flow with mass transfer over a stretchingsurface with suction or injectionrdquo Mathematical Problems inEngineering vol 2010 Article ID 264901 14 pages 2010
[33] M A A Hamad M J Uddin and A I M Ismail ldquoRadiationeffects on heat and mass transfer in MHD stagnation-pointflow over a permeable flat plate with thermal convectivesurface boundary condition temperature dependent viscosityand thermal conductivityrdquoNuclear Engineering and Design vol242 pp 194ndash200 2012
Mathematical Problems in Engineering 21
[34] A Aziz M J Uddin M A A Hamad and A I MdIsmail ldquoMHD flow over an inclined radiating plate with thetemperature-dependent thermal conductivity variable reactiveindex and heat generationrdquoHeat Transfer vol 41 no 3 pp 241ndash259 2012
[35] A G Hansen Similarity Analysis of Boundary Layer Problemsin Engineering Prentice Hall Englewood Cliffs NJ USA 1964
[36] W FAmesNonlinear Partial Differential Equations in Engineer-ing Academic Press New York NY USA 1972
[37] R Seshadri and T Y Na Group Invariance in EngineeringBoundary Value Problems Springer New York NY USA 1985
[38] D Shang Theory of Heat Transfer with Forced Convection FilmFlows vol 3 of Heat and Mass Transfer 2010
[40] E Erfani M M Rashidi and A B Parsa ldquoThe modifieddifferential transform method for solving off-centered stag-nation flow toward a rotating discrdquo International Journal ofComputational Methods vol 7 no 4 pp 655ndash670 2010
[41] M M Rashidi N Laraqi and A Basiri Parsa ldquoAnalyticalmodeling of heat convection in magnetized micropolar fluid byusing modified differential transform methodrdquo Heat Transfervol 40 no 3 pp 187ndash204 2011
[42] M M Rashidi and M Keimanesh ldquoUsing differential trans-form method and pade approximant for solving mhd flow ina laminar liquid film from a horizontal stretching surfacerdquoMathematical Problems in Engineering vol 2010 Article ID491319 14 pages 2010
[43] M M Rashidi and G Domairry ldquoNew analytical solution ofthe three-dimensional NavierStokes equationsrdquoModern PhysicsLetters B vol 23 no 26 pp 3147ndash3155 2009
[44] M M Rashidi M Keimanesh O A Beg and T K HungldquoMagnetohydrodynamic biorheological transport phenomenain a porous medium a simulation of magnetic blood flowcontrol and filtrationrdquo International Journal for NumericalMethods in Biomedical Engineering vol 27 no 6 pp 805ndash8212011
[45] S J Liao The proposed homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992
[46] M M Rashidi G Domairry and S Dinarvand ldquoApproximatesolutions for the Burger and regularized long wave equationsby means of the homotopy analysis methodrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 14 no 3 pp708ndash717 2009
[47] Z Ziabakhsh and G Domairry ldquoAnalytic solution of naturalconvection flow of a non-Newtonian fluid between two verticalflat plates using homotopy analysis methodrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 14 no 5pp 1868ndash1880 2009
[48] M M Rashidi S A Mohimanian Pour T Hayat and SObaidat ldquoAnalytic approximate solutions for steady flow over arotating disk in porousmediumwith heat transfer by homotopyanalysis methodrdquo Computers and Fluids vol 54 no 1 pp 1ndash92012
[49] M Sajid T Hayat and S Asghar ldquoComparison between theHAM and HPM solutions of thin film flows of non-Newtonianfluids on a moving beltrdquo Nonlinear Dynamics vol 50 no 1-2pp 27ndash35 2007
[50] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for the double-diffusive natural convective boundary
layer flow in a porous medium saturated by a nanofluidrdquoInternational Journal of Heat and Mass Transfer vol 54 no 1-3pp 374ndash378 2011
[51] K Vajravelu and A Hadjinicolaou ldquoConvective heat transferin an electrically conducting fluid at a stretching surfacewith uniform free streamrdquo International Journal of EngineeringScience vol 35 pp 1237ndash1244 1997
[52] S Mukhopadhyay and G C Layek ldquoEffects of variable fluidviscosity on flow past a heated stretching sheet embedded ina porous medium in presence of heat sourcesinkrdquo Meccanicavol 47 no 4 pp 863ndash876 2012
[53] S J Liao ldquoAn optimal homotopy-analysis approach for stronglynonlinear differential equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 8 pp 2003ndash20162010
[54] A Aziz ldquoA similarity solution for laminar thermal boundarylayer over a flat plate with a convective surface boundary con-ditionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1064ndash1068 2009
Table 6 Comparison of values of 1198911015840(120578) obtained by various ordersof the HAM with numerical solution when 119873119887 = 03119873119905 =
119887 = 00 If 119873119887 increases 1198911015840(120578) and 120579(120578) increase lightly and120601(120578) decreases The dimensionless velocity dimensionlesstemperature and dimensionless concentration profiles fordifferent values of thermophoresis 119873119905 with constant values119873119903 = 01 119873119887 = 01 119876 = 01 Le = 10 119870 = 10119899 = 10 119887 = 00 are presented in Figures 17 18 and19 It is observed that the velocity and temperature of thefluid are not impressible from119873119905 but concentration increasesslightly In Figures 20 21 and 22 respectively comparison ofsolutions of 1198911015840(120578) 120579(120578) and 120601(120578) for 119873119903 = 01 119873119887 = 01119873119905 = 01 Le = 10 119870 = 10 119899 = 10 119887 = 00 anddifferent values of heat generationabsorption parameter 119876are shown It is observed that in presence of heat source1198911015840(120578) 120579(120578) both increase while heat sink causes that 1198911015840(120578)
120579(120578) both decrease extremely These effects are more visible
Table 9 Values of reduced Nusselt number and Sherwood numberobtained by HAM for different values of the parameters Le 119870 119899 119887and 119876 when119873119887 = 119873119903 = 119873119905 = 01
Figure 17 Effect of thermophoresis 119873119905 on the dimensionlessvelocity profile
for temperature distribution For 120601(120578) the effect of changing119876 is seen to be almost insignificant This is in agreementwith the physical factThe dimensionless profiles for differentvalues of Lewis number Le with constant parameters 119873119903 =01 119873119887 = 01 119873119905 = 01 119876 = 01 119870 = 10 119899 = 10119887 = 00 are shown in Figures 23 24 and 25 It is clear that forthese constant parameters variation of Lewis number has anyconsiderable effect on the velocity and temperature profilesbut concentration profile decreases with the increase of Lewisnumber What is similar to these results can be detected inFigures 26 27 and 28 for chemical reaction parameter 119870Figures 29 30 and 31 respectively present the comparisonof solutions of 1198911015840(120578) 120579(120578) and 120601(120578) for 119873119903 = 05 119873119887 = 05119873119905 = 01 119876 = 001 Le = 10 119870 = 10 119887 = 00 and differentvalues of order of chemical reaction 119899 It is observed that
Mathematical Problems in Engineering 15
0 2 4 6 80
1
120578
Nt = 00
Nt = 005
Nt = 10NumericalNumerical
Numerical
08
06
04
02
10
Q = 01Le = 10K = 10n = 10b = 00
Nb = 01
120579(120578
)
Nr = 01
Figure 18 Effect of thermophoresis 119873119905 on the dimensionlesstemperature profile
0 2 4 6 80
1
120601(120578
)
120578
Nt = 00
Nt = 005
Nt = 10NumericalNumerical
Numerical
08
06
04
02
10
Q = 01Le = 10K = 10n = 10b = 00
Nb = 01Nr = 01
Figure 19 Effect of thermophoresis 119873119905 on the dimensionlessconcentration profile
for higher order of chemical reaction velocity decreases andtemperature and concentration profiles increase extremelyUltimately Figures 32 33 and 34 depict the effect of thermalslip parameter 119887 on velocity temperature and concentrationfunctions when 119873119903 = 01 119873119887 = 01 119873119905 = 01 119876 = 01Le = 20 119870 = 15 119899 = 10 It is seen that the velocityand temperature of the fluid decrease with the increase of the
0 2 4 6 80
1
12
10
02
04
06
08
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numericalf998400 (120578
)
120578
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 20 Effect of generationabsorption heat parameter119876 on thedimensionless velocity profile
0 2 4 6 8 100
02
04
06
08
1
120579(120578
)
120578
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numerical
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 21 Effect of generationabsorption heat parameter 119876 on thedimensionless temperature profile
thermal slip parameter in the near of horizontal plate whilethe concentration of the fluid does not vary patently
Also for investigation of the parameters of physicalinterest Table 9 is presented In this table numerical valuesof reduced Nusselt number and Sherwood number obtainedby HAM for different values of the parameters 119890 119870 119899 119887 and119876 can be compared when119873119887 = 119873119903 = 119873119905 = 01
16 Mathematical Problems in Engineering
10
02
04
06
08
0 2 4 6 80
1
120578
120601(120578
)
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numerical
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 22 Effect of generationabsorption heat parameter119876 on thedimensionless concentration profile
0 2 4 6 80
1
12
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 23 Effect of Lewis number Le on the dimensionless velocityprofile
9 Conclusions
In this paper we studied the steady laminar incompressiblefree convective flow of a nanofluid past a chemically reactingupward facing horizontal plate in porous medium takinginto account heat generation and the thermal slip boundarycondition
0 2 4 6 80
1
120579(120578
)
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 24 Effect of Lewis number Le on the dimensionlesstemperature profile
0 2 4 6 80
1
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 25 Effect of Lewis number Le on the dimensionlessconcentration profile
The governing partial differential equations have beentransformed by similarity transformations into a system ofordinary differential equations which are solved by OHAMand numerical method (fourth-order Runge-Kutta schemewith the shooting method) Dimensionless velocity temper-ature and concentration functions are presented for various
Mathematical Problems in Engineering 17
0 2 4 6 80
1
12
K = 00
K = 05
K = 10NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 26 Effect of chemical reaction parameter 119870 on the dimen-sionless velocity profile
0 2 4 6 80
1
K = 00
K = 05
K = 10NumericalNumerical
Numerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 27 Effect of chemical reaction parameter 119870 on the dimen-sionless temperature profile
values of parameters for the problem for example buoy-ancy ratio 119873119903 Brownian motion 119873119887 thermophoresis 119873119905heat generationabsorption parameter 119876 Lewis number Lechemical reaction parameter119870 order of chemical reaction 119899and thermal slip parameter 119887 From the present investigationthe following may be concluded
(i) The velocity 1198911015840(120578) increases with increasing 119873119887 anddecreases with increasing119873119903 119899 and 119887
0 2 4 6 80
1
K = 00
K = 05
K = 10NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 28 Effect of chemical reaction parameter 119870 on the dimen-sionless concentration profile
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 29 Effect of order of chemical reaction 119899 on the dimension-less velocity profile
(ii) The temperature 120579(120578) increases with 119873119903 119873119887 119899 anddecreases with 119887
(iii) Parameters of119873119905 Le and119870 have no effect on velocityand temperature profiles
(iv) The concentration distribution 120601(120578) increases withincreasing N119905 119899 and decreases with 119873119887 Le and 119870but119873119903 119876 and 119887 are not affected by it
18 Mathematical Problems in Engineering
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 30 Effect of order of chemical reaction 119899 on the dimension-less temperature profile
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 31 Effect of order of chemical reaction 119899 on the dimension-less concentration profile
(v) In presence of heat source (119876 gt 0) 1198911015840(120578) 120579(120578) bothincrease while heat sink (119876 lt 0) causes that 1198911015840(120578)120579(120578) both decrease extremely These effects are morevisible for temperature distribution
In addition numerical results for the reducedNusselt andSherwood numbers are tabulated It is concluded that thereduced Nusselt number increases with the Lewis numberand reaction parameter whist it decreases with the order of
0 2 4 6 8
1
12
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 32 Effect of order of thermal slip parameter 119887 on thedimensionless velocity profile
0 2 4 6 80
1
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 33 Effect of order of thermal slip parameter 119887 on thedimensionless temperature profile
chemical reaction thermal slip and generation parametersIt is further concluded that the reduced Sherwood numberenhances with the Lewis number generation and reactionparameters whist it suppresses with thermal slip and orderof chemical reaction
The convergence of the solution series and the power ofHAM in controlling and adjusting the convergence regionand rate of solution series were discussed The proper range
Mathematical Problems in Engineering 19
0 2 4 6 80
1
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 34 Effect of order of thermal slip parameter 119887 on thedimensionless concentration profile
of auxiliary parameter ℏ to ensure the convergence of thesolution series was obtained through the so called ℏ-curveThe HAM provides us with a convenient way to control theconvergence of approximation series which is a fundamentalqualitative difference in analysis between the HAM andother methods The numerical results of the above problemsdisplay a fast convergence with minimal calculations Thisshows that the HAM is a very efficient method Finally theagreement between analytical and numerical results of thepresent study with previous published results is excellent
Nomenclature
119887 Thermal slip parameter119862 Dimensional concentration119862119891 Skin friction factor1198631 Thermal slip factor119863119861 Brownian diffusion coefficient119863119879 Thermophoretic diffusion coefficient119891 Dimensionless velocity functions119892 Gravitation acceleration119870 Chemical reaction parameter119896 Permeability of the porous media1198960 The constant reaction rate119896119898 Effective thermal conductivity of the porous
medium119896(119909) Variable reaction rateL Linear operator of the HAM119871 Length of horizontal plateLe Lewis numberN Nonlinear operator of the HAM
119899 Order of chemical reaction119873119887 Brownian motion119873119903 Buoyancy ratio119873119905 ThermophoresisNu Nusselt number119875 PressurePr Prandtl number119876 Heat generationabsorption parameter1198760 Heat generationabsorption constant119902 Internal heat generation rate
Ra Rayleigh numberSh Sherwood number119879 Temperature119905 Time119906 Velocity in 119909-direction119880119903 Reference velocity Velocity vectorV Velocity in 119910-direction119909 Distance along the surface119910 Distance normal to the surface
Greek Letters
120572119898 Thermal diffusivity120573 Volumetric expansion coefficient of nanofluid120576 Porosity of the porous media120601 Dimensionless concentration120578 Similarity variable120583 Dynamic viscosity120579 Dimensionless temperature120588119891 Density of the base fluid120588119901 Density of the nanoparticles(120588119862)119891 Effective heat capacity of the fluid(120588119862)119875 Effective heat capacity of the nanoparticle material120595 Stream function
Subscript Superscript
infin Conditions far away from the surface1015840 Differentiation with respect to 120578
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
M M Rashidi wishes to thank the Centre for DifferentialEquations Continuum Mechanics and Applications Schoolof Computational and Applied Mathematics University ofthe Witwatersrand for the hospitality during his visit EMomoniat thanks theNational Research Foundation of SouthAfrica for their support
20 Mathematical Problems in Engineering
References
[1] E Abu-Nada F O Hakan and I Pop ldquoBuoyancy inducedflow in a nanofluid filled enclosure partially exposed to forcedconvectionrdquo Superlattices and Microstructures vol 51 no 3 pp381ndash395 2012
[2] S Kakac and A Pramuanjaroenkij ldquoReview of convective heattransfer enhancement with nanofluidsrdquo International Journal ofHeat and Mass Transfer vol 52 no 13-14 pp 3187ndash3196 2009
[3] J A Eastman S U S Choi S Li W Yu and L J ThompsonldquoAnomalously increased effective thermal conductivities ofethylene glycol-based nanofluids containing copper nanoparti-clesrdquo Applied Physics Letters vol 78 no 6 pp 718ndash720 2001
[4] Y Xuan and Q Li ldquoHeat transfer enhancement of nanofluidsrdquoInternational Journal of Heat and Fluid Flow vol 21 no 1 pp58ndash64 2000
[5] S K Das N Putra P Thiesen and W Roetzel ldquoTemperaturedependence of thermal conductivity enhancement for nanoflu-idsrdquo Journal of Heat Transfer vol 125 no 4 pp 567ndash574 2003
[6] D H Kumar H E Patel V R R Kumar T SundararajanT Pradeep and S K Das ldquoModel for heat conduction innanofluidsrdquo Physical Review Letters vol 93 no 14 Article ID144301 2004
[7] H E Patel T Sundararajan T Pradeep A Dasgupta NDasgupta and S K Das ldquoA micro-convection model forthermal conductivity of nanofluidsrdquo Pramana vol 65 no 5 pp863ndash869 2005
[8] A V Kuznetsov ldquoNon-oscillatory and oscillatory nanofluidbio-thermal convection in a horizontal layer of finite depthrdquoEuropean Journal of Mechanics vol 30 no 2 pp 156ndash165 2011
[9] M Napoli J C T Eijkel and S Pennathur ldquoNanofluidictechnology for biomolecule applications a critical reviewrdquo Labon a Chip vol 10 no 8 pp 957ndash985 2010
[10] J C Crepeau and R Clarksean ldquoSimilarity solutions of naturalconvection with internal heat generationrdquo Journal of HeatTransfer vol 119 no 1 pp 183ndash185 1997
[11] A Z Sahin ldquoTransient heat conduction in semi-infinite solidwith spatially decaying exponential heat generationrdquo Interna-tional Communications in Heat andMass Transfer vol 19 no 3pp 349ndash358 1992
[12] O D Makind and A Aziz ldquoMixed convection from a con-vectively heated vertical plate to a fluid with internal heatgenerationrdquo Journal of Heat Transfer vol 133 Article ID 1225016 pages 2011
[13] J Buongiorno ldquoConvective transport in nanofluidsrdquo Journal ofHeat Transfer vol 128 no 3 pp 240ndash250 2006
[14] AV Kuznetsov andDANield ldquoNatural convective boundary-layer flow of a nanofluid past a vertical platerdquo InternationalJournal of Thermal Sciences vol 49 no 2 pp 243ndash247 2010
[15] L B Godson D Raja L D Mohan and S WongwisescldquoEnhancement of heat transfer using nanofluids-an overviewrdquoRenewable and Sustainable Energy Reviews vol 14 no 2 pp629ndash641 2010
[16] D A Nield and A BejanConvection in PorousMedia SpringerNew York NY USA 3rd edition 2006
[17] P Vadasz Emerging Topics in Heat and Mass Transfer in PorousMedia Springer New York NY USA 2008
[18] K Vafai Porous Media Applications in Biological Systems andBiotechnology CRC Press New York NY USA 2010
[19] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application to
heat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977
[20] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for natural convective boundary-layer flow in a porousmedium saturated by a nanofluidrdquo International Journal of Heatand Mass Transfer vol 52 no 25-26 pp 5792ndash5795 2009
[21] P Cheng and I D Chang ldquoBuoyancy induced flows in asaturated porous medium adjacent to impermeable horizontalsurfacesrdquo International Journal of Heat and Mass Transfer vol19 no 11 pp 1267ndash1272 1976
[22] I D Chang and P Cheng ldquoMatched asymptotic expansionsfor free convection about an impermeable horizontal surfacein a porous mediumrdquo International Journal of Heat and MassTransfer vol 26 no 2 pp 163ndash174 1983
[23] D S Shiunlin and B Gebhart ldquoBuoyancy-induced flow adja-cent to a horizontal surface submerged in porous mediumsaturated with cold waterrdquo International Journal of Heat andMass Transfer vol 29 no 4 pp 611ndash623 1986
[24] J H Merkin and G Zhang ldquoOn the similarity solutions for freeconvection in a saturated porousmedium adjacent to imperme-able horizontal surfacesrdquoWarme- und Stoffubertragung vol 25no 3 pp 179ndash184 1990
[25] M A Chaudhary J H Merkin and I Pop ldquoNatural convectionfrom a horizontal permeable surface in a porous mediummdashnumerical and asymptotic solutionsrdquo Transport in PorousMedia vol 22 no 3 pp 327ndash344 1996
[26] R S R Gorla and A Chamkha ldquoNatural convective boundarylayer flow over a nonisothermal vertical plate embedded ina porous medium saturated with a nanofluidrdquo Nanoscale andMicroscale Thermophysical Engineering vol 15 no 2 pp 81ndash942011
[27] W A Khan and I Pop ldquoFree convection boundary layer flowpast a horizontal flat plate embedded in a porous medium filledwith a nanofluidrdquo Journal of Heat Transfer vol 133 no 9 ArticleID 094501 2011
[28] A Aziz W A Khan and I Pop ldquoFree convection boundarylayer flow past a horizontal flat plate embedded in porousmedium filled by nanofluid containing gyrotactic microorgan-ismsrdquo International Journal of Thermal Sciences vol 56 pp 48ndash57 2012
[29] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer New York NYUSA 2009
[30] A A Avramenko S G Kobzar I V Shevchuk A V Kuznetsovand L T Iwanisov ldquoSymmetry of turbulent boundary-layerflows investigation of different eddy viscosity modelsrdquo ActaMechanica vol 151 no 1-2 pp 1ndash14 2001
[31] A V Kuznetsov A A Avramenko and P Geng ldquoAnalyticalinvestigation of a falling plume caused by bioconvection ofoxytactic bacteria in a fluid saturated porous mediumrdquo Inter-national Journal of Engineering Science vol 42 no 5-6 pp 557ndash569 2004
[32] M Jalil S Asghar and M Mushtaq ldquoLie group analysis ofmixed convection flow with mass transfer over a stretchingsurface with suction or injectionrdquo Mathematical Problems inEngineering vol 2010 Article ID 264901 14 pages 2010
[33] M A A Hamad M J Uddin and A I M Ismail ldquoRadiationeffects on heat and mass transfer in MHD stagnation-pointflow over a permeable flat plate with thermal convectivesurface boundary condition temperature dependent viscosityand thermal conductivityrdquoNuclear Engineering and Design vol242 pp 194ndash200 2012
Mathematical Problems in Engineering 21
[34] A Aziz M J Uddin M A A Hamad and A I MdIsmail ldquoMHD flow over an inclined radiating plate with thetemperature-dependent thermal conductivity variable reactiveindex and heat generationrdquoHeat Transfer vol 41 no 3 pp 241ndash259 2012
[35] A G Hansen Similarity Analysis of Boundary Layer Problemsin Engineering Prentice Hall Englewood Cliffs NJ USA 1964
[36] W FAmesNonlinear Partial Differential Equations in Engineer-ing Academic Press New York NY USA 1972
[37] R Seshadri and T Y Na Group Invariance in EngineeringBoundary Value Problems Springer New York NY USA 1985
[38] D Shang Theory of Heat Transfer with Forced Convection FilmFlows vol 3 of Heat and Mass Transfer 2010
[40] E Erfani M M Rashidi and A B Parsa ldquoThe modifieddifferential transform method for solving off-centered stag-nation flow toward a rotating discrdquo International Journal ofComputational Methods vol 7 no 4 pp 655ndash670 2010
[41] M M Rashidi N Laraqi and A Basiri Parsa ldquoAnalyticalmodeling of heat convection in magnetized micropolar fluid byusing modified differential transform methodrdquo Heat Transfervol 40 no 3 pp 187ndash204 2011
[42] M M Rashidi and M Keimanesh ldquoUsing differential trans-form method and pade approximant for solving mhd flow ina laminar liquid film from a horizontal stretching surfacerdquoMathematical Problems in Engineering vol 2010 Article ID491319 14 pages 2010
[43] M M Rashidi and G Domairry ldquoNew analytical solution ofthe three-dimensional NavierStokes equationsrdquoModern PhysicsLetters B vol 23 no 26 pp 3147ndash3155 2009
[44] M M Rashidi M Keimanesh O A Beg and T K HungldquoMagnetohydrodynamic biorheological transport phenomenain a porous medium a simulation of magnetic blood flowcontrol and filtrationrdquo International Journal for NumericalMethods in Biomedical Engineering vol 27 no 6 pp 805ndash8212011
[45] S J Liao The proposed homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992
[46] M M Rashidi G Domairry and S Dinarvand ldquoApproximatesolutions for the Burger and regularized long wave equationsby means of the homotopy analysis methodrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 14 no 3 pp708ndash717 2009
[47] Z Ziabakhsh and G Domairry ldquoAnalytic solution of naturalconvection flow of a non-Newtonian fluid between two verticalflat plates using homotopy analysis methodrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 14 no 5pp 1868ndash1880 2009
[48] M M Rashidi S A Mohimanian Pour T Hayat and SObaidat ldquoAnalytic approximate solutions for steady flow over arotating disk in porousmediumwith heat transfer by homotopyanalysis methodrdquo Computers and Fluids vol 54 no 1 pp 1ndash92012
[49] M Sajid T Hayat and S Asghar ldquoComparison between theHAM and HPM solutions of thin film flows of non-Newtonianfluids on a moving beltrdquo Nonlinear Dynamics vol 50 no 1-2pp 27ndash35 2007
[50] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for the double-diffusive natural convective boundary
layer flow in a porous medium saturated by a nanofluidrdquoInternational Journal of Heat and Mass Transfer vol 54 no 1-3pp 374ndash378 2011
[51] K Vajravelu and A Hadjinicolaou ldquoConvective heat transferin an electrically conducting fluid at a stretching surfacewith uniform free streamrdquo International Journal of EngineeringScience vol 35 pp 1237ndash1244 1997
[52] S Mukhopadhyay and G C Layek ldquoEffects of variable fluidviscosity on flow past a heated stretching sheet embedded ina porous medium in presence of heat sourcesinkrdquo Meccanicavol 47 no 4 pp 863ndash876 2012
[53] S J Liao ldquoAn optimal homotopy-analysis approach for stronglynonlinear differential equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 8 pp 2003ndash20162010
[54] A Aziz ldquoA similarity solution for laminar thermal boundarylayer over a flat plate with a convective surface boundary con-ditionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1064ndash1068 2009
Figure 18 Effect of thermophoresis 119873119905 on the dimensionlesstemperature profile
0 2 4 6 80
1
120601(120578
)
120578
Nt = 00
Nt = 005
Nt = 10NumericalNumerical
Numerical
08
06
04
02
10
Q = 01Le = 10K = 10n = 10b = 00
Nb = 01Nr = 01
Figure 19 Effect of thermophoresis 119873119905 on the dimensionlessconcentration profile
for higher order of chemical reaction velocity decreases andtemperature and concentration profiles increase extremelyUltimately Figures 32 33 and 34 depict the effect of thermalslip parameter 119887 on velocity temperature and concentrationfunctions when 119873119903 = 01 119873119887 = 01 119873119905 = 01 119876 = 01Le = 20 119870 = 15 119899 = 10 It is seen that the velocityand temperature of the fluid decrease with the increase of the
0 2 4 6 80
1
12
10
02
04
06
08
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numericalf998400 (120578
)
120578
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 20 Effect of generationabsorption heat parameter119876 on thedimensionless velocity profile
0 2 4 6 8 100
02
04
06
08
1
120579(120578
)
120578
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numerical
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 21 Effect of generationabsorption heat parameter 119876 on thedimensionless temperature profile
thermal slip parameter in the near of horizontal plate whilethe concentration of the fluid does not vary patently
Also for investigation of the parameters of physicalinterest Table 9 is presented In this table numerical valuesof reduced Nusselt number and Sherwood number obtainedby HAM for different values of the parameters 119890 119870 119899 119887 and119876 can be compared when119873119887 = 119873119903 = 119873119905 = 01
16 Mathematical Problems in Engineering
10
02
04
06
08
0 2 4 6 80
1
120578
120601(120578
)
Q = minus01
Q = 00
Q = 01NumericalNumerical
Numerical
Nt = 01Nr = 01Nb = 01Le = 10K = 10n = 10b = 00
Figure 22 Effect of generationabsorption heat parameter119876 on thedimensionless concentration profile
0 2 4 6 80
1
12
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 23 Effect of Lewis number Le on the dimensionless velocityprofile
9 Conclusions
In this paper we studied the steady laminar incompressiblefree convective flow of a nanofluid past a chemically reactingupward facing horizontal plate in porous medium takinginto account heat generation and the thermal slip boundarycondition
0 2 4 6 80
1
120579(120578
)
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 24 Effect of Lewis number Le on the dimensionlesstemperature profile
0 2 4 6 80
1
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 25 Effect of Lewis number Le on the dimensionlessconcentration profile
The governing partial differential equations have beentransformed by similarity transformations into a system ofordinary differential equations which are solved by OHAMand numerical method (fourth-order Runge-Kutta schemewith the shooting method) Dimensionless velocity temper-ature and concentration functions are presented for various
Mathematical Problems in Engineering 17
0 2 4 6 80
1
12
K = 00
K = 05
K = 10NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 26 Effect of chemical reaction parameter 119870 on the dimen-sionless velocity profile
0 2 4 6 80
1
K = 00
K = 05
K = 10NumericalNumerical
Numerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 27 Effect of chemical reaction parameter 119870 on the dimen-sionless temperature profile
values of parameters for the problem for example buoy-ancy ratio 119873119903 Brownian motion 119873119887 thermophoresis 119873119905heat generationabsorption parameter 119876 Lewis number Lechemical reaction parameter119870 order of chemical reaction 119899and thermal slip parameter 119887 From the present investigationthe following may be concluded
(i) The velocity 1198911015840(120578) increases with increasing 119873119887 anddecreases with increasing119873119903 119899 and 119887
0 2 4 6 80
1
K = 00
K = 05
K = 10NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 28 Effect of chemical reaction parameter 119870 on the dimen-sionless concentration profile
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 29 Effect of order of chemical reaction 119899 on the dimension-less velocity profile
(ii) The temperature 120579(120578) increases with 119873119903 119873119887 119899 anddecreases with 119887
(iii) Parameters of119873119905 Le and119870 have no effect on velocityand temperature profiles
(iv) The concentration distribution 120601(120578) increases withincreasing N119905 119899 and decreases with 119873119887 Le and 119870but119873119903 119876 and 119887 are not affected by it
18 Mathematical Problems in Engineering
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 30 Effect of order of chemical reaction 119899 on the dimension-less temperature profile
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 31 Effect of order of chemical reaction 119899 on the dimension-less concentration profile
(v) In presence of heat source (119876 gt 0) 1198911015840(120578) 120579(120578) bothincrease while heat sink (119876 lt 0) causes that 1198911015840(120578)120579(120578) both decrease extremely These effects are morevisible for temperature distribution
In addition numerical results for the reducedNusselt andSherwood numbers are tabulated It is concluded that thereduced Nusselt number increases with the Lewis numberand reaction parameter whist it decreases with the order of
0 2 4 6 8
1
12
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 32 Effect of order of thermal slip parameter 119887 on thedimensionless velocity profile
0 2 4 6 80
1
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 33 Effect of order of thermal slip parameter 119887 on thedimensionless temperature profile
chemical reaction thermal slip and generation parametersIt is further concluded that the reduced Sherwood numberenhances with the Lewis number generation and reactionparameters whist it suppresses with thermal slip and orderof chemical reaction
The convergence of the solution series and the power ofHAM in controlling and adjusting the convergence regionand rate of solution series were discussed The proper range
Mathematical Problems in Engineering 19
0 2 4 6 80
1
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 34 Effect of order of thermal slip parameter 119887 on thedimensionless concentration profile
of auxiliary parameter ℏ to ensure the convergence of thesolution series was obtained through the so called ℏ-curveThe HAM provides us with a convenient way to control theconvergence of approximation series which is a fundamentalqualitative difference in analysis between the HAM andother methods The numerical results of the above problemsdisplay a fast convergence with minimal calculations Thisshows that the HAM is a very efficient method Finally theagreement between analytical and numerical results of thepresent study with previous published results is excellent
Nomenclature
119887 Thermal slip parameter119862 Dimensional concentration119862119891 Skin friction factor1198631 Thermal slip factor119863119861 Brownian diffusion coefficient119863119879 Thermophoretic diffusion coefficient119891 Dimensionless velocity functions119892 Gravitation acceleration119870 Chemical reaction parameter119896 Permeability of the porous media1198960 The constant reaction rate119896119898 Effective thermal conductivity of the porous
medium119896(119909) Variable reaction rateL Linear operator of the HAM119871 Length of horizontal plateLe Lewis numberN Nonlinear operator of the HAM
119899 Order of chemical reaction119873119887 Brownian motion119873119903 Buoyancy ratio119873119905 ThermophoresisNu Nusselt number119875 PressurePr Prandtl number119876 Heat generationabsorption parameter1198760 Heat generationabsorption constant119902 Internal heat generation rate
Ra Rayleigh numberSh Sherwood number119879 Temperature119905 Time119906 Velocity in 119909-direction119880119903 Reference velocity Velocity vectorV Velocity in 119910-direction119909 Distance along the surface119910 Distance normal to the surface
Greek Letters
120572119898 Thermal diffusivity120573 Volumetric expansion coefficient of nanofluid120576 Porosity of the porous media120601 Dimensionless concentration120578 Similarity variable120583 Dynamic viscosity120579 Dimensionless temperature120588119891 Density of the base fluid120588119901 Density of the nanoparticles(120588119862)119891 Effective heat capacity of the fluid(120588119862)119875 Effective heat capacity of the nanoparticle material120595 Stream function
Subscript Superscript
infin Conditions far away from the surface1015840 Differentiation with respect to 120578
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
M M Rashidi wishes to thank the Centre for DifferentialEquations Continuum Mechanics and Applications Schoolof Computational and Applied Mathematics University ofthe Witwatersrand for the hospitality during his visit EMomoniat thanks theNational Research Foundation of SouthAfrica for their support
20 Mathematical Problems in Engineering
References
[1] E Abu-Nada F O Hakan and I Pop ldquoBuoyancy inducedflow in a nanofluid filled enclosure partially exposed to forcedconvectionrdquo Superlattices and Microstructures vol 51 no 3 pp381ndash395 2012
[2] S Kakac and A Pramuanjaroenkij ldquoReview of convective heattransfer enhancement with nanofluidsrdquo International Journal ofHeat and Mass Transfer vol 52 no 13-14 pp 3187ndash3196 2009
[3] J A Eastman S U S Choi S Li W Yu and L J ThompsonldquoAnomalously increased effective thermal conductivities ofethylene glycol-based nanofluids containing copper nanoparti-clesrdquo Applied Physics Letters vol 78 no 6 pp 718ndash720 2001
[4] Y Xuan and Q Li ldquoHeat transfer enhancement of nanofluidsrdquoInternational Journal of Heat and Fluid Flow vol 21 no 1 pp58ndash64 2000
[5] S K Das N Putra P Thiesen and W Roetzel ldquoTemperaturedependence of thermal conductivity enhancement for nanoflu-idsrdquo Journal of Heat Transfer vol 125 no 4 pp 567ndash574 2003
[6] D H Kumar H E Patel V R R Kumar T SundararajanT Pradeep and S K Das ldquoModel for heat conduction innanofluidsrdquo Physical Review Letters vol 93 no 14 Article ID144301 2004
[7] H E Patel T Sundararajan T Pradeep A Dasgupta NDasgupta and S K Das ldquoA micro-convection model forthermal conductivity of nanofluidsrdquo Pramana vol 65 no 5 pp863ndash869 2005
[8] A V Kuznetsov ldquoNon-oscillatory and oscillatory nanofluidbio-thermal convection in a horizontal layer of finite depthrdquoEuropean Journal of Mechanics vol 30 no 2 pp 156ndash165 2011
[9] M Napoli J C T Eijkel and S Pennathur ldquoNanofluidictechnology for biomolecule applications a critical reviewrdquo Labon a Chip vol 10 no 8 pp 957ndash985 2010
[10] J C Crepeau and R Clarksean ldquoSimilarity solutions of naturalconvection with internal heat generationrdquo Journal of HeatTransfer vol 119 no 1 pp 183ndash185 1997
[11] A Z Sahin ldquoTransient heat conduction in semi-infinite solidwith spatially decaying exponential heat generationrdquo Interna-tional Communications in Heat andMass Transfer vol 19 no 3pp 349ndash358 1992
[12] O D Makind and A Aziz ldquoMixed convection from a con-vectively heated vertical plate to a fluid with internal heatgenerationrdquo Journal of Heat Transfer vol 133 Article ID 1225016 pages 2011
[13] J Buongiorno ldquoConvective transport in nanofluidsrdquo Journal ofHeat Transfer vol 128 no 3 pp 240ndash250 2006
[14] AV Kuznetsov andDANield ldquoNatural convective boundary-layer flow of a nanofluid past a vertical platerdquo InternationalJournal of Thermal Sciences vol 49 no 2 pp 243ndash247 2010
[15] L B Godson D Raja L D Mohan and S WongwisescldquoEnhancement of heat transfer using nanofluids-an overviewrdquoRenewable and Sustainable Energy Reviews vol 14 no 2 pp629ndash641 2010
[16] D A Nield and A BejanConvection in PorousMedia SpringerNew York NY USA 3rd edition 2006
[17] P Vadasz Emerging Topics in Heat and Mass Transfer in PorousMedia Springer New York NY USA 2008
[18] K Vafai Porous Media Applications in Biological Systems andBiotechnology CRC Press New York NY USA 2010
[19] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application to
heat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977
[20] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for natural convective boundary-layer flow in a porousmedium saturated by a nanofluidrdquo International Journal of Heatand Mass Transfer vol 52 no 25-26 pp 5792ndash5795 2009
[21] P Cheng and I D Chang ldquoBuoyancy induced flows in asaturated porous medium adjacent to impermeable horizontalsurfacesrdquo International Journal of Heat and Mass Transfer vol19 no 11 pp 1267ndash1272 1976
[22] I D Chang and P Cheng ldquoMatched asymptotic expansionsfor free convection about an impermeable horizontal surfacein a porous mediumrdquo International Journal of Heat and MassTransfer vol 26 no 2 pp 163ndash174 1983
[23] D S Shiunlin and B Gebhart ldquoBuoyancy-induced flow adja-cent to a horizontal surface submerged in porous mediumsaturated with cold waterrdquo International Journal of Heat andMass Transfer vol 29 no 4 pp 611ndash623 1986
[24] J H Merkin and G Zhang ldquoOn the similarity solutions for freeconvection in a saturated porousmedium adjacent to imperme-able horizontal surfacesrdquoWarme- und Stoffubertragung vol 25no 3 pp 179ndash184 1990
[25] M A Chaudhary J H Merkin and I Pop ldquoNatural convectionfrom a horizontal permeable surface in a porous mediummdashnumerical and asymptotic solutionsrdquo Transport in PorousMedia vol 22 no 3 pp 327ndash344 1996
[26] R S R Gorla and A Chamkha ldquoNatural convective boundarylayer flow over a nonisothermal vertical plate embedded ina porous medium saturated with a nanofluidrdquo Nanoscale andMicroscale Thermophysical Engineering vol 15 no 2 pp 81ndash942011
[27] W A Khan and I Pop ldquoFree convection boundary layer flowpast a horizontal flat plate embedded in a porous medium filledwith a nanofluidrdquo Journal of Heat Transfer vol 133 no 9 ArticleID 094501 2011
[28] A Aziz W A Khan and I Pop ldquoFree convection boundarylayer flow past a horizontal flat plate embedded in porousmedium filled by nanofluid containing gyrotactic microorgan-ismsrdquo International Journal of Thermal Sciences vol 56 pp 48ndash57 2012
[29] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer New York NYUSA 2009
[30] A A Avramenko S G Kobzar I V Shevchuk A V Kuznetsovand L T Iwanisov ldquoSymmetry of turbulent boundary-layerflows investigation of different eddy viscosity modelsrdquo ActaMechanica vol 151 no 1-2 pp 1ndash14 2001
[31] A V Kuznetsov A A Avramenko and P Geng ldquoAnalyticalinvestigation of a falling plume caused by bioconvection ofoxytactic bacteria in a fluid saturated porous mediumrdquo Inter-national Journal of Engineering Science vol 42 no 5-6 pp 557ndash569 2004
[32] M Jalil S Asghar and M Mushtaq ldquoLie group analysis ofmixed convection flow with mass transfer over a stretchingsurface with suction or injectionrdquo Mathematical Problems inEngineering vol 2010 Article ID 264901 14 pages 2010
[33] M A A Hamad M J Uddin and A I M Ismail ldquoRadiationeffects on heat and mass transfer in MHD stagnation-pointflow over a permeable flat plate with thermal convectivesurface boundary condition temperature dependent viscosityand thermal conductivityrdquoNuclear Engineering and Design vol242 pp 194ndash200 2012
Mathematical Problems in Engineering 21
[34] A Aziz M J Uddin M A A Hamad and A I MdIsmail ldquoMHD flow over an inclined radiating plate with thetemperature-dependent thermal conductivity variable reactiveindex and heat generationrdquoHeat Transfer vol 41 no 3 pp 241ndash259 2012
[35] A G Hansen Similarity Analysis of Boundary Layer Problemsin Engineering Prentice Hall Englewood Cliffs NJ USA 1964
[36] W FAmesNonlinear Partial Differential Equations in Engineer-ing Academic Press New York NY USA 1972
[37] R Seshadri and T Y Na Group Invariance in EngineeringBoundary Value Problems Springer New York NY USA 1985
[38] D Shang Theory of Heat Transfer with Forced Convection FilmFlows vol 3 of Heat and Mass Transfer 2010
[40] E Erfani M M Rashidi and A B Parsa ldquoThe modifieddifferential transform method for solving off-centered stag-nation flow toward a rotating discrdquo International Journal ofComputational Methods vol 7 no 4 pp 655ndash670 2010
[41] M M Rashidi N Laraqi and A Basiri Parsa ldquoAnalyticalmodeling of heat convection in magnetized micropolar fluid byusing modified differential transform methodrdquo Heat Transfervol 40 no 3 pp 187ndash204 2011
[42] M M Rashidi and M Keimanesh ldquoUsing differential trans-form method and pade approximant for solving mhd flow ina laminar liquid film from a horizontal stretching surfacerdquoMathematical Problems in Engineering vol 2010 Article ID491319 14 pages 2010
[43] M M Rashidi and G Domairry ldquoNew analytical solution ofthe three-dimensional NavierStokes equationsrdquoModern PhysicsLetters B vol 23 no 26 pp 3147ndash3155 2009
[44] M M Rashidi M Keimanesh O A Beg and T K HungldquoMagnetohydrodynamic biorheological transport phenomenain a porous medium a simulation of magnetic blood flowcontrol and filtrationrdquo International Journal for NumericalMethods in Biomedical Engineering vol 27 no 6 pp 805ndash8212011
[45] S J Liao The proposed homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992
[46] M M Rashidi G Domairry and S Dinarvand ldquoApproximatesolutions for the Burger and regularized long wave equationsby means of the homotopy analysis methodrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 14 no 3 pp708ndash717 2009
[47] Z Ziabakhsh and G Domairry ldquoAnalytic solution of naturalconvection flow of a non-Newtonian fluid between two verticalflat plates using homotopy analysis methodrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 14 no 5pp 1868ndash1880 2009
[48] M M Rashidi S A Mohimanian Pour T Hayat and SObaidat ldquoAnalytic approximate solutions for steady flow over arotating disk in porousmediumwith heat transfer by homotopyanalysis methodrdquo Computers and Fluids vol 54 no 1 pp 1ndash92012
[49] M Sajid T Hayat and S Asghar ldquoComparison between theHAM and HPM solutions of thin film flows of non-Newtonianfluids on a moving beltrdquo Nonlinear Dynamics vol 50 no 1-2pp 27ndash35 2007
[50] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for the double-diffusive natural convective boundary
layer flow in a porous medium saturated by a nanofluidrdquoInternational Journal of Heat and Mass Transfer vol 54 no 1-3pp 374ndash378 2011
[51] K Vajravelu and A Hadjinicolaou ldquoConvective heat transferin an electrically conducting fluid at a stretching surfacewith uniform free streamrdquo International Journal of EngineeringScience vol 35 pp 1237ndash1244 1997
[52] S Mukhopadhyay and G C Layek ldquoEffects of variable fluidviscosity on flow past a heated stretching sheet embedded ina porous medium in presence of heat sourcesinkrdquo Meccanicavol 47 no 4 pp 863ndash876 2012
[53] S J Liao ldquoAn optimal homotopy-analysis approach for stronglynonlinear differential equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 8 pp 2003ndash20162010
[54] A Aziz ldquoA similarity solution for laminar thermal boundarylayer over a flat plate with a convective surface boundary con-ditionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1064ndash1068 2009
Figure 22 Effect of generationabsorption heat parameter119876 on thedimensionless concentration profile
0 2 4 6 80
1
12
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 23 Effect of Lewis number Le on the dimensionless velocityprofile
9 Conclusions
In this paper we studied the steady laminar incompressiblefree convective flow of a nanofluid past a chemically reactingupward facing horizontal plate in porous medium takinginto account heat generation and the thermal slip boundarycondition
0 2 4 6 80
1
120579(120578
)
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 24 Effect of Lewis number Le on the dimensionlesstemperature profile
0 2 4 6 80
1
Le = 10
Le = 20Le = 50NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01K = 10n = 10b = 00
Figure 25 Effect of Lewis number Le on the dimensionlessconcentration profile
The governing partial differential equations have beentransformed by similarity transformations into a system ofordinary differential equations which are solved by OHAMand numerical method (fourth-order Runge-Kutta schemewith the shooting method) Dimensionless velocity temper-ature and concentration functions are presented for various
Mathematical Problems in Engineering 17
0 2 4 6 80
1
12
K = 00
K = 05
K = 10NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 26 Effect of chemical reaction parameter 119870 on the dimen-sionless velocity profile
0 2 4 6 80
1
K = 00
K = 05
K = 10NumericalNumerical
Numerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 27 Effect of chemical reaction parameter 119870 on the dimen-sionless temperature profile
values of parameters for the problem for example buoy-ancy ratio 119873119903 Brownian motion 119873119887 thermophoresis 119873119905heat generationabsorption parameter 119876 Lewis number Lechemical reaction parameter119870 order of chemical reaction 119899and thermal slip parameter 119887 From the present investigationthe following may be concluded
(i) The velocity 1198911015840(120578) increases with increasing 119873119887 anddecreases with increasing119873119903 119899 and 119887
0 2 4 6 80
1
K = 00
K = 05
K = 10NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 28 Effect of chemical reaction parameter 119870 on the dimen-sionless concentration profile
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 29 Effect of order of chemical reaction 119899 on the dimension-less velocity profile
(ii) The temperature 120579(120578) increases with 119873119903 119873119887 119899 anddecreases with 119887
(iii) Parameters of119873119905 Le and119870 have no effect on velocityand temperature profiles
(iv) The concentration distribution 120601(120578) increases withincreasing N119905 119899 and decreases with 119873119887 Le and 119870but119873119903 119876 and 119887 are not affected by it
18 Mathematical Problems in Engineering
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 30 Effect of order of chemical reaction 119899 on the dimension-less temperature profile
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 31 Effect of order of chemical reaction 119899 on the dimension-less concentration profile
(v) In presence of heat source (119876 gt 0) 1198911015840(120578) 120579(120578) bothincrease while heat sink (119876 lt 0) causes that 1198911015840(120578)120579(120578) both decrease extremely These effects are morevisible for temperature distribution
In addition numerical results for the reducedNusselt andSherwood numbers are tabulated It is concluded that thereduced Nusselt number increases with the Lewis numberand reaction parameter whist it decreases with the order of
0 2 4 6 8
1
12
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 32 Effect of order of thermal slip parameter 119887 on thedimensionless velocity profile
0 2 4 6 80
1
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 33 Effect of order of thermal slip parameter 119887 on thedimensionless temperature profile
chemical reaction thermal slip and generation parametersIt is further concluded that the reduced Sherwood numberenhances with the Lewis number generation and reactionparameters whist it suppresses with thermal slip and orderof chemical reaction
The convergence of the solution series and the power ofHAM in controlling and adjusting the convergence regionand rate of solution series were discussed The proper range
Mathematical Problems in Engineering 19
0 2 4 6 80
1
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 34 Effect of order of thermal slip parameter 119887 on thedimensionless concentration profile
of auxiliary parameter ℏ to ensure the convergence of thesolution series was obtained through the so called ℏ-curveThe HAM provides us with a convenient way to control theconvergence of approximation series which is a fundamentalqualitative difference in analysis between the HAM andother methods The numerical results of the above problemsdisplay a fast convergence with minimal calculations Thisshows that the HAM is a very efficient method Finally theagreement between analytical and numerical results of thepresent study with previous published results is excellent
Nomenclature
119887 Thermal slip parameter119862 Dimensional concentration119862119891 Skin friction factor1198631 Thermal slip factor119863119861 Brownian diffusion coefficient119863119879 Thermophoretic diffusion coefficient119891 Dimensionless velocity functions119892 Gravitation acceleration119870 Chemical reaction parameter119896 Permeability of the porous media1198960 The constant reaction rate119896119898 Effective thermal conductivity of the porous
medium119896(119909) Variable reaction rateL Linear operator of the HAM119871 Length of horizontal plateLe Lewis numberN Nonlinear operator of the HAM
119899 Order of chemical reaction119873119887 Brownian motion119873119903 Buoyancy ratio119873119905 ThermophoresisNu Nusselt number119875 PressurePr Prandtl number119876 Heat generationabsorption parameter1198760 Heat generationabsorption constant119902 Internal heat generation rate
Ra Rayleigh numberSh Sherwood number119879 Temperature119905 Time119906 Velocity in 119909-direction119880119903 Reference velocity Velocity vectorV Velocity in 119910-direction119909 Distance along the surface119910 Distance normal to the surface
Greek Letters
120572119898 Thermal diffusivity120573 Volumetric expansion coefficient of nanofluid120576 Porosity of the porous media120601 Dimensionless concentration120578 Similarity variable120583 Dynamic viscosity120579 Dimensionless temperature120588119891 Density of the base fluid120588119901 Density of the nanoparticles(120588119862)119891 Effective heat capacity of the fluid(120588119862)119875 Effective heat capacity of the nanoparticle material120595 Stream function
Subscript Superscript
infin Conditions far away from the surface1015840 Differentiation with respect to 120578
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
M M Rashidi wishes to thank the Centre for DifferentialEquations Continuum Mechanics and Applications Schoolof Computational and Applied Mathematics University ofthe Witwatersrand for the hospitality during his visit EMomoniat thanks theNational Research Foundation of SouthAfrica for their support
20 Mathematical Problems in Engineering
References
[1] E Abu-Nada F O Hakan and I Pop ldquoBuoyancy inducedflow in a nanofluid filled enclosure partially exposed to forcedconvectionrdquo Superlattices and Microstructures vol 51 no 3 pp381ndash395 2012
[2] S Kakac and A Pramuanjaroenkij ldquoReview of convective heattransfer enhancement with nanofluidsrdquo International Journal ofHeat and Mass Transfer vol 52 no 13-14 pp 3187ndash3196 2009
[3] J A Eastman S U S Choi S Li W Yu and L J ThompsonldquoAnomalously increased effective thermal conductivities ofethylene glycol-based nanofluids containing copper nanoparti-clesrdquo Applied Physics Letters vol 78 no 6 pp 718ndash720 2001
[4] Y Xuan and Q Li ldquoHeat transfer enhancement of nanofluidsrdquoInternational Journal of Heat and Fluid Flow vol 21 no 1 pp58ndash64 2000
[5] S K Das N Putra P Thiesen and W Roetzel ldquoTemperaturedependence of thermal conductivity enhancement for nanoflu-idsrdquo Journal of Heat Transfer vol 125 no 4 pp 567ndash574 2003
[6] D H Kumar H E Patel V R R Kumar T SundararajanT Pradeep and S K Das ldquoModel for heat conduction innanofluidsrdquo Physical Review Letters vol 93 no 14 Article ID144301 2004
[7] H E Patel T Sundararajan T Pradeep A Dasgupta NDasgupta and S K Das ldquoA micro-convection model forthermal conductivity of nanofluidsrdquo Pramana vol 65 no 5 pp863ndash869 2005
[8] A V Kuznetsov ldquoNon-oscillatory and oscillatory nanofluidbio-thermal convection in a horizontal layer of finite depthrdquoEuropean Journal of Mechanics vol 30 no 2 pp 156ndash165 2011
[9] M Napoli J C T Eijkel and S Pennathur ldquoNanofluidictechnology for biomolecule applications a critical reviewrdquo Labon a Chip vol 10 no 8 pp 957ndash985 2010
[10] J C Crepeau and R Clarksean ldquoSimilarity solutions of naturalconvection with internal heat generationrdquo Journal of HeatTransfer vol 119 no 1 pp 183ndash185 1997
[11] A Z Sahin ldquoTransient heat conduction in semi-infinite solidwith spatially decaying exponential heat generationrdquo Interna-tional Communications in Heat andMass Transfer vol 19 no 3pp 349ndash358 1992
[12] O D Makind and A Aziz ldquoMixed convection from a con-vectively heated vertical plate to a fluid with internal heatgenerationrdquo Journal of Heat Transfer vol 133 Article ID 1225016 pages 2011
[13] J Buongiorno ldquoConvective transport in nanofluidsrdquo Journal ofHeat Transfer vol 128 no 3 pp 240ndash250 2006
[14] AV Kuznetsov andDANield ldquoNatural convective boundary-layer flow of a nanofluid past a vertical platerdquo InternationalJournal of Thermal Sciences vol 49 no 2 pp 243ndash247 2010
[15] L B Godson D Raja L D Mohan and S WongwisescldquoEnhancement of heat transfer using nanofluids-an overviewrdquoRenewable and Sustainable Energy Reviews vol 14 no 2 pp629ndash641 2010
[16] D A Nield and A BejanConvection in PorousMedia SpringerNew York NY USA 3rd edition 2006
[17] P Vadasz Emerging Topics in Heat and Mass Transfer in PorousMedia Springer New York NY USA 2008
[18] K Vafai Porous Media Applications in Biological Systems andBiotechnology CRC Press New York NY USA 2010
[19] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application to
heat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977
[20] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for natural convective boundary-layer flow in a porousmedium saturated by a nanofluidrdquo International Journal of Heatand Mass Transfer vol 52 no 25-26 pp 5792ndash5795 2009
[21] P Cheng and I D Chang ldquoBuoyancy induced flows in asaturated porous medium adjacent to impermeable horizontalsurfacesrdquo International Journal of Heat and Mass Transfer vol19 no 11 pp 1267ndash1272 1976
[22] I D Chang and P Cheng ldquoMatched asymptotic expansionsfor free convection about an impermeable horizontal surfacein a porous mediumrdquo International Journal of Heat and MassTransfer vol 26 no 2 pp 163ndash174 1983
[23] D S Shiunlin and B Gebhart ldquoBuoyancy-induced flow adja-cent to a horizontal surface submerged in porous mediumsaturated with cold waterrdquo International Journal of Heat andMass Transfer vol 29 no 4 pp 611ndash623 1986
[24] J H Merkin and G Zhang ldquoOn the similarity solutions for freeconvection in a saturated porousmedium adjacent to imperme-able horizontal surfacesrdquoWarme- und Stoffubertragung vol 25no 3 pp 179ndash184 1990
[25] M A Chaudhary J H Merkin and I Pop ldquoNatural convectionfrom a horizontal permeable surface in a porous mediummdashnumerical and asymptotic solutionsrdquo Transport in PorousMedia vol 22 no 3 pp 327ndash344 1996
[26] R S R Gorla and A Chamkha ldquoNatural convective boundarylayer flow over a nonisothermal vertical plate embedded ina porous medium saturated with a nanofluidrdquo Nanoscale andMicroscale Thermophysical Engineering vol 15 no 2 pp 81ndash942011
[27] W A Khan and I Pop ldquoFree convection boundary layer flowpast a horizontal flat plate embedded in a porous medium filledwith a nanofluidrdquo Journal of Heat Transfer vol 133 no 9 ArticleID 094501 2011
[28] A Aziz W A Khan and I Pop ldquoFree convection boundarylayer flow past a horizontal flat plate embedded in porousmedium filled by nanofluid containing gyrotactic microorgan-ismsrdquo International Journal of Thermal Sciences vol 56 pp 48ndash57 2012
[29] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer New York NYUSA 2009
[30] A A Avramenko S G Kobzar I V Shevchuk A V Kuznetsovand L T Iwanisov ldquoSymmetry of turbulent boundary-layerflows investigation of different eddy viscosity modelsrdquo ActaMechanica vol 151 no 1-2 pp 1ndash14 2001
[31] A V Kuznetsov A A Avramenko and P Geng ldquoAnalyticalinvestigation of a falling plume caused by bioconvection ofoxytactic bacteria in a fluid saturated porous mediumrdquo Inter-national Journal of Engineering Science vol 42 no 5-6 pp 557ndash569 2004
[32] M Jalil S Asghar and M Mushtaq ldquoLie group analysis ofmixed convection flow with mass transfer over a stretchingsurface with suction or injectionrdquo Mathematical Problems inEngineering vol 2010 Article ID 264901 14 pages 2010
[33] M A A Hamad M J Uddin and A I M Ismail ldquoRadiationeffects on heat and mass transfer in MHD stagnation-pointflow over a permeable flat plate with thermal convectivesurface boundary condition temperature dependent viscosityand thermal conductivityrdquoNuclear Engineering and Design vol242 pp 194ndash200 2012
Mathematical Problems in Engineering 21
[34] A Aziz M J Uddin M A A Hamad and A I MdIsmail ldquoMHD flow over an inclined radiating plate with thetemperature-dependent thermal conductivity variable reactiveindex and heat generationrdquoHeat Transfer vol 41 no 3 pp 241ndash259 2012
[35] A G Hansen Similarity Analysis of Boundary Layer Problemsin Engineering Prentice Hall Englewood Cliffs NJ USA 1964
[36] W FAmesNonlinear Partial Differential Equations in Engineer-ing Academic Press New York NY USA 1972
[37] R Seshadri and T Y Na Group Invariance in EngineeringBoundary Value Problems Springer New York NY USA 1985
[38] D Shang Theory of Heat Transfer with Forced Convection FilmFlows vol 3 of Heat and Mass Transfer 2010
[40] E Erfani M M Rashidi and A B Parsa ldquoThe modifieddifferential transform method for solving off-centered stag-nation flow toward a rotating discrdquo International Journal ofComputational Methods vol 7 no 4 pp 655ndash670 2010
[41] M M Rashidi N Laraqi and A Basiri Parsa ldquoAnalyticalmodeling of heat convection in magnetized micropolar fluid byusing modified differential transform methodrdquo Heat Transfervol 40 no 3 pp 187ndash204 2011
[42] M M Rashidi and M Keimanesh ldquoUsing differential trans-form method and pade approximant for solving mhd flow ina laminar liquid film from a horizontal stretching surfacerdquoMathematical Problems in Engineering vol 2010 Article ID491319 14 pages 2010
[43] M M Rashidi and G Domairry ldquoNew analytical solution ofthe three-dimensional NavierStokes equationsrdquoModern PhysicsLetters B vol 23 no 26 pp 3147ndash3155 2009
[44] M M Rashidi M Keimanesh O A Beg and T K HungldquoMagnetohydrodynamic biorheological transport phenomenain a porous medium a simulation of magnetic blood flowcontrol and filtrationrdquo International Journal for NumericalMethods in Biomedical Engineering vol 27 no 6 pp 805ndash8212011
[45] S J Liao The proposed homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992
[46] M M Rashidi G Domairry and S Dinarvand ldquoApproximatesolutions for the Burger and regularized long wave equationsby means of the homotopy analysis methodrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 14 no 3 pp708ndash717 2009
[47] Z Ziabakhsh and G Domairry ldquoAnalytic solution of naturalconvection flow of a non-Newtonian fluid between two verticalflat plates using homotopy analysis methodrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 14 no 5pp 1868ndash1880 2009
[48] M M Rashidi S A Mohimanian Pour T Hayat and SObaidat ldquoAnalytic approximate solutions for steady flow over arotating disk in porousmediumwith heat transfer by homotopyanalysis methodrdquo Computers and Fluids vol 54 no 1 pp 1ndash92012
[49] M Sajid T Hayat and S Asghar ldquoComparison between theHAM and HPM solutions of thin film flows of non-Newtonianfluids on a moving beltrdquo Nonlinear Dynamics vol 50 no 1-2pp 27ndash35 2007
[50] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for the double-diffusive natural convective boundary
layer flow in a porous medium saturated by a nanofluidrdquoInternational Journal of Heat and Mass Transfer vol 54 no 1-3pp 374ndash378 2011
[51] K Vajravelu and A Hadjinicolaou ldquoConvective heat transferin an electrically conducting fluid at a stretching surfacewith uniform free streamrdquo International Journal of EngineeringScience vol 35 pp 1237ndash1244 1997
[52] S Mukhopadhyay and G C Layek ldquoEffects of variable fluidviscosity on flow past a heated stretching sheet embedded ina porous medium in presence of heat sourcesinkrdquo Meccanicavol 47 no 4 pp 863ndash876 2012
[53] S J Liao ldquoAn optimal homotopy-analysis approach for stronglynonlinear differential equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 8 pp 2003ndash20162010
[54] A Aziz ldquoA similarity solution for laminar thermal boundarylayer over a flat plate with a convective surface boundary con-ditionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1064ndash1068 2009
Figure 26 Effect of chemical reaction parameter 119870 on the dimen-sionless velocity profile
0 2 4 6 80
1
K = 00
K = 05
K = 10NumericalNumerical
Numerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 27 Effect of chemical reaction parameter 119870 on the dimen-sionless temperature profile
values of parameters for the problem for example buoy-ancy ratio 119873119903 Brownian motion 119873119887 thermophoresis 119873119905heat generationabsorption parameter 119876 Lewis number Lechemical reaction parameter119870 order of chemical reaction 119899and thermal slip parameter 119887 From the present investigationthe following may be concluded
(i) The velocity 1198911015840(120578) increases with increasing 119873119887 anddecreases with increasing119873119903 119899 and 119887
0 2 4 6 80
1
K = 00
K = 05
K = 10NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 01Nb = 01Q = 01Le = 10n = 10b = 00
Figure 28 Effect of chemical reaction parameter 119870 on the dimen-sionless concentration profile
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 29 Effect of order of chemical reaction 119899 on the dimension-less velocity profile
(ii) The temperature 120579(120578) increases with 119873119903 119873119887 119899 anddecreases with 119887
(iii) Parameters of119873119905 Le and119870 have no effect on velocityand temperature profiles
(iv) The concentration distribution 120601(120578) increases withincreasing N119905 119899 and decreases with 119873119887 Le and 119870but119873119903 119876 and 119887 are not affected by it
18 Mathematical Problems in Engineering
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 30 Effect of order of chemical reaction 119899 on the dimension-less temperature profile
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 31 Effect of order of chemical reaction 119899 on the dimension-less concentration profile
(v) In presence of heat source (119876 gt 0) 1198911015840(120578) 120579(120578) bothincrease while heat sink (119876 lt 0) causes that 1198911015840(120578)120579(120578) both decrease extremely These effects are morevisible for temperature distribution
In addition numerical results for the reducedNusselt andSherwood numbers are tabulated It is concluded that thereduced Nusselt number increases with the Lewis numberand reaction parameter whist it decreases with the order of
0 2 4 6 8
1
12
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 32 Effect of order of thermal slip parameter 119887 on thedimensionless velocity profile
0 2 4 6 80
1
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 33 Effect of order of thermal slip parameter 119887 on thedimensionless temperature profile
chemical reaction thermal slip and generation parametersIt is further concluded that the reduced Sherwood numberenhances with the Lewis number generation and reactionparameters whist it suppresses with thermal slip and orderof chemical reaction
The convergence of the solution series and the power ofHAM in controlling and adjusting the convergence regionand rate of solution series were discussed The proper range
Mathematical Problems in Engineering 19
0 2 4 6 80
1
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 34 Effect of order of thermal slip parameter 119887 on thedimensionless concentration profile
of auxiliary parameter ℏ to ensure the convergence of thesolution series was obtained through the so called ℏ-curveThe HAM provides us with a convenient way to control theconvergence of approximation series which is a fundamentalqualitative difference in analysis between the HAM andother methods The numerical results of the above problemsdisplay a fast convergence with minimal calculations Thisshows that the HAM is a very efficient method Finally theagreement between analytical and numerical results of thepresent study with previous published results is excellent
Nomenclature
119887 Thermal slip parameter119862 Dimensional concentration119862119891 Skin friction factor1198631 Thermal slip factor119863119861 Brownian diffusion coefficient119863119879 Thermophoretic diffusion coefficient119891 Dimensionless velocity functions119892 Gravitation acceleration119870 Chemical reaction parameter119896 Permeability of the porous media1198960 The constant reaction rate119896119898 Effective thermal conductivity of the porous
medium119896(119909) Variable reaction rateL Linear operator of the HAM119871 Length of horizontal plateLe Lewis numberN Nonlinear operator of the HAM
119899 Order of chemical reaction119873119887 Brownian motion119873119903 Buoyancy ratio119873119905 ThermophoresisNu Nusselt number119875 PressurePr Prandtl number119876 Heat generationabsorption parameter1198760 Heat generationabsorption constant119902 Internal heat generation rate
Ra Rayleigh numberSh Sherwood number119879 Temperature119905 Time119906 Velocity in 119909-direction119880119903 Reference velocity Velocity vectorV Velocity in 119910-direction119909 Distance along the surface119910 Distance normal to the surface
Greek Letters
120572119898 Thermal diffusivity120573 Volumetric expansion coefficient of nanofluid120576 Porosity of the porous media120601 Dimensionless concentration120578 Similarity variable120583 Dynamic viscosity120579 Dimensionless temperature120588119891 Density of the base fluid120588119901 Density of the nanoparticles(120588119862)119891 Effective heat capacity of the fluid(120588119862)119875 Effective heat capacity of the nanoparticle material120595 Stream function
Subscript Superscript
infin Conditions far away from the surface1015840 Differentiation with respect to 120578
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
M M Rashidi wishes to thank the Centre for DifferentialEquations Continuum Mechanics and Applications Schoolof Computational and Applied Mathematics University ofthe Witwatersrand for the hospitality during his visit EMomoniat thanks theNational Research Foundation of SouthAfrica for their support
20 Mathematical Problems in Engineering
References
[1] E Abu-Nada F O Hakan and I Pop ldquoBuoyancy inducedflow in a nanofluid filled enclosure partially exposed to forcedconvectionrdquo Superlattices and Microstructures vol 51 no 3 pp381ndash395 2012
[2] S Kakac and A Pramuanjaroenkij ldquoReview of convective heattransfer enhancement with nanofluidsrdquo International Journal ofHeat and Mass Transfer vol 52 no 13-14 pp 3187ndash3196 2009
[3] J A Eastman S U S Choi S Li W Yu and L J ThompsonldquoAnomalously increased effective thermal conductivities ofethylene glycol-based nanofluids containing copper nanoparti-clesrdquo Applied Physics Letters vol 78 no 6 pp 718ndash720 2001
[4] Y Xuan and Q Li ldquoHeat transfer enhancement of nanofluidsrdquoInternational Journal of Heat and Fluid Flow vol 21 no 1 pp58ndash64 2000
[5] S K Das N Putra P Thiesen and W Roetzel ldquoTemperaturedependence of thermal conductivity enhancement for nanoflu-idsrdquo Journal of Heat Transfer vol 125 no 4 pp 567ndash574 2003
[6] D H Kumar H E Patel V R R Kumar T SundararajanT Pradeep and S K Das ldquoModel for heat conduction innanofluidsrdquo Physical Review Letters vol 93 no 14 Article ID144301 2004
[7] H E Patel T Sundararajan T Pradeep A Dasgupta NDasgupta and S K Das ldquoA micro-convection model forthermal conductivity of nanofluidsrdquo Pramana vol 65 no 5 pp863ndash869 2005
[8] A V Kuznetsov ldquoNon-oscillatory and oscillatory nanofluidbio-thermal convection in a horizontal layer of finite depthrdquoEuropean Journal of Mechanics vol 30 no 2 pp 156ndash165 2011
[9] M Napoli J C T Eijkel and S Pennathur ldquoNanofluidictechnology for biomolecule applications a critical reviewrdquo Labon a Chip vol 10 no 8 pp 957ndash985 2010
[10] J C Crepeau and R Clarksean ldquoSimilarity solutions of naturalconvection with internal heat generationrdquo Journal of HeatTransfer vol 119 no 1 pp 183ndash185 1997
[11] A Z Sahin ldquoTransient heat conduction in semi-infinite solidwith spatially decaying exponential heat generationrdquo Interna-tional Communications in Heat andMass Transfer vol 19 no 3pp 349ndash358 1992
[12] O D Makind and A Aziz ldquoMixed convection from a con-vectively heated vertical plate to a fluid with internal heatgenerationrdquo Journal of Heat Transfer vol 133 Article ID 1225016 pages 2011
[13] J Buongiorno ldquoConvective transport in nanofluidsrdquo Journal ofHeat Transfer vol 128 no 3 pp 240ndash250 2006
[14] AV Kuznetsov andDANield ldquoNatural convective boundary-layer flow of a nanofluid past a vertical platerdquo InternationalJournal of Thermal Sciences vol 49 no 2 pp 243ndash247 2010
[15] L B Godson D Raja L D Mohan and S WongwisescldquoEnhancement of heat transfer using nanofluids-an overviewrdquoRenewable and Sustainable Energy Reviews vol 14 no 2 pp629ndash641 2010
[16] D A Nield and A BejanConvection in PorousMedia SpringerNew York NY USA 3rd edition 2006
[17] P Vadasz Emerging Topics in Heat and Mass Transfer in PorousMedia Springer New York NY USA 2008
[18] K Vafai Porous Media Applications in Biological Systems andBiotechnology CRC Press New York NY USA 2010
[19] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application to
heat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977
[20] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for natural convective boundary-layer flow in a porousmedium saturated by a nanofluidrdquo International Journal of Heatand Mass Transfer vol 52 no 25-26 pp 5792ndash5795 2009
[21] P Cheng and I D Chang ldquoBuoyancy induced flows in asaturated porous medium adjacent to impermeable horizontalsurfacesrdquo International Journal of Heat and Mass Transfer vol19 no 11 pp 1267ndash1272 1976
[22] I D Chang and P Cheng ldquoMatched asymptotic expansionsfor free convection about an impermeable horizontal surfacein a porous mediumrdquo International Journal of Heat and MassTransfer vol 26 no 2 pp 163ndash174 1983
[23] D S Shiunlin and B Gebhart ldquoBuoyancy-induced flow adja-cent to a horizontal surface submerged in porous mediumsaturated with cold waterrdquo International Journal of Heat andMass Transfer vol 29 no 4 pp 611ndash623 1986
[24] J H Merkin and G Zhang ldquoOn the similarity solutions for freeconvection in a saturated porousmedium adjacent to imperme-able horizontal surfacesrdquoWarme- und Stoffubertragung vol 25no 3 pp 179ndash184 1990
[25] M A Chaudhary J H Merkin and I Pop ldquoNatural convectionfrom a horizontal permeable surface in a porous mediummdashnumerical and asymptotic solutionsrdquo Transport in PorousMedia vol 22 no 3 pp 327ndash344 1996
[26] R S R Gorla and A Chamkha ldquoNatural convective boundarylayer flow over a nonisothermal vertical plate embedded ina porous medium saturated with a nanofluidrdquo Nanoscale andMicroscale Thermophysical Engineering vol 15 no 2 pp 81ndash942011
[27] W A Khan and I Pop ldquoFree convection boundary layer flowpast a horizontal flat plate embedded in a porous medium filledwith a nanofluidrdquo Journal of Heat Transfer vol 133 no 9 ArticleID 094501 2011
[28] A Aziz W A Khan and I Pop ldquoFree convection boundarylayer flow past a horizontal flat plate embedded in porousmedium filled by nanofluid containing gyrotactic microorgan-ismsrdquo International Journal of Thermal Sciences vol 56 pp 48ndash57 2012
[29] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer New York NYUSA 2009
[30] A A Avramenko S G Kobzar I V Shevchuk A V Kuznetsovand L T Iwanisov ldquoSymmetry of turbulent boundary-layerflows investigation of different eddy viscosity modelsrdquo ActaMechanica vol 151 no 1-2 pp 1ndash14 2001
[31] A V Kuznetsov A A Avramenko and P Geng ldquoAnalyticalinvestigation of a falling plume caused by bioconvection ofoxytactic bacteria in a fluid saturated porous mediumrdquo Inter-national Journal of Engineering Science vol 42 no 5-6 pp 557ndash569 2004
[32] M Jalil S Asghar and M Mushtaq ldquoLie group analysis ofmixed convection flow with mass transfer over a stretchingsurface with suction or injectionrdquo Mathematical Problems inEngineering vol 2010 Article ID 264901 14 pages 2010
[33] M A A Hamad M J Uddin and A I M Ismail ldquoRadiationeffects on heat and mass transfer in MHD stagnation-pointflow over a permeable flat plate with thermal convectivesurface boundary condition temperature dependent viscosityand thermal conductivityrdquoNuclear Engineering and Design vol242 pp 194ndash200 2012
Mathematical Problems in Engineering 21
[34] A Aziz M J Uddin M A A Hamad and A I MdIsmail ldquoMHD flow over an inclined radiating plate with thetemperature-dependent thermal conductivity variable reactiveindex and heat generationrdquoHeat Transfer vol 41 no 3 pp 241ndash259 2012
[35] A G Hansen Similarity Analysis of Boundary Layer Problemsin Engineering Prentice Hall Englewood Cliffs NJ USA 1964
[36] W FAmesNonlinear Partial Differential Equations in Engineer-ing Academic Press New York NY USA 1972
[37] R Seshadri and T Y Na Group Invariance in EngineeringBoundary Value Problems Springer New York NY USA 1985
[38] D Shang Theory of Heat Transfer with Forced Convection FilmFlows vol 3 of Heat and Mass Transfer 2010
[40] E Erfani M M Rashidi and A B Parsa ldquoThe modifieddifferential transform method for solving off-centered stag-nation flow toward a rotating discrdquo International Journal ofComputational Methods vol 7 no 4 pp 655ndash670 2010
[41] M M Rashidi N Laraqi and A Basiri Parsa ldquoAnalyticalmodeling of heat convection in magnetized micropolar fluid byusing modified differential transform methodrdquo Heat Transfervol 40 no 3 pp 187ndash204 2011
[42] M M Rashidi and M Keimanesh ldquoUsing differential trans-form method and pade approximant for solving mhd flow ina laminar liquid film from a horizontal stretching surfacerdquoMathematical Problems in Engineering vol 2010 Article ID491319 14 pages 2010
[43] M M Rashidi and G Domairry ldquoNew analytical solution ofthe three-dimensional NavierStokes equationsrdquoModern PhysicsLetters B vol 23 no 26 pp 3147ndash3155 2009
[44] M M Rashidi M Keimanesh O A Beg and T K HungldquoMagnetohydrodynamic biorheological transport phenomenain a porous medium a simulation of magnetic blood flowcontrol and filtrationrdquo International Journal for NumericalMethods in Biomedical Engineering vol 27 no 6 pp 805ndash8212011
[45] S J Liao The proposed homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992
[46] M M Rashidi G Domairry and S Dinarvand ldquoApproximatesolutions for the Burger and regularized long wave equationsby means of the homotopy analysis methodrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 14 no 3 pp708ndash717 2009
[47] Z Ziabakhsh and G Domairry ldquoAnalytic solution of naturalconvection flow of a non-Newtonian fluid between two verticalflat plates using homotopy analysis methodrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 14 no 5pp 1868ndash1880 2009
[48] M M Rashidi S A Mohimanian Pour T Hayat and SObaidat ldquoAnalytic approximate solutions for steady flow over arotating disk in porousmediumwith heat transfer by homotopyanalysis methodrdquo Computers and Fluids vol 54 no 1 pp 1ndash92012
[49] M Sajid T Hayat and S Asghar ldquoComparison between theHAM and HPM solutions of thin film flows of non-Newtonianfluids on a moving beltrdquo Nonlinear Dynamics vol 50 no 1-2pp 27ndash35 2007
[50] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for the double-diffusive natural convective boundary
layer flow in a porous medium saturated by a nanofluidrdquoInternational Journal of Heat and Mass Transfer vol 54 no 1-3pp 374ndash378 2011
[51] K Vajravelu and A Hadjinicolaou ldquoConvective heat transferin an electrically conducting fluid at a stretching surfacewith uniform free streamrdquo International Journal of EngineeringScience vol 35 pp 1237ndash1244 1997
[52] S Mukhopadhyay and G C Layek ldquoEffects of variable fluidviscosity on flow past a heated stretching sheet embedded ina porous medium in presence of heat sourcesinkrdquo Meccanicavol 47 no 4 pp 863ndash876 2012
[53] S J Liao ldquoAn optimal homotopy-analysis approach for stronglynonlinear differential equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 8 pp 2003ndash20162010
[54] A Aziz ldquoA similarity solution for laminar thermal boundarylayer over a flat plate with a convective surface boundary con-ditionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1064ndash1068 2009
Figure 30 Effect of order of chemical reaction 119899 on the dimension-less temperature profile
0 2 4 6 80
1
n = 10
n = 20
n = 30
Numerical
NumericalNumerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01Nr = 05Nb = 05Q = 01Le = 10K = 10b = 00
Figure 31 Effect of order of chemical reaction 119899 on the dimension-less concentration profile
(v) In presence of heat source (119876 gt 0) 1198911015840(120578) 120579(120578) bothincrease while heat sink (119876 lt 0) causes that 1198911015840(120578)120579(120578) both decrease extremely These effects are morevisible for temperature distribution
In addition numerical results for the reducedNusselt andSherwood numbers are tabulated It is concluded that thereduced Nusselt number increases with the Lewis numberand reaction parameter whist it decreases with the order of
0 2 4 6 8
1
12
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
f998400 (120578
)
120578
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 32 Effect of order of thermal slip parameter 119887 on thedimensionless velocity profile
0 2 4 6 80
1
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
120579(120578
)
120578
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 33 Effect of order of thermal slip parameter 119887 on thedimensionless temperature profile
chemical reaction thermal slip and generation parametersIt is further concluded that the reduced Sherwood numberenhances with the Lewis number generation and reactionparameters whist it suppresses with thermal slip and orderof chemical reaction
The convergence of the solution series and the power ofHAM in controlling and adjusting the convergence regionand rate of solution series were discussed The proper range
Mathematical Problems in Engineering 19
0 2 4 6 80
1
b = 10
b = 20
b = 00
NumericalNumerical
Numerical
120578
120601(120578
)
08
06
04
02
10
Nt = 01
Nr = 01Nb = 01
Q = 01Le = 10K = 10n = 10
Figure 34 Effect of order of thermal slip parameter 119887 on thedimensionless concentration profile
of auxiliary parameter ℏ to ensure the convergence of thesolution series was obtained through the so called ℏ-curveThe HAM provides us with a convenient way to control theconvergence of approximation series which is a fundamentalqualitative difference in analysis between the HAM andother methods The numerical results of the above problemsdisplay a fast convergence with minimal calculations Thisshows that the HAM is a very efficient method Finally theagreement between analytical and numerical results of thepresent study with previous published results is excellent
Nomenclature
119887 Thermal slip parameter119862 Dimensional concentration119862119891 Skin friction factor1198631 Thermal slip factor119863119861 Brownian diffusion coefficient119863119879 Thermophoretic diffusion coefficient119891 Dimensionless velocity functions119892 Gravitation acceleration119870 Chemical reaction parameter119896 Permeability of the porous media1198960 The constant reaction rate119896119898 Effective thermal conductivity of the porous
medium119896(119909) Variable reaction rateL Linear operator of the HAM119871 Length of horizontal plateLe Lewis numberN Nonlinear operator of the HAM
119899 Order of chemical reaction119873119887 Brownian motion119873119903 Buoyancy ratio119873119905 ThermophoresisNu Nusselt number119875 PressurePr Prandtl number119876 Heat generationabsorption parameter1198760 Heat generationabsorption constant119902 Internal heat generation rate
Ra Rayleigh numberSh Sherwood number119879 Temperature119905 Time119906 Velocity in 119909-direction119880119903 Reference velocity Velocity vectorV Velocity in 119910-direction119909 Distance along the surface119910 Distance normal to the surface
Greek Letters
120572119898 Thermal diffusivity120573 Volumetric expansion coefficient of nanofluid120576 Porosity of the porous media120601 Dimensionless concentration120578 Similarity variable120583 Dynamic viscosity120579 Dimensionless temperature120588119891 Density of the base fluid120588119901 Density of the nanoparticles(120588119862)119891 Effective heat capacity of the fluid(120588119862)119875 Effective heat capacity of the nanoparticle material120595 Stream function
Subscript Superscript
infin Conditions far away from the surface1015840 Differentiation with respect to 120578
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
M M Rashidi wishes to thank the Centre for DifferentialEquations Continuum Mechanics and Applications Schoolof Computational and Applied Mathematics University ofthe Witwatersrand for the hospitality during his visit EMomoniat thanks theNational Research Foundation of SouthAfrica for their support
20 Mathematical Problems in Engineering
References
[1] E Abu-Nada F O Hakan and I Pop ldquoBuoyancy inducedflow in a nanofluid filled enclosure partially exposed to forcedconvectionrdquo Superlattices and Microstructures vol 51 no 3 pp381ndash395 2012
[2] S Kakac and A Pramuanjaroenkij ldquoReview of convective heattransfer enhancement with nanofluidsrdquo International Journal ofHeat and Mass Transfer vol 52 no 13-14 pp 3187ndash3196 2009
[3] J A Eastman S U S Choi S Li W Yu and L J ThompsonldquoAnomalously increased effective thermal conductivities ofethylene glycol-based nanofluids containing copper nanoparti-clesrdquo Applied Physics Letters vol 78 no 6 pp 718ndash720 2001
[4] Y Xuan and Q Li ldquoHeat transfer enhancement of nanofluidsrdquoInternational Journal of Heat and Fluid Flow vol 21 no 1 pp58ndash64 2000
[5] S K Das N Putra P Thiesen and W Roetzel ldquoTemperaturedependence of thermal conductivity enhancement for nanoflu-idsrdquo Journal of Heat Transfer vol 125 no 4 pp 567ndash574 2003
[6] D H Kumar H E Patel V R R Kumar T SundararajanT Pradeep and S K Das ldquoModel for heat conduction innanofluidsrdquo Physical Review Letters vol 93 no 14 Article ID144301 2004
[7] H E Patel T Sundararajan T Pradeep A Dasgupta NDasgupta and S K Das ldquoA micro-convection model forthermal conductivity of nanofluidsrdquo Pramana vol 65 no 5 pp863ndash869 2005
[8] A V Kuznetsov ldquoNon-oscillatory and oscillatory nanofluidbio-thermal convection in a horizontal layer of finite depthrdquoEuropean Journal of Mechanics vol 30 no 2 pp 156ndash165 2011
[9] M Napoli J C T Eijkel and S Pennathur ldquoNanofluidictechnology for biomolecule applications a critical reviewrdquo Labon a Chip vol 10 no 8 pp 957ndash985 2010
[10] J C Crepeau and R Clarksean ldquoSimilarity solutions of naturalconvection with internal heat generationrdquo Journal of HeatTransfer vol 119 no 1 pp 183ndash185 1997
[11] A Z Sahin ldquoTransient heat conduction in semi-infinite solidwith spatially decaying exponential heat generationrdquo Interna-tional Communications in Heat andMass Transfer vol 19 no 3pp 349ndash358 1992
[12] O D Makind and A Aziz ldquoMixed convection from a con-vectively heated vertical plate to a fluid with internal heatgenerationrdquo Journal of Heat Transfer vol 133 Article ID 1225016 pages 2011
[13] J Buongiorno ldquoConvective transport in nanofluidsrdquo Journal ofHeat Transfer vol 128 no 3 pp 240ndash250 2006
[14] AV Kuznetsov andDANield ldquoNatural convective boundary-layer flow of a nanofluid past a vertical platerdquo InternationalJournal of Thermal Sciences vol 49 no 2 pp 243ndash247 2010
[15] L B Godson D Raja L D Mohan and S WongwisescldquoEnhancement of heat transfer using nanofluids-an overviewrdquoRenewable and Sustainable Energy Reviews vol 14 no 2 pp629ndash641 2010
[16] D A Nield and A BejanConvection in PorousMedia SpringerNew York NY USA 3rd edition 2006
[17] P Vadasz Emerging Topics in Heat and Mass Transfer in PorousMedia Springer New York NY USA 2008
[18] K Vafai Porous Media Applications in Biological Systems andBiotechnology CRC Press New York NY USA 2010
[19] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application to
heat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977
[20] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for natural convective boundary-layer flow in a porousmedium saturated by a nanofluidrdquo International Journal of Heatand Mass Transfer vol 52 no 25-26 pp 5792ndash5795 2009
[21] P Cheng and I D Chang ldquoBuoyancy induced flows in asaturated porous medium adjacent to impermeable horizontalsurfacesrdquo International Journal of Heat and Mass Transfer vol19 no 11 pp 1267ndash1272 1976
[22] I D Chang and P Cheng ldquoMatched asymptotic expansionsfor free convection about an impermeable horizontal surfacein a porous mediumrdquo International Journal of Heat and MassTransfer vol 26 no 2 pp 163ndash174 1983
[23] D S Shiunlin and B Gebhart ldquoBuoyancy-induced flow adja-cent to a horizontal surface submerged in porous mediumsaturated with cold waterrdquo International Journal of Heat andMass Transfer vol 29 no 4 pp 611ndash623 1986
[24] J H Merkin and G Zhang ldquoOn the similarity solutions for freeconvection in a saturated porousmedium adjacent to imperme-able horizontal surfacesrdquoWarme- und Stoffubertragung vol 25no 3 pp 179ndash184 1990
[25] M A Chaudhary J H Merkin and I Pop ldquoNatural convectionfrom a horizontal permeable surface in a porous mediummdashnumerical and asymptotic solutionsrdquo Transport in PorousMedia vol 22 no 3 pp 327ndash344 1996
[26] R S R Gorla and A Chamkha ldquoNatural convective boundarylayer flow over a nonisothermal vertical plate embedded ina porous medium saturated with a nanofluidrdquo Nanoscale andMicroscale Thermophysical Engineering vol 15 no 2 pp 81ndash942011
[27] W A Khan and I Pop ldquoFree convection boundary layer flowpast a horizontal flat plate embedded in a porous medium filledwith a nanofluidrdquo Journal of Heat Transfer vol 133 no 9 ArticleID 094501 2011
[28] A Aziz W A Khan and I Pop ldquoFree convection boundarylayer flow past a horizontal flat plate embedded in porousmedium filled by nanofluid containing gyrotactic microorgan-ismsrdquo International Journal of Thermal Sciences vol 56 pp 48ndash57 2012
[29] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer New York NYUSA 2009
[30] A A Avramenko S G Kobzar I V Shevchuk A V Kuznetsovand L T Iwanisov ldquoSymmetry of turbulent boundary-layerflows investigation of different eddy viscosity modelsrdquo ActaMechanica vol 151 no 1-2 pp 1ndash14 2001
[31] A V Kuznetsov A A Avramenko and P Geng ldquoAnalyticalinvestigation of a falling plume caused by bioconvection ofoxytactic bacteria in a fluid saturated porous mediumrdquo Inter-national Journal of Engineering Science vol 42 no 5-6 pp 557ndash569 2004
[32] M Jalil S Asghar and M Mushtaq ldquoLie group analysis ofmixed convection flow with mass transfer over a stretchingsurface with suction or injectionrdquo Mathematical Problems inEngineering vol 2010 Article ID 264901 14 pages 2010
[33] M A A Hamad M J Uddin and A I M Ismail ldquoRadiationeffects on heat and mass transfer in MHD stagnation-pointflow over a permeable flat plate with thermal convectivesurface boundary condition temperature dependent viscosityand thermal conductivityrdquoNuclear Engineering and Design vol242 pp 194ndash200 2012
Mathematical Problems in Engineering 21
[34] A Aziz M J Uddin M A A Hamad and A I MdIsmail ldquoMHD flow over an inclined radiating plate with thetemperature-dependent thermal conductivity variable reactiveindex and heat generationrdquoHeat Transfer vol 41 no 3 pp 241ndash259 2012
[35] A G Hansen Similarity Analysis of Boundary Layer Problemsin Engineering Prentice Hall Englewood Cliffs NJ USA 1964
[36] W FAmesNonlinear Partial Differential Equations in Engineer-ing Academic Press New York NY USA 1972
[37] R Seshadri and T Y Na Group Invariance in EngineeringBoundary Value Problems Springer New York NY USA 1985
[38] D Shang Theory of Heat Transfer with Forced Convection FilmFlows vol 3 of Heat and Mass Transfer 2010
[40] E Erfani M M Rashidi and A B Parsa ldquoThe modifieddifferential transform method for solving off-centered stag-nation flow toward a rotating discrdquo International Journal ofComputational Methods vol 7 no 4 pp 655ndash670 2010
[41] M M Rashidi N Laraqi and A Basiri Parsa ldquoAnalyticalmodeling of heat convection in magnetized micropolar fluid byusing modified differential transform methodrdquo Heat Transfervol 40 no 3 pp 187ndash204 2011
[42] M M Rashidi and M Keimanesh ldquoUsing differential trans-form method and pade approximant for solving mhd flow ina laminar liquid film from a horizontal stretching surfacerdquoMathematical Problems in Engineering vol 2010 Article ID491319 14 pages 2010
[43] M M Rashidi and G Domairry ldquoNew analytical solution ofthe three-dimensional NavierStokes equationsrdquoModern PhysicsLetters B vol 23 no 26 pp 3147ndash3155 2009
[44] M M Rashidi M Keimanesh O A Beg and T K HungldquoMagnetohydrodynamic biorheological transport phenomenain a porous medium a simulation of magnetic blood flowcontrol and filtrationrdquo International Journal for NumericalMethods in Biomedical Engineering vol 27 no 6 pp 805ndash8212011
[45] S J Liao The proposed homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992
[46] M M Rashidi G Domairry and S Dinarvand ldquoApproximatesolutions for the Burger and regularized long wave equationsby means of the homotopy analysis methodrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 14 no 3 pp708ndash717 2009
[47] Z Ziabakhsh and G Domairry ldquoAnalytic solution of naturalconvection flow of a non-Newtonian fluid between two verticalflat plates using homotopy analysis methodrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 14 no 5pp 1868ndash1880 2009
[48] M M Rashidi S A Mohimanian Pour T Hayat and SObaidat ldquoAnalytic approximate solutions for steady flow over arotating disk in porousmediumwith heat transfer by homotopyanalysis methodrdquo Computers and Fluids vol 54 no 1 pp 1ndash92012
[49] M Sajid T Hayat and S Asghar ldquoComparison between theHAM and HPM solutions of thin film flows of non-Newtonianfluids on a moving beltrdquo Nonlinear Dynamics vol 50 no 1-2pp 27ndash35 2007
[50] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for the double-diffusive natural convective boundary
layer flow in a porous medium saturated by a nanofluidrdquoInternational Journal of Heat and Mass Transfer vol 54 no 1-3pp 374ndash378 2011
[51] K Vajravelu and A Hadjinicolaou ldquoConvective heat transferin an electrically conducting fluid at a stretching surfacewith uniform free streamrdquo International Journal of EngineeringScience vol 35 pp 1237ndash1244 1997
[52] S Mukhopadhyay and G C Layek ldquoEffects of variable fluidviscosity on flow past a heated stretching sheet embedded ina porous medium in presence of heat sourcesinkrdquo Meccanicavol 47 no 4 pp 863ndash876 2012
[53] S J Liao ldquoAn optimal homotopy-analysis approach for stronglynonlinear differential equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 8 pp 2003ndash20162010
[54] A Aziz ldquoA similarity solution for laminar thermal boundarylayer over a flat plate with a convective surface boundary con-ditionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1064ndash1068 2009
Figure 34 Effect of order of thermal slip parameter 119887 on thedimensionless concentration profile
of auxiliary parameter ℏ to ensure the convergence of thesolution series was obtained through the so called ℏ-curveThe HAM provides us with a convenient way to control theconvergence of approximation series which is a fundamentalqualitative difference in analysis between the HAM andother methods The numerical results of the above problemsdisplay a fast convergence with minimal calculations Thisshows that the HAM is a very efficient method Finally theagreement between analytical and numerical results of thepresent study with previous published results is excellent
Nomenclature
119887 Thermal slip parameter119862 Dimensional concentration119862119891 Skin friction factor1198631 Thermal slip factor119863119861 Brownian diffusion coefficient119863119879 Thermophoretic diffusion coefficient119891 Dimensionless velocity functions119892 Gravitation acceleration119870 Chemical reaction parameter119896 Permeability of the porous media1198960 The constant reaction rate119896119898 Effective thermal conductivity of the porous
medium119896(119909) Variable reaction rateL Linear operator of the HAM119871 Length of horizontal plateLe Lewis numberN Nonlinear operator of the HAM
119899 Order of chemical reaction119873119887 Brownian motion119873119903 Buoyancy ratio119873119905 ThermophoresisNu Nusselt number119875 PressurePr Prandtl number119876 Heat generationabsorption parameter1198760 Heat generationabsorption constant119902 Internal heat generation rate
Ra Rayleigh numberSh Sherwood number119879 Temperature119905 Time119906 Velocity in 119909-direction119880119903 Reference velocity Velocity vectorV Velocity in 119910-direction119909 Distance along the surface119910 Distance normal to the surface
Greek Letters
120572119898 Thermal diffusivity120573 Volumetric expansion coefficient of nanofluid120576 Porosity of the porous media120601 Dimensionless concentration120578 Similarity variable120583 Dynamic viscosity120579 Dimensionless temperature120588119891 Density of the base fluid120588119901 Density of the nanoparticles(120588119862)119891 Effective heat capacity of the fluid(120588119862)119875 Effective heat capacity of the nanoparticle material120595 Stream function
Subscript Superscript
infin Conditions far away from the surface1015840 Differentiation with respect to 120578
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
M M Rashidi wishes to thank the Centre for DifferentialEquations Continuum Mechanics and Applications Schoolof Computational and Applied Mathematics University ofthe Witwatersrand for the hospitality during his visit EMomoniat thanks theNational Research Foundation of SouthAfrica for their support
20 Mathematical Problems in Engineering
References
[1] E Abu-Nada F O Hakan and I Pop ldquoBuoyancy inducedflow in a nanofluid filled enclosure partially exposed to forcedconvectionrdquo Superlattices and Microstructures vol 51 no 3 pp381ndash395 2012
[2] S Kakac and A Pramuanjaroenkij ldquoReview of convective heattransfer enhancement with nanofluidsrdquo International Journal ofHeat and Mass Transfer vol 52 no 13-14 pp 3187ndash3196 2009
[3] J A Eastman S U S Choi S Li W Yu and L J ThompsonldquoAnomalously increased effective thermal conductivities ofethylene glycol-based nanofluids containing copper nanoparti-clesrdquo Applied Physics Letters vol 78 no 6 pp 718ndash720 2001
[4] Y Xuan and Q Li ldquoHeat transfer enhancement of nanofluidsrdquoInternational Journal of Heat and Fluid Flow vol 21 no 1 pp58ndash64 2000
[5] S K Das N Putra P Thiesen and W Roetzel ldquoTemperaturedependence of thermal conductivity enhancement for nanoflu-idsrdquo Journal of Heat Transfer vol 125 no 4 pp 567ndash574 2003
[6] D H Kumar H E Patel V R R Kumar T SundararajanT Pradeep and S K Das ldquoModel for heat conduction innanofluidsrdquo Physical Review Letters vol 93 no 14 Article ID144301 2004
[7] H E Patel T Sundararajan T Pradeep A Dasgupta NDasgupta and S K Das ldquoA micro-convection model forthermal conductivity of nanofluidsrdquo Pramana vol 65 no 5 pp863ndash869 2005
[8] A V Kuznetsov ldquoNon-oscillatory and oscillatory nanofluidbio-thermal convection in a horizontal layer of finite depthrdquoEuropean Journal of Mechanics vol 30 no 2 pp 156ndash165 2011
[9] M Napoli J C T Eijkel and S Pennathur ldquoNanofluidictechnology for biomolecule applications a critical reviewrdquo Labon a Chip vol 10 no 8 pp 957ndash985 2010
[10] J C Crepeau and R Clarksean ldquoSimilarity solutions of naturalconvection with internal heat generationrdquo Journal of HeatTransfer vol 119 no 1 pp 183ndash185 1997
[11] A Z Sahin ldquoTransient heat conduction in semi-infinite solidwith spatially decaying exponential heat generationrdquo Interna-tional Communications in Heat andMass Transfer vol 19 no 3pp 349ndash358 1992
[12] O D Makind and A Aziz ldquoMixed convection from a con-vectively heated vertical plate to a fluid with internal heatgenerationrdquo Journal of Heat Transfer vol 133 Article ID 1225016 pages 2011
[13] J Buongiorno ldquoConvective transport in nanofluidsrdquo Journal ofHeat Transfer vol 128 no 3 pp 240ndash250 2006
[14] AV Kuznetsov andDANield ldquoNatural convective boundary-layer flow of a nanofluid past a vertical platerdquo InternationalJournal of Thermal Sciences vol 49 no 2 pp 243ndash247 2010
[15] L B Godson D Raja L D Mohan and S WongwisescldquoEnhancement of heat transfer using nanofluids-an overviewrdquoRenewable and Sustainable Energy Reviews vol 14 no 2 pp629ndash641 2010
[16] D A Nield and A BejanConvection in PorousMedia SpringerNew York NY USA 3rd edition 2006
[17] P Vadasz Emerging Topics in Heat and Mass Transfer in PorousMedia Springer New York NY USA 2008
[18] K Vafai Porous Media Applications in Biological Systems andBiotechnology CRC Press New York NY USA 2010
[19] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application to
heat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977
[20] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for natural convective boundary-layer flow in a porousmedium saturated by a nanofluidrdquo International Journal of Heatand Mass Transfer vol 52 no 25-26 pp 5792ndash5795 2009
[21] P Cheng and I D Chang ldquoBuoyancy induced flows in asaturated porous medium adjacent to impermeable horizontalsurfacesrdquo International Journal of Heat and Mass Transfer vol19 no 11 pp 1267ndash1272 1976
[22] I D Chang and P Cheng ldquoMatched asymptotic expansionsfor free convection about an impermeable horizontal surfacein a porous mediumrdquo International Journal of Heat and MassTransfer vol 26 no 2 pp 163ndash174 1983
[23] D S Shiunlin and B Gebhart ldquoBuoyancy-induced flow adja-cent to a horizontal surface submerged in porous mediumsaturated with cold waterrdquo International Journal of Heat andMass Transfer vol 29 no 4 pp 611ndash623 1986
[24] J H Merkin and G Zhang ldquoOn the similarity solutions for freeconvection in a saturated porousmedium adjacent to imperme-able horizontal surfacesrdquoWarme- und Stoffubertragung vol 25no 3 pp 179ndash184 1990
[25] M A Chaudhary J H Merkin and I Pop ldquoNatural convectionfrom a horizontal permeable surface in a porous mediummdashnumerical and asymptotic solutionsrdquo Transport in PorousMedia vol 22 no 3 pp 327ndash344 1996
[26] R S R Gorla and A Chamkha ldquoNatural convective boundarylayer flow over a nonisothermal vertical plate embedded ina porous medium saturated with a nanofluidrdquo Nanoscale andMicroscale Thermophysical Engineering vol 15 no 2 pp 81ndash942011
[27] W A Khan and I Pop ldquoFree convection boundary layer flowpast a horizontal flat plate embedded in a porous medium filledwith a nanofluidrdquo Journal of Heat Transfer vol 133 no 9 ArticleID 094501 2011
[28] A Aziz W A Khan and I Pop ldquoFree convection boundarylayer flow past a horizontal flat plate embedded in porousmedium filled by nanofluid containing gyrotactic microorgan-ismsrdquo International Journal of Thermal Sciences vol 56 pp 48ndash57 2012
[29] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer New York NYUSA 2009
[30] A A Avramenko S G Kobzar I V Shevchuk A V Kuznetsovand L T Iwanisov ldquoSymmetry of turbulent boundary-layerflows investigation of different eddy viscosity modelsrdquo ActaMechanica vol 151 no 1-2 pp 1ndash14 2001
[31] A V Kuznetsov A A Avramenko and P Geng ldquoAnalyticalinvestigation of a falling plume caused by bioconvection ofoxytactic bacteria in a fluid saturated porous mediumrdquo Inter-national Journal of Engineering Science vol 42 no 5-6 pp 557ndash569 2004
[32] M Jalil S Asghar and M Mushtaq ldquoLie group analysis ofmixed convection flow with mass transfer over a stretchingsurface with suction or injectionrdquo Mathematical Problems inEngineering vol 2010 Article ID 264901 14 pages 2010
[33] M A A Hamad M J Uddin and A I M Ismail ldquoRadiationeffects on heat and mass transfer in MHD stagnation-pointflow over a permeable flat plate with thermal convectivesurface boundary condition temperature dependent viscosityand thermal conductivityrdquoNuclear Engineering and Design vol242 pp 194ndash200 2012
Mathematical Problems in Engineering 21
[34] A Aziz M J Uddin M A A Hamad and A I MdIsmail ldquoMHD flow over an inclined radiating plate with thetemperature-dependent thermal conductivity variable reactiveindex and heat generationrdquoHeat Transfer vol 41 no 3 pp 241ndash259 2012
[35] A G Hansen Similarity Analysis of Boundary Layer Problemsin Engineering Prentice Hall Englewood Cliffs NJ USA 1964
[36] W FAmesNonlinear Partial Differential Equations in Engineer-ing Academic Press New York NY USA 1972
[37] R Seshadri and T Y Na Group Invariance in EngineeringBoundary Value Problems Springer New York NY USA 1985
[38] D Shang Theory of Heat Transfer with Forced Convection FilmFlows vol 3 of Heat and Mass Transfer 2010
[40] E Erfani M M Rashidi and A B Parsa ldquoThe modifieddifferential transform method for solving off-centered stag-nation flow toward a rotating discrdquo International Journal ofComputational Methods vol 7 no 4 pp 655ndash670 2010
[41] M M Rashidi N Laraqi and A Basiri Parsa ldquoAnalyticalmodeling of heat convection in magnetized micropolar fluid byusing modified differential transform methodrdquo Heat Transfervol 40 no 3 pp 187ndash204 2011
[42] M M Rashidi and M Keimanesh ldquoUsing differential trans-form method and pade approximant for solving mhd flow ina laminar liquid film from a horizontal stretching surfacerdquoMathematical Problems in Engineering vol 2010 Article ID491319 14 pages 2010
[43] M M Rashidi and G Domairry ldquoNew analytical solution ofthe three-dimensional NavierStokes equationsrdquoModern PhysicsLetters B vol 23 no 26 pp 3147ndash3155 2009
[44] M M Rashidi M Keimanesh O A Beg and T K HungldquoMagnetohydrodynamic biorheological transport phenomenain a porous medium a simulation of magnetic blood flowcontrol and filtrationrdquo International Journal for NumericalMethods in Biomedical Engineering vol 27 no 6 pp 805ndash8212011
[45] S J Liao The proposed homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992
[46] M M Rashidi G Domairry and S Dinarvand ldquoApproximatesolutions for the Burger and regularized long wave equationsby means of the homotopy analysis methodrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 14 no 3 pp708ndash717 2009
[47] Z Ziabakhsh and G Domairry ldquoAnalytic solution of naturalconvection flow of a non-Newtonian fluid between two verticalflat plates using homotopy analysis methodrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 14 no 5pp 1868ndash1880 2009
[48] M M Rashidi S A Mohimanian Pour T Hayat and SObaidat ldquoAnalytic approximate solutions for steady flow over arotating disk in porousmediumwith heat transfer by homotopyanalysis methodrdquo Computers and Fluids vol 54 no 1 pp 1ndash92012
[49] M Sajid T Hayat and S Asghar ldquoComparison between theHAM and HPM solutions of thin film flows of non-Newtonianfluids on a moving beltrdquo Nonlinear Dynamics vol 50 no 1-2pp 27ndash35 2007
[50] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for the double-diffusive natural convective boundary
layer flow in a porous medium saturated by a nanofluidrdquoInternational Journal of Heat and Mass Transfer vol 54 no 1-3pp 374ndash378 2011
[51] K Vajravelu and A Hadjinicolaou ldquoConvective heat transferin an electrically conducting fluid at a stretching surfacewith uniform free streamrdquo International Journal of EngineeringScience vol 35 pp 1237ndash1244 1997
[52] S Mukhopadhyay and G C Layek ldquoEffects of variable fluidviscosity on flow past a heated stretching sheet embedded ina porous medium in presence of heat sourcesinkrdquo Meccanicavol 47 no 4 pp 863ndash876 2012
[53] S J Liao ldquoAn optimal homotopy-analysis approach for stronglynonlinear differential equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 8 pp 2003ndash20162010
[54] A Aziz ldquoA similarity solution for laminar thermal boundarylayer over a flat plate with a convective surface boundary con-ditionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1064ndash1068 2009
[1] E Abu-Nada F O Hakan and I Pop ldquoBuoyancy inducedflow in a nanofluid filled enclosure partially exposed to forcedconvectionrdquo Superlattices and Microstructures vol 51 no 3 pp381ndash395 2012
[2] S Kakac and A Pramuanjaroenkij ldquoReview of convective heattransfer enhancement with nanofluidsrdquo International Journal ofHeat and Mass Transfer vol 52 no 13-14 pp 3187ndash3196 2009
[3] J A Eastman S U S Choi S Li W Yu and L J ThompsonldquoAnomalously increased effective thermal conductivities ofethylene glycol-based nanofluids containing copper nanoparti-clesrdquo Applied Physics Letters vol 78 no 6 pp 718ndash720 2001
[4] Y Xuan and Q Li ldquoHeat transfer enhancement of nanofluidsrdquoInternational Journal of Heat and Fluid Flow vol 21 no 1 pp58ndash64 2000
[5] S K Das N Putra P Thiesen and W Roetzel ldquoTemperaturedependence of thermal conductivity enhancement for nanoflu-idsrdquo Journal of Heat Transfer vol 125 no 4 pp 567ndash574 2003
[6] D H Kumar H E Patel V R R Kumar T SundararajanT Pradeep and S K Das ldquoModel for heat conduction innanofluidsrdquo Physical Review Letters vol 93 no 14 Article ID144301 2004
[7] H E Patel T Sundararajan T Pradeep A Dasgupta NDasgupta and S K Das ldquoA micro-convection model forthermal conductivity of nanofluidsrdquo Pramana vol 65 no 5 pp863ndash869 2005
[8] A V Kuznetsov ldquoNon-oscillatory and oscillatory nanofluidbio-thermal convection in a horizontal layer of finite depthrdquoEuropean Journal of Mechanics vol 30 no 2 pp 156ndash165 2011
[9] M Napoli J C T Eijkel and S Pennathur ldquoNanofluidictechnology for biomolecule applications a critical reviewrdquo Labon a Chip vol 10 no 8 pp 957ndash985 2010
[10] J C Crepeau and R Clarksean ldquoSimilarity solutions of naturalconvection with internal heat generationrdquo Journal of HeatTransfer vol 119 no 1 pp 183ndash185 1997
[11] A Z Sahin ldquoTransient heat conduction in semi-infinite solidwith spatially decaying exponential heat generationrdquo Interna-tional Communications in Heat andMass Transfer vol 19 no 3pp 349ndash358 1992
[12] O D Makind and A Aziz ldquoMixed convection from a con-vectively heated vertical plate to a fluid with internal heatgenerationrdquo Journal of Heat Transfer vol 133 Article ID 1225016 pages 2011
[13] J Buongiorno ldquoConvective transport in nanofluidsrdquo Journal ofHeat Transfer vol 128 no 3 pp 240ndash250 2006
[14] AV Kuznetsov andDANield ldquoNatural convective boundary-layer flow of a nanofluid past a vertical platerdquo InternationalJournal of Thermal Sciences vol 49 no 2 pp 243ndash247 2010
[15] L B Godson D Raja L D Mohan and S WongwisescldquoEnhancement of heat transfer using nanofluids-an overviewrdquoRenewable and Sustainable Energy Reviews vol 14 no 2 pp629ndash641 2010
[16] D A Nield and A BejanConvection in PorousMedia SpringerNew York NY USA 3rd edition 2006
[17] P Vadasz Emerging Topics in Heat and Mass Transfer in PorousMedia Springer New York NY USA 2008
[18] K Vafai Porous Media Applications in Biological Systems andBiotechnology CRC Press New York NY USA 2010
[19] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application to
heat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977
[20] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for natural convective boundary-layer flow in a porousmedium saturated by a nanofluidrdquo International Journal of Heatand Mass Transfer vol 52 no 25-26 pp 5792ndash5795 2009
[21] P Cheng and I D Chang ldquoBuoyancy induced flows in asaturated porous medium adjacent to impermeable horizontalsurfacesrdquo International Journal of Heat and Mass Transfer vol19 no 11 pp 1267ndash1272 1976
[22] I D Chang and P Cheng ldquoMatched asymptotic expansionsfor free convection about an impermeable horizontal surfacein a porous mediumrdquo International Journal of Heat and MassTransfer vol 26 no 2 pp 163ndash174 1983
[23] D S Shiunlin and B Gebhart ldquoBuoyancy-induced flow adja-cent to a horizontal surface submerged in porous mediumsaturated with cold waterrdquo International Journal of Heat andMass Transfer vol 29 no 4 pp 611ndash623 1986
[24] J H Merkin and G Zhang ldquoOn the similarity solutions for freeconvection in a saturated porousmedium adjacent to imperme-able horizontal surfacesrdquoWarme- und Stoffubertragung vol 25no 3 pp 179ndash184 1990
[25] M A Chaudhary J H Merkin and I Pop ldquoNatural convectionfrom a horizontal permeable surface in a porous mediummdashnumerical and asymptotic solutionsrdquo Transport in PorousMedia vol 22 no 3 pp 327ndash344 1996
[26] R S R Gorla and A Chamkha ldquoNatural convective boundarylayer flow over a nonisothermal vertical plate embedded ina porous medium saturated with a nanofluidrdquo Nanoscale andMicroscale Thermophysical Engineering vol 15 no 2 pp 81ndash942011
[27] W A Khan and I Pop ldquoFree convection boundary layer flowpast a horizontal flat plate embedded in a porous medium filledwith a nanofluidrdquo Journal of Heat Transfer vol 133 no 9 ArticleID 094501 2011
[28] A Aziz W A Khan and I Pop ldquoFree convection boundarylayer flow past a horizontal flat plate embedded in porousmedium filled by nanofluid containing gyrotactic microorgan-ismsrdquo International Journal of Thermal Sciences vol 56 pp 48ndash57 2012
[29] G W Bluman and S C Anco Symmetry and IntegrationMethods for Differential Equations Springer New York NYUSA 2009
[30] A A Avramenko S G Kobzar I V Shevchuk A V Kuznetsovand L T Iwanisov ldquoSymmetry of turbulent boundary-layerflows investigation of different eddy viscosity modelsrdquo ActaMechanica vol 151 no 1-2 pp 1ndash14 2001
[31] A V Kuznetsov A A Avramenko and P Geng ldquoAnalyticalinvestigation of a falling plume caused by bioconvection ofoxytactic bacteria in a fluid saturated porous mediumrdquo Inter-national Journal of Engineering Science vol 42 no 5-6 pp 557ndash569 2004
[32] M Jalil S Asghar and M Mushtaq ldquoLie group analysis ofmixed convection flow with mass transfer over a stretchingsurface with suction or injectionrdquo Mathematical Problems inEngineering vol 2010 Article ID 264901 14 pages 2010
[33] M A A Hamad M J Uddin and A I M Ismail ldquoRadiationeffects on heat and mass transfer in MHD stagnation-pointflow over a permeable flat plate with thermal convectivesurface boundary condition temperature dependent viscosityand thermal conductivityrdquoNuclear Engineering and Design vol242 pp 194ndash200 2012
Mathematical Problems in Engineering 21
[34] A Aziz M J Uddin M A A Hamad and A I MdIsmail ldquoMHD flow over an inclined radiating plate with thetemperature-dependent thermal conductivity variable reactiveindex and heat generationrdquoHeat Transfer vol 41 no 3 pp 241ndash259 2012
[35] A G Hansen Similarity Analysis of Boundary Layer Problemsin Engineering Prentice Hall Englewood Cliffs NJ USA 1964
[36] W FAmesNonlinear Partial Differential Equations in Engineer-ing Academic Press New York NY USA 1972
[37] R Seshadri and T Y Na Group Invariance in EngineeringBoundary Value Problems Springer New York NY USA 1985
[38] D Shang Theory of Heat Transfer with Forced Convection FilmFlows vol 3 of Heat and Mass Transfer 2010
[40] E Erfani M M Rashidi and A B Parsa ldquoThe modifieddifferential transform method for solving off-centered stag-nation flow toward a rotating discrdquo International Journal ofComputational Methods vol 7 no 4 pp 655ndash670 2010
[41] M M Rashidi N Laraqi and A Basiri Parsa ldquoAnalyticalmodeling of heat convection in magnetized micropolar fluid byusing modified differential transform methodrdquo Heat Transfervol 40 no 3 pp 187ndash204 2011
[42] M M Rashidi and M Keimanesh ldquoUsing differential trans-form method and pade approximant for solving mhd flow ina laminar liquid film from a horizontal stretching surfacerdquoMathematical Problems in Engineering vol 2010 Article ID491319 14 pages 2010
[43] M M Rashidi and G Domairry ldquoNew analytical solution ofthe three-dimensional NavierStokes equationsrdquoModern PhysicsLetters B vol 23 no 26 pp 3147ndash3155 2009
[44] M M Rashidi M Keimanesh O A Beg and T K HungldquoMagnetohydrodynamic biorheological transport phenomenain a porous medium a simulation of magnetic blood flowcontrol and filtrationrdquo International Journal for NumericalMethods in Biomedical Engineering vol 27 no 6 pp 805ndash8212011
[45] S J Liao The proposed homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992
[46] M M Rashidi G Domairry and S Dinarvand ldquoApproximatesolutions for the Burger and regularized long wave equationsby means of the homotopy analysis methodrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 14 no 3 pp708ndash717 2009
[47] Z Ziabakhsh and G Domairry ldquoAnalytic solution of naturalconvection flow of a non-Newtonian fluid between two verticalflat plates using homotopy analysis methodrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 14 no 5pp 1868ndash1880 2009
[48] M M Rashidi S A Mohimanian Pour T Hayat and SObaidat ldquoAnalytic approximate solutions for steady flow over arotating disk in porousmediumwith heat transfer by homotopyanalysis methodrdquo Computers and Fluids vol 54 no 1 pp 1ndash92012
[49] M Sajid T Hayat and S Asghar ldquoComparison between theHAM and HPM solutions of thin film flows of non-Newtonianfluids on a moving beltrdquo Nonlinear Dynamics vol 50 no 1-2pp 27ndash35 2007
[50] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for the double-diffusive natural convective boundary
layer flow in a porous medium saturated by a nanofluidrdquoInternational Journal of Heat and Mass Transfer vol 54 no 1-3pp 374ndash378 2011
[51] K Vajravelu and A Hadjinicolaou ldquoConvective heat transferin an electrically conducting fluid at a stretching surfacewith uniform free streamrdquo International Journal of EngineeringScience vol 35 pp 1237ndash1244 1997
[52] S Mukhopadhyay and G C Layek ldquoEffects of variable fluidviscosity on flow past a heated stretching sheet embedded ina porous medium in presence of heat sourcesinkrdquo Meccanicavol 47 no 4 pp 863ndash876 2012
[53] S J Liao ldquoAn optimal homotopy-analysis approach for stronglynonlinear differential equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 8 pp 2003ndash20162010
[54] A Aziz ldquoA similarity solution for laminar thermal boundarylayer over a flat plate with a convective surface boundary con-ditionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1064ndash1068 2009
[34] A Aziz M J Uddin M A A Hamad and A I MdIsmail ldquoMHD flow over an inclined radiating plate with thetemperature-dependent thermal conductivity variable reactiveindex and heat generationrdquoHeat Transfer vol 41 no 3 pp 241ndash259 2012
[35] A G Hansen Similarity Analysis of Boundary Layer Problemsin Engineering Prentice Hall Englewood Cliffs NJ USA 1964
[36] W FAmesNonlinear Partial Differential Equations in Engineer-ing Academic Press New York NY USA 1972
[37] R Seshadri and T Y Na Group Invariance in EngineeringBoundary Value Problems Springer New York NY USA 1985
[38] D Shang Theory of Heat Transfer with Forced Convection FilmFlows vol 3 of Heat and Mass Transfer 2010
[40] E Erfani M M Rashidi and A B Parsa ldquoThe modifieddifferential transform method for solving off-centered stag-nation flow toward a rotating discrdquo International Journal ofComputational Methods vol 7 no 4 pp 655ndash670 2010
[41] M M Rashidi N Laraqi and A Basiri Parsa ldquoAnalyticalmodeling of heat convection in magnetized micropolar fluid byusing modified differential transform methodrdquo Heat Transfervol 40 no 3 pp 187ndash204 2011
[42] M M Rashidi and M Keimanesh ldquoUsing differential trans-form method and pade approximant for solving mhd flow ina laminar liquid film from a horizontal stretching surfacerdquoMathematical Problems in Engineering vol 2010 Article ID491319 14 pages 2010
[43] M M Rashidi and G Domairry ldquoNew analytical solution ofthe three-dimensional NavierStokes equationsrdquoModern PhysicsLetters B vol 23 no 26 pp 3147ndash3155 2009
[44] M M Rashidi M Keimanesh O A Beg and T K HungldquoMagnetohydrodynamic biorheological transport phenomenain a porous medium a simulation of magnetic blood flowcontrol and filtrationrdquo International Journal for NumericalMethods in Biomedical Engineering vol 27 no 6 pp 805ndash8212011
[45] S J Liao The proposed homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992
[46] M M Rashidi G Domairry and S Dinarvand ldquoApproximatesolutions for the Burger and regularized long wave equationsby means of the homotopy analysis methodrdquo CommunicationsinNonlinear Science andNumerical Simulation vol 14 no 3 pp708ndash717 2009
[47] Z Ziabakhsh and G Domairry ldquoAnalytic solution of naturalconvection flow of a non-Newtonian fluid between two verticalflat plates using homotopy analysis methodrdquo Communicationsin Nonlinear Science and Numerical Simulation vol 14 no 5pp 1868ndash1880 2009
[48] M M Rashidi S A Mohimanian Pour T Hayat and SObaidat ldquoAnalytic approximate solutions for steady flow over arotating disk in porousmediumwith heat transfer by homotopyanalysis methodrdquo Computers and Fluids vol 54 no 1 pp 1ndash92012
[49] M Sajid T Hayat and S Asghar ldquoComparison between theHAM and HPM solutions of thin film flows of non-Newtonianfluids on a moving beltrdquo Nonlinear Dynamics vol 50 no 1-2pp 27ndash35 2007
[50] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for the double-diffusive natural convective boundary
layer flow in a porous medium saturated by a nanofluidrdquoInternational Journal of Heat and Mass Transfer vol 54 no 1-3pp 374ndash378 2011
[51] K Vajravelu and A Hadjinicolaou ldquoConvective heat transferin an electrically conducting fluid at a stretching surfacewith uniform free streamrdquo International Journal of EngineeringScience vol 35 pp 1237ndash1244 1997
[52] S Mukhopadhyay and G C Layek ldquoEffects of variable fluidviscosity on flow past a heated stretching sheet embedded ina porous medium in presence of heat sourcesinkrdquo Meccanicavol 47 no 4 pp 863ndash876 2012
[53] S J Liao ldquoAn optimal homotopy-analysis approach for stronglynonlinear differential equationsrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 8 pp 2003ndash20162010
[54] A Aziz ldquoA similarity solution for laminar thermal boundarylayer over a flat plate with a convective surface boundary con-ditionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1064ndash1068 2009
![Chapter 7 Lie Groups, Lie Algebras and the Exponential Mapcis610/cis61005sl8.pdf · Lie Groups, Lie Algebras and the Exponential Map 7.1 Lie Groups and Lie Algebras In Gallier [?],](https://static.documents.pub/doc/80x56/5f0c1a337e708231d433c07b/chapter-7-lie-groups-lie-algebras-and-the-exponential-map-cis610-lie-groups.jpg)
