Hindawi Publishing CorporationInternational Journal of Chemical EngineeringVolume 2013 Article ID 296834 10 pageshttpdxdoiorg1011552013296834
Research ArticleMHD Heat and Mass Transfer of Chemical ReactionFluid Flow over a Moving Vertical Plate in Presence of HeatSource with Convective Surface Boundary Condition
B R Rout1 S K Parida2 and S Panda3
1 Department of Mathematics Krupajal Engineering College Prasanti Vihar Pubasasan KausalyagangaBhubaneswar Odisha 751002 India
2Department of Mathematics Institute of Technical Education and Research (ITER) SOA UniversityBhubaneswar Odisha 751019 India
3 Department of Mathematics and Civil Engineering National Institute of Technology (NIT) Calicut Calicut 673601 India
Correspondence should be addressed to S K Parida sparidamath2007rediffmailcom
Received 1 January 2013 Revised 5 March 2013 Accepted 6 March 2013
Academic Editor Jose C Merchuk
Copyright copy 2013 B R Rout 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
This paper aims to investigate the influence of chemical reaction and the combined effects of internal heat generation and aconvective boundary condition on the laminar boundary layer MHD heat and mass transfer flow over a moving vertical flat plateThe lower surface of the plate is in contact with a hot fluid while the stream of cold fluid flows over the upper surface with heatsource and chemical reaction The basic equations governing the flow heat transfer and concentration are reduced to a set ofordinary differential equations by using appropriate transformation for variables and solved numerically by Runge-Kutta fourth-order integration scheme in association with shooting methodThe effects of physical parameters on the velocity temperature andconcentration profiles are illustrated graphically A table recording the values of skin friction heat transfer and mass transfer at theplate is also presentedThe discussion focuses on the physical interpretation of the results as well as their comparison with previousstudies which shows good agreement as a special case of the problem
1 Introduction
The study of convective flow with heat and mass transferunder the influence of magnetic field and chemical reactionwith heat source has practical applications in many areasof science and engineering This phenomenon plays animportant role in chemical industry petroleum industrycooling of nuclear reactors and packed-bed catalytic reactorsNatural convection flows occur frequently in nature due totemperature differences concentration differences and alsodue to combined effects The concentration difference maysometimes produce qualitative changes to the rate of heattransfer The study of heat generation in many fluids due toexothermic and endothermic chemical reactions and naturalconvection with heat generenation can be added to combus-tionmodeling In this direction Vajrvelu andNayfeh [1] stud-ied the hydromagnetic convection at a cone and at a wedge
in presence of temperature-dependent heat generation andabsorption effect Chamkha [2] later examined the effectof heat generation or absorption on hydromagnetic three-dimensional free convection flow over a vertical stretchingsurface The flow through porous media is a subject of mostcommon interest and has emerged as a separate intensiveresearch area because heat and mass transfer in porousmedium is very much prevalent in nature and can also beencountered in many technological processes In this contextthe effect of temperature-dependent heat sources has beenstudied by Moalem [3] taking into account the steady stateheat transfer within porous medium Rahman and Sattar [4]have investigated the effect of heat generation or absorptionon convective flow of a micropolar fluid past a continuouslymoving vertical porous plate in presence of a magnetic fieldAnalysis of transport processes and their interaction withchemical reaction has the greatest contributions to many
2 International Journal of Chemical Engineering
areas of chemical science The effect of chemical reactionon different geometry of the problem has been investigatedby many authors Das et al [5] have studied the effect ofmass transfer flow past an impulsively started infinite verticalplate with heat flux and chemical reaction The chemicalreaction effect on heat and mass transfer flow along a semi-infinite horizontal plate has been studied by Anjalidevi andKandaswamy [6] and later it was extended for Hiemenzflow by Seddeek et al [7] and for polar fluid by Patil andKulkarni [8] Salem and Abd El-Aziz [9] have reported theeffect of hall currents and chemical reaction on hydromag-netic flow of a stretching vertical surface with internal heatgeneration or absorption Ibrahim et al [10] studied theeffect of chemical reaction and radiation absorption on theunsteady MHD free convection flow past a semi-infinitevertical permeablemoving platewith heat source and suctionA detailed numerical study has been carried out for unsteadyhydromagnetic natural convection heat and mass transferwith chemical reaction over a vertical plate in rotating systemwith periodic suction by Parida et al [11] Rajeswari etal [12] have investigated chemical reaction heat and masstransfer on nonlinear MHD boundary layer flow through avertical porous surface in presence of suctionMahdy [13] hasstudied the effect of chemical reaction and heat generationor absorption on double diffussive convection from verticaltruncated cone in a porous media with variable viscosityPal and Talukdar [14] have studied perturbation analysisof unsteady magnetohydrodynamic convective heat masstransfer in boundary layer slip flow past a vertical permeableplate with a thermal radiation and chemical reaction Furtherthe effect of thermal radiation heat and mass transfer flowof a variable viscosity fluid past a vertical porous plate inpresence of transverse magnetic field was investigated byMakinde and Ogulu [15] The analysis of MHD mixed-convection interaction with thermal radiation and higherorder chemical reaction is carried out by Makinde [16] Aziz[17] theoretically examined a similarity solution for a laminarthermal boundary layer over a flat plate with a convectivesurface boundary condition He found an interesting resultthat a similarity solution is possible if the convective heattransfer along with the hot fluid on the lower surface of theplate is inversely proportional to the square root of the axialdistance Recently the combined effects of an exponentiallydecaying internal heat generation and a convective boundarycondition on the thermal boundary layer over a flat plate areinvestigated by Olanrewaju et al [18] In their study authorshave negelected the Sherwood effect Similar analysis hasbeen carried out by Makinde [19 20] without heat sourceandwith heat source [21] neglecting chemical reaction effectThere has been considerable interest in studying the effectof chemical reaction [22] and heat source effect on theboundary layer flow problem with heat and mass transfer ofan electrically conducting fluid in different geometry [23ndash25]
Heat source and chemical reaction effects are crucial incontrolling the heat and mass transfer The present paperattempts to investigate the influence of chemical reactionand the combined effects of internal heat generation and theconvective boundary condition on the MHD heat and masstransfer flow To the best of the authorsrsquo knowledge so far no
one has considered the combined effect of chemical reactionand heat source along with convective surface boundarycondition on MHD flow and the heat and mass transfer overa moving vertical plate This fact motivated us to propose thesimilar study We extend the recent work of Makinde [19]Olanrewaju et al [18] and Gangadhar et al [22] to expose theeffect of chemical reaction on MHD heat and mass transferover a moving vertical plate in presence of heat source alongwith convective surface boundary condition The couplednonlinear partial differential equations governing the flowheat and mass transfer have been reduced to a set of couplednonlinear ordinary differential equations by using similaritytransformation Following [19] the similarity solutions existif the convective heat transfer associated with the hot fluidon lower surface of the plate is proportional to the inversesquare root of the axial distance The reduced equations aresolved numerically using Runge-Kutta fourth-order integra-tion scheme together with shooting method The effect ofvarious physical parameters on the velocity temperature andconcentration fields is studied
2 Mathematical Formulation
A typical flow scenario is illustrated in Figure 1 it showsa steady two-dimensional boundary layer flow of a streamof cold incompressible electrically conducting fluid over amoving vertical flat plate at temperature 119879
infinin presence of
heat source and chemical reactionThe left surface of the plateis being heated by convection from a hot fluid at temperature119879119891that gives a heat transfer coefficient ℎ
119891 and 119879
infinis the
temperature of the fluid away from the plate The cold fluidin contact with the upper surface of the plate generates heatinternally at the volumetric rate 119876
0 Here the 119909-axis is taken
along the direction of plate and 119910-axis is normal to it Amagnetic field of uniform field strength 119861
0is applied in the
negative direction of 119910-axisThe continuity momentum energy and concentration
equations describing the flow under the Boussinesq approxi-mation can be written as
120597119906
120597119909+120597V
120597119910= 0 (1)
119906120597119906
120597119909+ V
120597119906
120597119910= ]
1205972
119906
120597 1199102minus120590 1198612
0
120588119906 + 119892120573 (119879 minus 119879
infin)
+ 119892120573lowast
(119862 minus 119862infin)
(2)
119906120597119879
120597119909+ V
120597119879
120597119910= 120572
1205972
119879
1205971199102+
1198760
120588119862119901
(119879 minus 119879infin) (3)
119906120597119862
120597119909+ V
120597119862
120597119910= 119863
1205972
119862
1205971199102minus Kr1015840119862 (4)
The symbols 119906 and V denote the fluid velocity in the 119909-and 119910-direction Here 119879 and 119862 are the temperature andconcentration variables ] is the kinematic viscosity 120572 is thethermal diffusivity 119863 is the mass diffusivity 120573 is the thermalexpansion coefficient 120573lowast is the solutal expansion coefficient
International Journal of Chemical Engineering 3
119906 = 1198800
120592 = 0
119879119891
119862119908
119879
119862
119906
1198760
119892
1198610
119910
119862infin 119879infin
119909
Figure 1 Sketch of flow geometry
120588 is the fluid density 119892 is the gravitational acceleration 120590 isthe electrical conductivity 119876
0is the heat source 119862
119901is the
specific heat at constant pressure and Kr1015840 is the chemicalreaction rate on the species concentration In the aboveequations several assumptions have been made First theplate is nonconducting and the effects of radiant heatingviscous dissipation Hall effects and induced fields areneglected Second the physical properties that is viscosityheat capacity thermal diffusivity and the mass diffusivity ofthe fluid remain invariant throughout the fluid
The appropriate boundary conditions at the plate surfaceand far into the cold fluid are
where 119862119908is the species concentration at the plate surface 119860
is the constant 120582 is the power index of the concentration 1198800
is the plate velocity 119896 is the thermal conductivity coefficientand 119862
infinis the concentration of the fluid away from the
plate The boundary layer equations presented are nonlinearpartial differential equations and are in general difficult tosolve However the equations admit of a self-similar solutionTherefore transformation allows them to be reduced to asystem of ordinary differential equations that are relativelyeasy to solve numerically We look for solution compatiblewith (1) of the form
119906 = 11988001198911015840
(120578) V = minus1
2
radic]1198800
119909119891 (120578) +
1198800119910
2 1199091198911015840
(120578) (6)
where 120578 = 119910radic1198800(]119909) and prime denotes the differentiation
with respect to 120578
1
08
06
04
02
0
0 5 10 15120578
119891998400
(120578)
Ha = 01 119878 = 0 Kr = 0
Ha = 03 119878 = 0 Kr = 0
Ha = 03 119878 = 002 Kr = 01
Ha = 03 119878 = 002 Kr = 05
Figure 2 Velocity profiles for different values of Ha 119878 and Kr forGr = 01 Gc = 01 Sc = 062 Bi = 01 Nc = 001 and Pr = 072
Let us introduce the dimensionless quantities that is
Here Ha119909is the local magnetic field parameter Gr
119909is the
local thermal Grashof number Gc119909is the modified Grashof
number Bi119909is the local convective heat transfer parameter
Pr is the Prandtl number Sc is the Schmidt number 119878119909is the
local heat source parameter Kr119909is the local chemical reaction
parameter and Nc is the concentration difference parameterUsing (6) and (7) in (2)ndash(4) we get the following equations
119891101584010158401015840
+1
211989111989110158401015840
minusHa1199091198911015840
+ Gr119909120579 + Gc
119909120601 = 0 (8)
12057910158401015840
+1
2Pr119891 120579
1015840
+ Pr119878119909120579 = 0 (9)
12060110158401015840
+1
2Sc1198911206011015840 minus ScKr
119909(120601 +Nc) = 0 (10)
4 International Journal of Chemical Engineering
The corresponding boundary conditions equation (5) forvelocity temperature and concentration fields in terms ofnondimensional variables are
119891 (0) = 0 1198911015840
(0) = 1
1205791015840
(0) = Bi119909[120579 (0) minus 1] 120601 (0) = 1
1198911015840
(infin) = 0 120579 (infin) = 0 120601 (infin) = 0
(11)
It is observed that in the absence of local source parameterand chemical reaction parameter that is for 119878
119909= 0 and
Kr119909= 0 (8) (9) and (10) together with boundary condition
(11) are the same as those obtained by Makinde [19] It isnoticed that the concentration equation (10) in presence ofthe chemical reaction parameter (Kr
119909) in the fluid yields
nonhomogeneous differential equationwhich is coupledwithmomentum equation (8) and in general difficult to solveanalytically In order to overcome this difficulty we solvethese equations numerically by fourth-order Runge-Kuttamethod in association with shooting technique Firstly theseequations together with associated boundary conditions arereduced to first-order differential equations Since equationsto be solved are the third order for the velocity and secondorder for the temperature and concentration the values of1198911015840 1205791015840 and 120601
1015840 are needed at 120578 = 0 Therefore the shootingmethod is used to solve this boundary value problem Thelocal skin friction coefficient the local Nusselt number thelocal Sherwood number and the plate surface temperatureare computed in terms of 11989110158401015840(0) minus1205791015840(0) minus1206011015840(0) and 120579(0)respectively It can be noted that the local parameters Ha
119909
Gr119909 Gc119909 Bi119909 119878119909 and Kr
119909in (8)ndash(10) are functions of 119909
and generate local similarity solution In order to have a truesimilarity solution we assume the following relation [19]
ℎ119891=
119886
radic119909 120590 =
119887
119909 120573 =
119888
119909
120573lowast
=119889
119909 119876
0=119890
119909 Kr1015840 = 119898
119909
(12)
where 119886 119887 119888 119889 119890 and 119898 are the constants with appropriatedimensions In view of relation (12) the parameters Ha
119909
Gr119909 Gc119909 Bi119909 119878119909 and Kr
119909are now independent of 119909 and
henceforth we drop the index ldquo119909rdquo for simplicity
3 Result Discussion
The numerical solutions of the boundary value problemfor system of ordinary differential equations were obtainedby Runge-Kutta method along with shooting techniqueSince the physical domain of the underlying problem isunbounded the computational domain is chosen sufficientlylarge in order to meet the far field boundary condition atinfinity Here the transverse distance is fixed to 10 and suitablymore than 10 depending upon the choice of the parametersTo demonstrate successful implementation of the numericalscheme the numerical results are compared to those obtainedby a previous published paper (see [19]) for the local skinfriction coefficient plate surface temperature and the local
1
08
06
04
02
00 2 4 6 8 10
Bi = 01 119878 = 0 Kr = 0
Bi = 10 119878 = 0 Kr = 0
Bi = 10 119878 = 01 Kr = 005
Bi = 10 119878 = 01 Kr = 05
119891998400
(120578)
120578
Figure 3 Velocity profiles for different values of Bi 119878 and Kr forHa = 01 Gr = 01 Gc = 01 Sc = 062 Nc = 001 and Pr = 072
Figure 4 Velocity profiles for different values of Gc 119878 and Kr forHa = 01 Gr = 01 Sc = 062 Bi = 01 Nc = 001 and Pr = 072
Sherwood and Nusselt numbers in Table 1 for the parametersembedded in absence of local chemical reaction parameter(Kr) and local heat source parameter (119878) Table 1 presents acomparison of 11989110158401015840(0) minus1205791015840(0) 120579(0) and minus120601
1015840
(0) between thepresent results and the results obtained by Makinde [19] forvarious values of Ha Gr Gc Bi Pr and Sc when 119878 = Kr =0 The results are found to be in excellent agreement It isimportant to note that the momentum equation is coupledwith heat and mass transfer equations and hence the Prandtlnumber Schmidt number chemical reaction parameter andsource term have an influence on skin friction in our present
International Journal of Chemical Engineering 5
Table 1 Comparison of 11989110158401015840(0) minus1205791015840(0) 120579(0) and minus1206011015840(0) for various values of Bi Ha Gr Gc Pr and Sc when Kr = 119878 = 0
Bi Gr Gc Ha Pr Sc Makinde [19] Present study11989110158401015840
Figure 5 Velocity profiles for different values of Gr 119878 and Kr forHa = 01 Gc = 01 Sc = 062 Bi = 01 Nc = 001 and Pr = 072
problem The influence of heat source and chemical reactionparameters on local skin friction local Nusselt number platesurface temperature and Sherwood number are highlightedin Tables 2 and 3 for various nondimensional flow param-eters It is clearly seen from Table 2 that the magnitude ofskin friction and Nusselt number increase whereas the platesurface temperature and Sherwood number decrease withthe increase of source parameter Furthermore increasingthe strength of chemical reacting substances is to increasethe local skin friction the plate surface temperature andSherwood number but opposite behavior is seen for localNusselt numberThe obvious observation fromTable 2 is thatthe fluid low Prandtl number increases the magnitude oflocal skin friction and local Nusselt number while decreasingthe plate surface temperature and local Sherwood numberAgain the fluid with low convective resistance (or externalresistance) decreases the magnitude of local skin frictionwhile the Nusselt number the plate surface temperatureand local Sherwood number increase It is also observed
Sc = 062 119878 = 0 Kr = 0
Sc = 262 119878 = 0 Kr = 0
Sc = 062 119878 = 005 Kr = 005
Sc = 062 119878 = 005 Kr = 05
1
08
06
04
02
00 2 4 6 8 10
119891998400 (120578)
120578
Figure 6 Velocity profiles for different values of Sc 119878 and Kr forHa = 01 Gr = 01 Gc = 01 Bi = 01 Nc = 001 and Pr = 072
Table 2 Computation of skin friction (11989110158401015840(0)) Nusselt number(minus1205791015840(0)) plate surface temperature (120579(0)) and Sherwood number(minus1206011015840(0)) for different values of Bi Pr 119878 andKrTheother parametersare Ha = 01 Gr = 01 Gc = 01 Sc = 062 and Nc = 001
from Table 3 that the increase of Schimdt number resultsin increase of the magnitude of the local skin friction butopposite behavior is marked in case of Nusselt number andthe plate surface temperature
6 International Journal of Chemical Engineering
Table 3 Computation of skin friction (11989110158401015840(0)) Nusselt number(minus1205791015840(0)) plate surface temperature (120579(0)) and Sherwood number(minus1206011015840(0)) for different values of Sc 119878 and Kr The other parametersare fixed at Bi = Gc = Gr = 01 Pr = 072 and Nc = 001
31 Velocity Profiles Figures 2ndash7 exhibit the velocity pro-files obtained by the numerical simulations for variousflow parameters involved in the problem The simulatedparameters are reported in the figure caption The effectsof magnetic parameter on the velocity field in presenceand absence of source and chemical reaction parameterare shown in Figure 2 It illustrates that the velocity profiledecreases with the increase of magnetic parameter in absenceof source and chemical reaction parameter because Lorentzforce acts against the flow if the magnetic field is applied inthe normal direction In presence of source and chemicalreaction parameter no such appreciable change is observedin Figure 2This corresponds to Figure (2) in [19] and therebyagain validating our numerical scheme A little increase in thevelocity profile near the boundary layer is marked in Figure 3with the increase in the convective heat parameter becausethe fluid adjacent to the right surface of the plate becomeslighter by hot fluid and rises faster The boundary layer flowsdevelop adjacent to vertical surface and velocity reaches amaximum in the boundary layer It is evident from Figures4 and 5 that greater cooling of surface an increase in Gc andincrease in Gr result in an increase in the velocity It is dueto the fact that the increase in the values of Grashof numberand modified Grashof number has the tendency to increasethe thermal and mass buoyancy effect The increase is alsoevident due to the presence of source and chemical reactionparameters Furthermore the velocity increases rapidly andsuddenly falls near the boundary and then approaches thefar field boundary condition due to favorable buoyancy forcewith the increase of both Gr and Gc It can be seen that theincrease in the Prandtl number and Schmidt number leads toa fall in the velocity as shown in Figures 6 and 7
32 Temperature Profiles Figures 8ndash13 show the temperatureprofiles obtained by the numerical simulations for variousvalues of flow parameters Figure 8 clearly demonstrates thatthe temperature profiles increase with the increase of themagnetic field parameter which implies that the appliedmagnetic field tends to heat the fluid and thus reducesthe heat transfer from the wall Further it can be seenthat temperature profile increases due to increase of heatsource as well as chemical reaction parameter The thermalboundary layer thickness increases with an increase in the
Figure 8 Temperature profiles for different values of Ha 119878 and Krfor Gr = 01 Gc = 01 Sc = 062 Bi = 01 Nc = 001 and Pr = 072
plate surface convective heat parameter (Figure 9) and asimilar effect is also observed in Figure 12 with the increaseof Schmidt number It can be observed that the amplitudeof fluid temperature in presence of heat source and chemicalreacting substances is more in comparison to in absence ofthese parameters The steady state temperatures for differentGrashof number modified Grashof number internal heatsource and chemical reaction parameters are shown inFigures 10 and 11 The thermal boundary layer decreases withincreasing Grashof number and modified Grashof numberbut reverse effect is observed with the presence of chemicalreaction parameter This is also revealed in Figure 13 which
Figure 10 Temperature profiles for different values of Gc 119878 and Krfor Ha = 01 Gr = 01 Sc = 062 Bi = 01 Nc = 001 and Pr = 072
shows that thermal boundary layer thickness decreases asthe Prandtl number increases implying higher heat transferIt is due to that smaller values of Pr means increasing thethermal conductivity and therefore heat is able to diffuseaway from the plate more quickly than higher values of Prhence the rate of heat transfer is reduced It is noted thatowing to the presence of heat source effect (119878 gt 0) andchemical reaction parameter (Kr gt 0) the thermal stateof the fluid increases Hence the temperature of the fluidincreases within the boundary layers In the event that thestrength of the heat source and chemical reaction parametersare relatively large a remarkable change is observed in thetemperature profiles within the thermal boundary layer ascan be seen in Figure 13 Further the effect of heat generation
Figure 11 Temperature profiles for different values of Gr 119878 and KrforHa = 01 Gc = 01 119878c = 062 Bi = 01 Nc = 001 and Pr = 072
Sc = 062 119878 = 0 Kr = 0
Sc = 262 119878 = 0 Kr = 0
Sc = 062 119878 = 005 Kr = 005
Sc = 062 119878 = 005 Kr = 05
035
03
025
02
015
005
01
00 2 4 6 8 10
120578
120579(120578)
Figure 12 Temperature profiles for different values of Sc 119878 and Krfor Ha = 01 Gr = 01 Gc = 01 Bi = 01 Nc = 001 and Pr = 072
is more pronounced on temperature profiles for high Prandtlnumber fluids
33 Concentration Profile Figures 14ndash19 show the concen-tration profiles obtained by the numerical simulations forvarious values of nondimensional parameters Bi Ha GrGc Kr Pr Nc Sc and 119878 In Figure 14 the effect of anapplied magnetic field is found to increase the concentrationboundary layer However it is interesting to note that theconcentration profiles decrease with the increase of both heatsource and chemical reaction parameters (Figures 14 and 15)Figures 16 and 17 reveal the concentration variations with Grand Gc respectively It is due to the fact that an increase in thevalues of Grashof number and modified Grashof number hasthe tendency to increase the mass buoyancy effect This gives
Figure 14 Concentration profiles for different values of Ha 119878 andKr for Gr = 01 Gc = 01 Sc = 062 Bi = 01 Nc = 001 andPr = 072
rise to an increase in the induced flow and thereby decreasesconcentration Figure 18 depicts the effect of Schmidt numberon the concentration Like temperature the concentrationvalue is higher at the surface and falls exponentiallyThe con-centration decreases with an increase in Sc and the decrease ismore with the increase in concentration parameter Figure 19displays the effect of Pr on concentration profile against 120578withthe variation of source and chemical reaction parametersThe magnitude of concentration is higher at the plate andthen decays to zero asymptotically this is due to the fact thatthermal conductivity of fluid decreases with the increase of Pr
Figure 16 Concentration profiles for different values of Gc 119878 andKr for Ha = 01 Gr = 01 Sc = 062 Bi = 01 Nc = 001 andPr = 072
resulting a decrease in thermal boundary layer thickness andsource term further influences the decrease of concentration
4 Conclusions
The present numerical study has been carried out for heatand mass transfer of MHD flow over a moving vertical platein presence of heat source and chemical reaction along withconvective surface boundary conditionThe shootingmethodwith Runge-Kutta fourth-order iteration scheme has beenimplemented to solve the dimensionless velocity thermaland mass boundary layer equations It has been shownthat the magnitude of local skin friction and local Nusseltnumber increase whereas the plate surface temperature and
Figure 18 Concentration profiles for different values of Sc 119878 andKr for Ha = 01 Gr = 01 Gc = 01 Bi = 01 Nc = 001 andPr = 072
Sherwood number decreases with an increase in sourceparameter The increase in the strength of chemical reactingsubstances causes an increase in the magnitude of localskin friction the plate surface temperature and Sherwoodnumber but opposite behavior is seen for local Nusseltnumber The velocity profile decreases by increasing themagnetic parameter and even the increase is more prominentwith the increase in source and chemical reaction parameterThe thermal boundary layer thickness increases with theincrease of source chemical reaction parameter plate surfaceconvective heat parameter and Schmidt number while themass flux boundary layer thickness decreases Moreover thethermal boundary layer thickness the mass boundary layerand velocity decrease as the Prandtl number increases
[1] K Vajrevelu and J Nayfeh ldquoHydromagnetic convection at acone and a wedgerdquo International Communications in Heat andMass Transfer vol 19 pp 701ndash710 1992
[2] A J Chamkha ldquoHydromagnetic three-dimensional free con-vection on a vertical stretching surface with heat generation orabsorptionrdquo International Journal of Heat and Fluid Flow vol20 no 1 pp 84ndash92 1999
[3] D Moalem ldquoSteady state heat transfer within porous mediumwith temperature dependent heat generationrdquo InternationalJournal of Heat and Mass Transfer vol 19 no 5 pp 529ndash5371976
[4] M M Rahman and M A Sattar ldquoMagnetohydrodynamicconvective flow of a micropolar fluid past a continuouslymoving vertical porous plate in the presence of heat genera-tionabsorptionrdquo Journal of Heat Transfer vol 128 no 2 pp142ndash152 2006
[5] U N Das R Deka and V M Soundalgekar ldquoEffects of masstransfer on flowpast an impulsively started infinite vertical platewith constant heat flux and chemical reactionrdquo Forschung imIngenieurwesenEngineering Research vol 60 no 10 pp 284ndash287 1994
[6] S P Anjalidevi andRKandasamy ldquoEffects of chemical reactionheat and mass transfer on laminar flow along a semi infinitehorizontal platerdquoHeat andMass Transfer vol 35 no 6 pp 465ndash467 1999
[7] M A Seddeek A A Darwish andM S Abdelmeguid ldquoEffectsof chemical reaction and variable viscosity on hydromagneticmixed convection heat and mass transfer for Hiemenz flowthrough porous media with radiationrdquo Communications inNonlinear Science and Numerical Simulation vol 12 no 2 pp195ndash213 2007
[8] P M Patil and P S Kulkarni ldquoEffects of chemical reaction onfree convective flowof a polar fluid through a porousmedium inthe presence of internal heat generationrdquo International Journalof Thermal Sciences vol 47 no 8 pp 1043ndash1054 2008
[9] A M Salem and M Abd El-Aziz ldquoEffect of Hall currents andchemical reaction on hydromagnetic flow of a stretching ver-tical surface with internal heat generationabsorptionrdquo AppliedMathematical Modelling vol 32 no 7 pp 1236ndash1254 2008
[10] F S IbrahimAM Elaiw andAA Bakr ldquoEffect of the chemicalreaction and radiation absorption on the unsteady MHDfree convection flow past a semi infinite vertical permeablemoving plate with heat source and suctionrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 6 pp1056ndash1066 2008
[11] S K Parida M Acharya G C Dash and S Panda ldquoMHD heatand mass transfer in a rotating system with periodic suctionrdquoArabian Journal for Science and Engineering vol 36 no 6 pp1139ndash1151 2011
[12] R Rajeswari B Jothiram andV K Nelson ldquoChemical reactionheat and mass transfer on nonlinear MHD boundary layer flowthrough a vertical porous surface in the presence of suctionrdquoAppliedMathematical Sciences vol 3 no 49-52 pp 2469ndash24802009
[13] A Mahdy ldquoEffect of chemical reaction and heat generationor absorption on double-diffusive convection from a verticaltruncated cone in porous media with variable viscosityrdquo Inter-national Communications in Heat andMass Transfer vol 37 no5 pp 548ndash554 2010
[14] D Pal and B Talukdar ldquoPerturbation analysis of unsteadymagnetohydrodynamic convective heat and mass transfer in aboundary layer slip flow past a vertical permeable plate withthermal radiation and chemical reactionrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 7 pp1813ndash1830 2010
[15] O D Makinde and A Ogulu ldquoThe effect of thermal radiationon the heat and mass transfer flow of a variable viscosity fluidpast a vertical porous plate permeated by a transverse magneticfieldrdquo Chemical Engineering Communications vol 195 no 12pp 1575ndash1584 2008
[16] O D Makinde ldquoMHD mixed-convection interaction withthermal radiation and nth order chemical reaction past avertical porous plate embedded in a porous mediumrdquo ChemicalEngineering Communications vol 198 no 4 pp 590ndash608 2011
[17] 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
[18] P O Olanrewaju O T Arulogun and K Adebimpe ldquoInternalheat generation effect on thermal boundary layer with a con-vective surface boundary conditionrdquo American Journal of FluidDynamics vol 2 no 1 pp 1ndash4 2012
[19] ODMakinde ldquoOnMHDheat andmass transfer over amovingvertical plate with a convective surface boundary conditionrdquoCanadian Journal of Chemical Engineering vol 88 no 6 pp983ndash990 2010
[20] O D Makinde ldquoSimilarity solution of hydromagnetic heat andmass transfer over a vertical plate with a convective surfaceboundary conditionrdquo International Journal of Physical Sciencesvol 5 no 6 pp 700ndash710 2010
[21] O D Makinde ldquoSimilarity solution for natural convectionfrom a moving vertical plate with internal heat generation anda convective boundary conditionrdquo Thermal Science vol 15supplement 1 pp S137ndashS143 2011
[22] K Gangadhar N B Reddy and P K Kameswaran ldquoSimilaritysolution of hydromagnetic heat andmass transfer over a verticalplate with convective surface boundary condition and chemicalreactionrdquo International Journal of Nonlinear Science vol 3 no3 pp 298ndash307 2012
[23] NFM Noor S Abbabandy and I Hasim ldquoHeat and MassTransfer of thermophoretic MHD flow over an inclined radiateisothermal permeable surface in presence of heat source sinkrdquoInternational Journal of Heat and Mass Transfer vol 55 no 7pp 2122ndash2128 2012
[24] V Bisht M Kumar and Z Uddin ldquoEffect of variable thermalconductivity and chemical reaction on steadymixed convectionboundary layer flow with heat and mass transfer inside a conedue to a pointrdquo Journal of Applied Fluid Mechanics vol 4 no 4pp 59ndash63 2011
[25] A A Bakr ldquoEffects of chemical reaction on MHD free convec-tion andmass transfer flowof amicropolar fluidwith oscillatoryplate velocity and constant heat source in a rotating frame ofreferencerdquoCommunications inNonlinear Science andNumericalSimulation vol 16 no 2 pp 698ndash710 2011
areas of chemical science The effect of chemical reactionon different geometry of the problem has been investigatedby many authors Das et al [5] have studied the effect ofmass transfer flow past an impulsively started infinite verticalplate with heat flux and chemical reaction The chemicalreaction effect on heat and mass transfer flow along a semi-infinite horizontal plate has been studied by Anjalidevi andKandaswamy [6] and later it was extended for Hiemenzflow by Seddeek et al [7] and for polar fluid by Patil andKulkarni [8] Salem and Abd El-Aziz [9] have reported theeffect of hall currents and chemical reaction on hydromag-netic flow of a stretching vertical surface with internal heatgeneration or absorption Ibrahim et al [10] studied theeffect of chemical reaction and radiation absorption on theunsteady MHD free convection flow past a semi-infinitevertical permeablemoving platewith heat source and suctionA detailed numerical study has been carried out for unsteadyhydromagnetic natural convection heat and mass transferwith chemical reaction over a vertical plate in rotating systemwith periodic suction by Parida et al [11] Rajeswari etal [12] have investigated chemical reaction heat and masstransfer on nonlinear MHD boundary layer flow through avertical porous surface in presence of suctionMahdy [13] hasstudied the effect of chemical reaction and heat generationor absorption on double diffussive convection from verticaltruncated cone in a porous media with variable viscosityPal and Talukdar [14] have studied perturbation analysisof unsteady magnetohydrodynamic convective heat masstransfer in boundary layer slip flow past a vertical permeableplate with a thermal radiation and chemical reaction Furtherthe effect of thermal radiation heat and mass transfer flowof a variable viscosity fluid past a vertical porous plate inpresence of transverse magnetic field was investigated byMakinde and Ogulu [15] The analysis of MHD mixed-convection interaction with thermal radiation and higherorder chemical reaction is carried out by Makinde [16] Aziz[17] theoretically examined a similarity solution for a laminarthermal boundary layer over a flat plate with a convectivesurface boundary condition He found an interesting resultthat a similarity solution is possible if the convective heattransfer along with the hot fluid on the lower surface of theplate is inversely proportional to the square root of the axialdistance Recently the combined effects of an exponentiallydecaying internal heat generation and a convective boundarycondition on the thermal boundary layer over a flat plate areinvestigated by Olanrewaju et al [18] In their study authorshave negelected the Sherwood effect Similar analysis hasbeen carried out by Makinde [19 20] without heat sourceandwith heat source [21] neglecting chemical reaction effectThere has been considerable interest in studying the effectof chemical reaction [22] and heat source effect on theboundary layer flow problem with heat and mass transfer ofan electrically conducting fluid in different geometry [23ndash25]
Heat source and chemical reaction effects are crucial incontrolling the heat and mass transfer The present paperattempts to investigate the influence of chemical reactionand the combined effects of internal heat generation and theconvective boundary condition on the MHD heat and masstransfer flow To the best of the authorsrsquo knowledge so far no
one has considered the combined effect of chemical reactionand heat source along with convective surface boundarycondition on MHD flow and the heat and mass transfer overa moving vertical plate This fact motivated us to propose thesimilar study We extend the recent work of Makinde [19]Olanrewaju et al [18] and Gangadhar et al [22] to expose theeffect of chemical reaction on MHD heat and mass transferover a moving vertical plate in presence of heat source alongwith convective surface boundary condition The couplednonlinear partial differential equations governing the flowheat and mass transfer have been reduced to a set of couplednonlinear ordinary differential equations by using similaritytransformation Following [19] the similarity solutions existif the convective heat transfer associated with the hot fluidon lower surface of the plate is proportional to the inversesquare root of the axial distance The reduced equations aresolved numerically using Runge-Kutta fourth-order integra-tion scheme together with shooting method The effect ofvarious physical parameters on the velocity temperature andconcentration fields is studied
2 Mathematical Formulation
A typical flow scenario is illustrated in Figure 1 it showsa steady two-dimensional boundary layer flow of a streamof cold incompressible electrically conducting fluid over amoving vertical flat plate at temperature 119879
infinin presence of
heat source and chemical reactionThe left surface of the plateis being heated by convection from a hot fluid at temperature119879119891that gives a heat transfer coefficient ℎ
119891 and 119879
infinis the
temperature of the fluid away from the plate The cold fluidin contact with the upper surface of the plate generates heatinternally at the volumetric rate 119876
0 Here the 119909-axis is taken
along the direction of plate and 119910-axis is normal to it Amagnetic field of uniform field strength 119861
0is applied in the
negative direction of 119910-axisThe continuity momentum energy and concentration
equations describing the flow under the Boussinesq approxi-mation can be written as
120597119906
120597119909+120597V
120597119910= 0 (1)
119906120597119906
120597119909+ V
120597119906
120597119910= ]
1205972
119906
120597 1199102minus120590 1198612
0
120588119906 + 119892120573 (119879 minus 119879
infin)
+ 119892120573lowast
(119862 minus 119862infin)
(2)
119906120597119879
120597119909+ V
120597119879
120597119910= 120572
1205972
119879
1205971199102+
1198760
120588119862119901
(119879 minus 119879infin) (3)
119906120597119862
120597119909+ V
120597119862
120597119910= 119863
1205972
119862
1205971199102minus Kr1015840119862 (4)
The symbols 119906 and V denote the fluid velocity in the 119909-and 119910-direction Here 119879 and 119862 are the temperature andconcentration variables ] is the kinematic viscosity 120572 is thethermal diffusivity 119863 is the mass diffusivity 120573 is the thermalexpansion coefficient 120573lowast is the solutal expansion coefficient
International Journal of Chemical Engineering 3
119906 = 1198800
120592 = 0
119879119891
119862119908
119879
119862
119906
1198760
119892
1198610
119910
119862infin 119879infin
119909
Figure 1 Sketch of flow geometry
120588 is the fluid density 119892 is the gravitational acceleration 120590 isthe electrical conductivity 119876
0is the heat source 119862
119901is the
specific heat at constant pressure and Kr1015840 is the chemicalreaction rate on the species concentration In the aboveequations several assumptions have been made First theplate is nonconducting and the effects of radiant heatingviscous dissipation Hall effects and induced fields areneglected Second the physical properties that is viscosityheat capacity thermal diffusivity and the mass diffusivity ofthe fluid remain invariant throughout the fluid
The appropriate boundary conditions at the plate surfaceand far into the cold fluid are
where 119862119908is the species concentration at the plate surface 119860
is the constant 120582 is the power index of the concentration 1198800
is the plate velocity 119896 is the thermal conductivity coefficientand 119862
infinis the concentration of the fluid away from the
plate The boundary layer equations presented are nonlinearpartial differential equations and are in general difficult tosolve However the equations admit of a self-similar solutionTherefore transformation allows them to be reduced to asystem of ordinary differential equations that are relativelyeasy to solve numerically We look for solution compatiblewith (1) of the form
119906 = 11988001198911015840
(120578) V = minus1
2
radic]1198800
119909119891 (120578) +
1198800119910
2 1199091198911015840
(120578) (6)
where 120578 = 119910radic1198800(]119909) and prime denotes the differentiation
with respect to 120578
1
08
06
04
02
0
0 5 10 15120578
119891998400
(120578)
Ha = 01 119878 = 0 Kr = 0
Ha = 03 119878 = 0 Kr = 0
Ha = 03 119878 = 002 Kr = 01
Ha = 03 119878 = 002 Kr = 05
Figure 2 Velocity profiles for different values of Ha 119878 and Kr forGr = 01 Gc = 01 Sc = 062 Bi = 01 Nc = 001 and Pr = 072
Let us introduce the dimensionless quantities that is
Here Ha119909is the local magnetic field parameter Gr
119909is the
local thermal Grashof number Gc119909is the modified Grashof
number Bi119909is the local convective heat transfer parameter
Pr is the Prandtl number Sc is the Schmidt number 119878119909is the
local heat source parameter Kr119909is the local chemical reaction
parameter and Nc is the concentration difference parameterUsing (6) and (7) in (2)ndash(4) we get the following equations
119891101584010158401015840
+1
211989111989110158401015840
minusHa1199091198911015840
+ Gr119909120579 + Gc
119909120601 = 0 (8)
12057910158401015840
+1
2Pr119891 120579
1015840
+ Pr119878119909120579 = 0 (9)
12060110158401015840
+1
2Sc1198911206011015840 minus ScKr
119909(120601 +Nc) = 0 (10)
4 International Journal of Chemical Engineering
The corresponding boundary conditions equation (5) forvelocity temperature and concentration fields in terms ofnondimensional variables are
119891 (0) = 0 1198911015840
(0) = 1
1205791015840
(0) = Bi119909[120579 (0) minus 1] 120601 (0) = 1
1198911015840
(infin) = 0 120579 (infin) = 0 120601 (infin) = 0
(11)
It is observed that in the absence of local source parameterand chemical reaction parameter that is for 119878
119909= 0 and
Kr119909= 0 (8) (9) and (10) together with boundary condition
(11) are the same as those obtained by Makinde [19] It isnoticed that the concentration equation (10) in presence ofthe chemical reaction parameter (Kr
119909) in the fluid yields
nonhomogeneous differential equationwhich is coupledwithmomentum equation (8) and in general difficult to solveanalytically In order to overcome this difficulty we solvethese equations numerically by fourth-order Runge-Kuttamethod in association with shooting technique Firstly theseequations together with associated boundary conditions arereduced to first-order differential equations Since equationsto be solved are the third order for the velocity and secondorder for the temperature and concentration the values of1198911015840 1205791015840 and 120601
1015840 are needed at 120578 = 0 Therefore the shootingmethod is used to solve this boundary value problem Thelocal skin friction coefficient the local Nusselt number thelocal Sherwood number and the plate surface temperatureare computed in terms of 11989110158401015840(0) minus1205791015840(0) minus1206011015840(0) and 120579(0)respectively It can be noted that the local parameters Ha
119909
Gr119909 Gc119909 Bi119909 119878119909 and Kr
119909in (8)ndash(10) are functions of 119909
and generate local similarity solution In order to have a truesimilarity solution we assume the following relation [19]
ℎ119891=
119886
radic119909 120590 =
119887
119909 120573 =
119888
119909
120573lowast
=119889
119909 119876
0=119890
119909 Kr1015840 = 119898
119909
(12)
where 119886 119887 119888 119889 119890 and 119898 are the constants with appropriatedimensions In view of relation (12) the parameters Ha
119909
Gr119909 Gc119909 Bi119909 119878119909 and Kr
119909are now independent of 119909 and
henceforth we drop the index ldquo119909rdquo for simplicity
3 Result Discussion
The numerical solutions of the boundary value problemfor system of ordinary differential equations were obtainedby Runge-Kutta method along with shooting techniqueSince the physical domain of the underlying problem isunbounded the computational domain is chosen sufficientlylarge in order to meet the far field boundary condition atinfinity Here the transverse distance is fixed to 10 and suitablymore than 10 depending upon the choice of the parametersTo demonstrate successful implementation of the numericalscheme the numerical results are compared to those obtainedby a previous published paper (see [19]) for the local skinfriction coefficient plate surface temperature and the local
1
08
06
04
02
00 2 4 6 8 10
Bi = 01 119878 = 0 Kr = 0
Bi = 10 119878 = 0 Kr = 0
Bi = 10 119878 = 01 Kr = 005
Bi = 10 119878 = 01 Kr = 05
119891998400
(120578)
120578
Figure 3 Velocity profiles for different values of Bi 119878 and Kr forHa = 01 Gr = 01 Gc = 01 Sc = 062 Nc = 001 and Pr = 072
Figure 4 Velocity profiles for different values of Gc 119878 and Kr forHa = 01 Gr = 01 Sc = 062 Bi = 01 Nc = 001 and Pr = 072
Sherwood and Nusselt numbers in Table 1 for the parametersembedded in absence of local chemical reaction parameter(Kr) and local heat source parameter (119878) Table 1 presents acomparison of 11989110158401015840(0) minus1205791015840(0) 120579(0) and minus120601
1015840
(0) between thepresent results and the results obtained by Makinde [19] forvarious values of Ha Gr Gc Bi Pr and Sc when 119878 = Kr =0 The results are found to be in excellent agreement It isimportant to note that the momentum equation is coupledwith heat and mass transfer equations and hence the Prandtlnumber Schmidt number chemical reaction parameter andsource term have an influence on skin friction in our present
International Journal of Chemical Engineering 5
Table 1 Comparison of 11989110158401015840(0) minus1205791015840(0) 120579(0) and minus1206011015840(0) for various values of Bi Ha Gr Gc Pr and Sc when Kr = 119878 = 0
Bi Gr Gc Ha Pr Sc Makinde [19] Present study11989110158401015840
Figure 5 Velocity profiles for different values of Gr 119878 and Kr forHa = 01 Gc = 01 Sc = 062 Bi = 01 Nc = 001 and Pr = 072
problem The influence of heat source and chemical reactionparameters on local skin friction local Nusselt number platesurface temperature and Sherwood number are highlightedin Tables 2 and 3 for various nondimensional flow param-eters It is clearly seen from Table 2 that the magnitude ofskin friction and Nusselt number increase whereas the platesurface temperature and Sherwood number decrease withthe increase of source parameter Furthermore increasingthe strength of chemical reacting substances is to increasethe local skin friction the plate surface temperature andSherwood number but opposite behavior is seen for localNusselt numberThe obvious observation fromTable 2 is thatthe fluid low Prandtl number increases the magnitude oflocal skin friction and local Nusselt number while decreasingthe plate surface temperature and local Sherwood numberAgain the fluid with low convective resistance (or externalresistance) decreases the magnitude of local skin frictionwhile the Nusselt number the plate surface temperatureand local Sherwood number increase It is also observed
Sc = 062 119878 = 0 Kr = 0
Sc = 262 119878 = 0 Kr = 0
Sc = 062 119878 = 005 Kr = 005
Sc = 062 119878 = 005 Kr = 05
1
08
06
04
02
00 2 4 6 8 10
119891998400 (120578)
120578
Figure 6 Velocity profiles for different values of Sc 119878 and Kr forHa = 01 Gr = 01 Gc = 01 Bi = 01 Nc = 001 and Pr = 072
Table 2 Computation of skin friction (11989110158401015840(0)) Nusselt number(minus1205791015840(0)) plate surface temperature (120579(0)) and Sherwood number(minus1206011015840(0)) for different values of Bi Pr 119878 andKrTheother parametersare Ha = 01 Gr = 01 Gc = 01 Sc = 062 and Nc = 001
from Table 3 that the increase of Schimdt number resultsin increase of the magnitude of the local skin friction butopposite behavior is marked in case of Nusselt number andthe plate surface temperature
6 International Journal of Chemical Engineering
Table 3 Computation of skin friction (11989110158401015840(0)) Nusselt number(minus1205791015840(0)) plate surface temperature (120579(0)) and Sherwood number(minus1206011015840(0)) for different values of Sc 119878 and Kr The other parametersare fixed at Bi = Gc = Gr = 01 Pr = 072 and Nc = 001
31 Velocity Profiles Figures 2ndash7 exhibit the velocity pro-files obtained by the numerical simulations for variousflow parameters involved in the problem The simulatedparameters are reported in the figure caption The effectsof magnetic parameter on the velocity field in presenceand absence of source and chemical reaction parameterare shown in Figure 2 It illustrates that the velocity profiledecreases with the increase of magnetic parameter in absenceof source and chemical reaction parameter because Lorentzforce acts against the flow if the magnetic field is applied inthe normal direction In presence of source and chemicalreaction parameter no such appreciable change is observedin Figure 2This corresponds to Figure (2) in [19] and therebyagain validating our numerical scheme A little increase in thevelocity profile near the boundary layer is marked in Figure 3with the increase in the convective heat parameter becausethe fluid adjacent to the right surface of the plate becomeslighter by hot fluid and rises faster The boundary layer flowsdevelop adjacent to vertical surface and velocity reaches amaximum in the boundary layer It is evident from Figures4 and 5 that greater cooling of surface an increase in Gc andincrease in Gr result in an increase in the velocity It is dueto the fact that the increase in the values of Grashof numberand modified Grashof number has the tendency to increasethe thermal and mass buoyancy effect The increase is alsoevident due to the presence of source and chemical reactionparameters Furthermore the velocity increases rapidly andsuddenly falls near the boundary and then approaches thefar field boundary condition due to favorable buoyancy forcewith the increase of both Gr and Gc It can be seen that theincrease in the Prandtl number and Schmidt number leads toa fall in the velocity as shown in Figures 6 and 7
32 Temperature Profiles Figures 8ndash13 show the temperatureprofiles obtained by the numerical simulations for variousvalues of flow parameters Figure 8 clearly demonstrates thatthe temperature profiles increase with the increase of themagnetic field parameter which implies that the appliedmagnetic field tends to heat the fluid and thus reducesthe heat transfer from the wall Further it can be seenthat temperature profile increases due to increase of heatsource as well as chemical reaction parameter The thermalboundary layer thickness increases with an increase in the
Figure 8 Temperature profiles for different values of Ha 119878 and Krfor Gr = 01 Gc = 01 Sc = 062 Bi = 01 Nc = 001 and Pr = 072
plate surface convective heat parameter (Figure 9) and asimilar effect is also observed in Figure 12 with the increaseof Schmidt number It can be observed that the amplitudeof fluid temperature in presence of heat source and chemicalreacting substances is more in comparison to in absence ofthese parameters The steady state temperatures for differentGrashof number modified Grashof number internal heatsource and chemical reaction parameters are shown inFigures 10 and 11 The thermal boundary layer decreases withincreasing Grashof number and modified Grashof numberbut reverse effect is observed with the presence of chemicalreaction parameter This is also revealed in Figure 13 which
Figure 10 Temperature profiles for different values of Gc 119878 and Krfor Ha = 01 Gr = 01 Sc = 062 Bi = 01 Nc = 001 and Pr = 072
shows that thermal boundary layer thickness decreases asthe Prandtl number increases implying higher heat transferIt is due to that smaller values of Pr means increasing thethermal conductivity and therefore heat is able to diffuseaway from the plate more quickly than higher values of Prhence the rate of heat transfer is reduced It is noted thatowing to the presence of heat source effect (119878 gt 0) andchemical reaction parameter (Kr gt 0) the thermal stateof the fluid increases Hence the temperature of the fluidincreases within the boundary layers In the event that thestrength of the heat source and chemical reaction parametersare relatively large a remarkable change is observed in thetemperature profiles within the thermal boundary layer ascan be seen in Figure 13 Further the effect of heat generation
Figure 11 Temperature profiles for different values of Gr 119878 and KrforHa = 01 Gc = 01 119878c = 062 Bi = 01 Nc = 001 and Pr = 072
Sc = 062 119878 = 0 Kr = 0
Sc = 262 119878 = 0 Kr = 0
Sc = 062 119878 = 005 Kr = 005
Sc = 062 119878 = 005 Kr = 05
035
03
025
02
015
005
01
00 2 4 6 8 10
120578
120579(120578)
Figure 12 Temperature profiles for different values of Sc 119878 and Krfor Ha = 01 Gr = 01 Gc = 01 Bi = 01 Nc = 001 and Pr = 072
is more pronounced on temperature profiles for high Prandtlnumber fluids
33 Concentration Profile Figures 14ndash19 show the concen-tration profiles obtained by the numerical simulations forvarious values of nondimensional parameters Bi Ha GrGc Kr Pr Nc Sc and 119878 In Figure 14 the effect of anapplied magnetic field is found to increase the concentrationboundary layer However it is interesting to note that theconcentration profiles decrease with the increase of both heatsource and chemical reaction parameters (Figures 14 and 15)Figures 16 and 17 reveal the concentration variations with Grand Gc respectively It is due to the fact that an increase in thevalues of Grashof number and modified Grashof number hasthe tendency to increase the mass buoyancy effect This gives
Figure 14 Concentration profiles for different values of Ha 119878 andKr for Gr = 01 Gc = 01 Sc = 062 Bi = 01 Nc = 001 andPr = 072
rise to an increase in the induced flow and thereby decreasesconcentration Figure 18 depicts the effect of Schmidt numberon the concentration Like temperature the concentrationvalue is higher at the surface and falls exponentiallyThe con-centration decreases with an increase in Sc and the decrease ismore with the increase in concentration parameter Figure 19displays the effect of Pr on concentration profile against 120578withthe variation of source and chemical reaction parametersThe magnitude of concentration is higher at the plate andthen decays to zero asymptotically this is due to the fact thatthermal conductivity of fluid decreases with the increase of Pr
Figure 16 Concentration profiles for different values of Gc 119878 andKr for Ha = 01 Gr = 01 Sc = 062 Bi = 01 Nc = 001 andPr = 072
resulting a decrease in thermal boundary layer thickness andsource term further influences the decrease of concentration
4 Conclusions
The present numerical study has been carried out for heatand mass transfer of MHD flow over a moving vertical platein presence of heat source and chemical reaction along withconvective surface boundary conditionThe shootingmethodwith Runge-Kutta fourth-order iteration scheme has beenimplemented to solve the dimensionless velocity thermaland mass boundary layer equations It has been shownthat the magnitude of local skin friction and local Nusseltnumber increase whereas the plate surface temperature and
Figure 18 Concentration profiles for different values of Sc 119878 andKr for Ha = 01 Gr = 01 Gc = 01 Bi = 01 Nc = 001 andPr = 072
Sherwood number decreases with an increase in sourceparameter The increase in the strength of chemical reactingsubstances causes an increase in the magnitude of localskin friction the plate surface temperature and Sherwoodnumber but opposite behavior is seen for local Nusseltnumber The velocity profile decreases by increasing themagnetic parameter and even the increase is more prominentwith the increase in source and chemical reaction parameterThe thermal boundary layer thickness increases with theincrease of source chemical reaction parameter plate surfaceconvective heat parameter and Schmidt number while themass flux boundary layer thickness decreases Moreover thethermal boundary layer thickness the mass boundary layerand velocity decrease as the Prandtl number increases
[1] K Vajrevelu and J Nayfeh ldquoHydromagnetic convection at acone and a wedgerdquo International Communications in Heat andMass Transfer vol 19 pp 701ndash710 1992
[2] A J Chamkha ldquoHydromagnetic three-dimensional free con-vection on a vertical stretching surface with heat generation orabsorptionrdquo International Journal of Heat and Fluid Flow vol20 no 1 pp 84ndash92 1999
[3] D Moalem ldquoSteady state heat transfer within porous mediumwith temperature dependent heat generationrdquo InternationalJournal of Heat and Mass Transfer vol 19 no 5 pp 529ndash5371976
[4] M M Rahman and M A Sattar ldquoMagnetohydrodynamicconvective flow of a micropolar fluid past a continuouslymoving vertical porous plate in the presence of heat genera-tionabsorptionrdquo Journal of Heat Transfer vol 128 no 2 pp142ndash152 2006
[5] U N Das R Deka and V M Soundalgekar ldquoEffects of masstransfer on flowpast an impulsively started infinite vertical platewith constant heat flux and chemical reactionrdquo Forschung imIngenieurwesenEngineering Research vol 60 no 10 pp 284ndash287 1994
[6] S P Anjalidevi andRKandasamy ldquoEffects of chemical reactionheat and mass transfer on laminar flow along a semi infinitehorizontal platerdquoHeat andMass Transfer vol 35 no 6 pp 465ndash467 1999
[7] M A Seddeek A A Darwish andM S Abdelmeguid ldquoEffectsof chemical reaction and variable viscosity on hydromagneticmixed convection heat and mass transfer for Hiemenz flowthrough porous media with radiationrdquo Communications inNonlinear Science and Numerical Simulation vol 12 no 2 pp195ndash213 2007
[8] P M Patil and P S Kulkarni ldquoEffects of chemical reaction onfree convective flowof a polar fluid through a porousmedium inthe presence of internal heat generationrdquo International Journalof Thermal Sciences vol 47 no 8 pp 1043ndash1054 2008
[9] A M Salem and M Abd El-Aziz ldquoEffect of Hall currents andchemical reaction on hydromagnetic flow of a stretching ver-tical surface with internal heat generationabsorptionrdquo AppliedMathematical Modelling vol 32 no 7 pp 1236ndash1254 2008
[10] F S IbrahimAM Elaiw andAA Bakr ldquoEffect of the chemicalreaction and radiation absorption on the unsteady MHDfree convection flow past a semi infinite vertical permeablemoving plate with heat source and suctionrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 6 pp1056ndash1066 2008
[11] S K Parida M Acharya G C Dash and S Panda ldquoMHD heatand mass transfer in a rotating system with periodic suctionrdquoArabian Journal for Science and Engineering vol 36 no 6 pp1139ndash1151 2011
[12] R Rajeswari B Jothiram andV K Nelson ldquoChemical reactionheat and mass transfer on nonlinear MHD boundary layer flowthrough a vertical porous surface in the presence of suctionrdquoAppliedMathematical Sciences vol 3 no 49-52 pp 2469ndash24802009
[13] A Mahdy ldquoEffect of chemical reaction and heat generationor absorption on double-diffusive convection from a verticaltruncated cone in porous media with variable viscosityrdquo Inter-national Communications in Heat andMass Transfer vol 37 no5 pp 548ndash554 2010
[14] D Pal and B Talukdar ldquoPerturbation analysis of unsteadymagnetohydrodynamic convective heat and mass transfer in aboundary layer slip flow past a vertical permeable plate withthermal radiation and chemical reactionrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 7 pp1813ndash1830 2010
[15] O D Makinde and A Ogulu ldquoThe effect of thermal radiationon the heat and mass transfer flow of a variable viscosity fluidpast a vertical porous plate permeated by a transverse magneticfieldrdquo Chemical Engineering Communications vol 195 no 12pp 1575ndash1584 2008
[16] O D Makinde ldquoMHD mixed-convection interaction withthermal radiation and nth order chemical reaction past avertical porous plate embedded in a porous mediumrdquo ChemicalEngineering Communications vol 198 no 4 pp 590ndash608 2011
[17] 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
[18] P O Olanrewaju O T Arulogun and K Adebimpe ldquoInternalheat generation effect on thermal boundary layer with a con-vective surface boundary conditionrdquo American Journal of FluidDynamics vol 2 no 1 pp 1ndash4 2012
[19] ODMakinde ldquoOnMHDheat andmass transfer over amovingvertical plate with a convective surface boundary conditionrdquoCanadian Journal of Chemical Engineering vol 88 no 6 pp983ndash990 2010
[20] O D Makinde ldquoSimilarity solution of hydromagnetic heat andmass transfer over a vertical plate with a convective surfaceboundary conditionrdquo International Journal of Physical Sciencesvol 5 no 6 pp 700ndash710 2010
[21] O D Makinde ldquoSimilarity solution for natural convectionfrom a moving vertical plate with internal heat generation anda convective boundary conditionrdquo Thermal Science vol 15supplement 1 pp S137ndashS143 2011
[22] K Gangadhar N B Reddy and P K Kameswaran ldquoSimilaritysolution of hydromagnetic heat andmass transfer over a verticalplate with convective surface boundary condition and chemicalreactionrdquo International Journal of Nonlinear Science vol 3 no3 pp 298ndash307 2012
[23] NFM Noor S Abbabandy and I Hasim ldquoHeat and MassTransfer of thermophoretic MHD flow over an inclined radiateisothermal permeable surface in presence of heat source sinkrdquoInternational Journal of Heat and Mass Transfer vol 55 no 7pp 2122ndash2128 2012
[24] V Bisht M Kumar and Z Uddin ldquoEffect of variable thermalconductivity and chemical reaction on steadymixed convectionboundary layer flow with heat and mass transfer inside a conedue to a pointrdquo Journal of Applied Fluid Mechanics vol 4 no 4pp 59ndash63 2011
[25] A A Bakr ldquoEffects of chemical reaction on MHD free convec-tion andmass transfer flowof amicropolar fluidwith oscillatoryplate velocity and constant heat source in a rotating frame ofreferencerdquoCommunications inNonlinear Science andNumericalSimulation vol 16 no 2 pp 698ndash710 2011
120588 is the fluid density 119892 is the gravitational acceleration 120590 isthe electrical conductivity 119876
0is the heat source 119862
119901is the
specific heat at constant pressure and Kr1015840 is the chemicalreaction rate on the species concentration In the aboveequations several assumptions have been made First theplate is nonconducting and the effects of radiant heatingviscous dissipation Hall effects and induced fields areneglected Second the physical properties that is viscosityheat capacity thermal diffusivity and the mass diffusivity ofthe fluid remain invariant throughout the fluid
The appropriate boundary conditions at the plate surfaceand far into the cold fluid are
where 119862119908is the species concentration at the plate surface 119860
is the constant 120582 is the power index of the concentration 1198800
is the plate velocity 119896 is the thermal conductivity coefficientand 119862
infinis the concentration of the fluid away from the
plate The boundary layer equations presented are nonlinearpartial differential equations and are in general difficult tosolve However the equations admit of a self-similar solutionTherefore transformation allows them to be reduced to asystem of ordinary differential equations that are relativelyeasy to solve numerically We look for solution compatiblewith (1) of the form
119906 = 11988001198911015840
(120578) V = minus1
2
radic]1198800
119909119891 (120578) +
1198800119910
2 1199091198911015840
(120578) (6)
where 120578 = 119910radic1198800(]119909) and prime denotes the differentiation
with respect to 120578
1
08
06
04
02
0
0 5 10 15120578
119891998400
(120578)
Ha = 01 119878 = 0 Kr = 0
Ha = 03 119878 = 0 Kr = 0
Ha = 03 119878 = 002 Kr = 01
Ha = 03 119878 = 002 Kr = 05
Figure 2 Velocity profiles for different values of Ha 119878 and Kr forGr = 01 Gc = 01 Sc = 062 Bi = 01 Nc = 001 and Pr = 072
Let us introduce the dimensionless quantities that is
Here Ha119909is the local magnetic field parameter Gr
119909is the
local thermal Grashof number Gc119909is the modified Grashof
number Bi119909is the local convective heat transfer parameter
Pr is the Prandtl number Sc is the Schmidt number 119878119909is the
local heat source parameter Kr119909is the local chemical reaction
parameter and Nc is the concentration difference parameterUsing (6) and (7) in (2)ndash(4) we get the following equations
119891101584010158401015840
+1
211989111989110158401015840
minusHa1199091198911015840
+ Gr119909120579 + Gc
119909120601 = 0 (8)
12057910158401015840
+1
2Pr119891 120579
1015840
+ Pr119878119909120579 = 0 (9)
12060110158401015840
+1
2Sc1198911206011015840 minus ScKr
119909(120601 +Nc) = 0 (10)
4 International Journal of Chemical Engineering
The corresponding boundary conditions equation (5) forvelocity temperature and concentration fields in terms ofnondimensional variables are
119891 (0) = 0 1198911015840
(0) = 1
1205791015840
(0) = Bi119909[120579 (0) minus 1] 120601 (0) = 1
1198911015840
(infin) = 0 120579 (infin) = 0 120601 (infin) = 0
(11)
It is observed that in the absence of local source parameterand chemical reaction parameter that is for 119878
119909= 0 and
Kr119909= 0 (8) (9) and (10) together with boundary condition
(11) are the same as those obtained by Makinde [19] It isnoticed that the concentration equation (10) in presence ofthe chemical reaction parameter (Kr
119909) in the fluid yields
nonhomogeneous differential equationwhich is coupledwithmomentum equation (8) and in general difficult to solveanalytically In order to overcome this difficulty we solvethese equations numerically by fourth-order Runge-Kuttamethod in association with shooting technique Firstly theseequations together with associated boundary conditions arereduced to first-order differential equations Since equationsto be solved are the third order for the velocity and secondorder for the temperature and concentration the values of1198911015840 1205791015840 and 120601
1015840 are needed at 120578 = 0 Therefore the shootingmethod is used to solve this boundary value problem Thelocal skin friction coefficient the local Nusselt number thelocal Sherwood number and the plate surface temperatureare computed in terms of 11989110158401015840(0) minus1205791015840(0) minus1206011015840(0) and 120579(0)respectively It can be noted that the local parameters Ha
119909
Gr119909 Gc119909 Bi119909 119878119909 and Kr
119909in (8)ndash(10) are functions of 119909
and generate local similarity solution In order to have a truesimilarity solution we assume the following relation [19]
ℎ119891=
119886
radic119909 120590 =
119887
119909 120573 =
119888
119909
120573lowast
=119889
119909 119876
0=119890
119909 Kr1015840 = 119898
119909
(12)
where 119886 119887 119888 119889 119890 and 119898 are the constants with appropriatedimensions In view of relation (12) the parameters Ha
119909
Gr119909 Gc119909 Bi119909 119878119909 and Kr
119909are now independent of 119909 and
henceforth we drop the index ldquo119909rdquo for simplicity
3 Result Discussion
The numerical solutions of the boundary value problemfor system of ordinary differential equations were obtainedby Runge-Kutta method along with shooting techniqueSince the physical domain of the underlying problem isunbounded the computational domain is chosen sufficientlylarge in order to meet the far field boundary condition atinfinity Here the transverse distance is fixed to 10 and suitablymore than 10 depending upon the choice of the parametersTo demonstrate successful implementation of the numericalscheme the numerical results are compared to those obtainedby a previous published paper (see [19]) for the local skinfriction coefficient plate surface temperature and the local
1
08
06
04
02
00 2 4 6 8 10
Bi = 01 119878 = 0 Kr = 0
Bi = 10 119878 = 0 Kr = 0
Bi = 10 119878 = 01 Kr = 005
Bi = 10 119878 = 01 Kr = 05
119891998400
(120578)
120578
Figure 3 Velocity profiles for different values of Bi 119878 and Kr forHa = 01 Gr = 01 Gc = 01 Sc = 062 Nc = 001 and Pr = 072
Figure 4 Velocity profiles for different values of Gc 119878 and Kr forHa = 01 Gr = 01 Sc = 062 Bi = 01 Nc = 001 and Pr = 072
Sherwood and Nusselt numbers in Table 1 for the parametersembedded in absence of local chemical reaction parameter(Kr) and local heat source parameter (119878) Table 1 presents acomparison of 11989110158401015840(0) minus1205791015840(0) 120579(0) and minus120601
1015840
(0) between thepresent results and the results obtained by Makinde [19] forvarious values of Ha Gr Gc Bi Pr and Sc when 119878 = Kr =0 The results are found to be in excellent agreement It isimportant to note that the momentum equation is coupledwith heat and mass transfer equations and hence the Prandtlnumber Schmidt number chemical reaction parameter andsource term have an influence on skin friction in our present
International Journal of Chemical Engineering 5
Table 1 Comparison of 11989110158401015840(0) minus1205791015840(0) 120579(0) and minus1206011015840(0) for various values of Bi Ha Gr Gc Pr and Sc when Kr = 119878 = 0
Bi Gr Gc Ha Pr Sc Makinde [19] Present study11989110158401015840
Figure 5 Velocity profiles for different values of Gr 119878 and Kr forHa = 01 Gc = 01 Sc = 062 Bi = 01 Nc = 001 and Pr = 072
problem The influence of heat source and chemical reactionparameters on local skin friction local Nusselt number platesurface temperature and Sherwood number are highlightedin Tables 2 and 3 for various nondimensional flow param-eters It is clearly seen from Table 2 that the magnitude ofskin friction and Nusselt number increase whereas the platesurface temperature and Sherwood number decrease withthe increase of source parameter Furthermore increasingthe strength of chemical reacting substances is to increasethe local skin friction the plate surface temperature andSherwood number but opposite behavior is seen for localNusselt numberThe obvious observation fromTable 2 is thatthe fluid low Prandtl number increases the magnitude oflocal skin friction and local Nusselt number while decreasingthe plate surface temperature and local Sherwood numberAgain the fluid with low convective resistance (or externalresistance) decreases the magnitude of local skin frictionwhile the Nusselt number the plate surface temperatureand local Sherwood number increase It is also observed
Sc = 062 119878 = 0 Kr = 0
Sc = 262 119878 = 0 Kr = 0
Sc = 062 119878 = 005 Kr = 005
Sc = 062 119878 = 005 Kr = 05
1
08
06
04
02
00 2 4 6 8 10
119891998400 (120578)
120578
Figure 6 Velocity profiles for different values of Sc 119878 and Kr forHa = 01 Gr = 01 Gc = 01 Bi = 01 Nc = 001 and Pr = 072
Table 2 Computation of skin friction (11989110158401015840(0)) Nusselt number(minus1205791015840(0)) plate surface temperature (120579(0)) and Sherwood number(minus1206011015840(0)) for different values of Bi Pr 119878 andKrTheother parametersare Ha = 01 Gr = 01 Gc = 01 Sc = 062 and Nc = 001
from Table 3 that the increase of Schimdt number resultsin increase of the magnitude of the local skin friction butopposite behavior is marked in case of Nusselt number andthe plate surface temperature
6 International Journal of Chemical Engineering
Table 3 Computation of skin friction (11989110158401015840(0)) Nusselt number(minus1205791015840(0)) plate surface temperature (120579(0)) and Sherwood number(minus1206011015840(0)) for different values of Sc 119878 and Kr The other parametersare fixed at Bi = Gc = Gr = 01 Pr = 072 and Nc = 001
31 Velocity Profiles Figures 2ndash7 exhibit the velocity pro-files obtained by the numerical simulations for variousflow parameters involved in the problem The simulatedparameters are reported in the figure caption The effectsof magnetic parameter on the velocity field in presenceand absence of source and chemical reaction parameterare shown in Figure 2 It illustrates that the velocity profiledecreases with the increase of magnetic parameter in absenceof source and chemical reaction parameter because Lorentzforce acts against the flow if the magnetic field is applied inthe normal direction In presence of source and chemicalreaction parameter no such appreciable change is observedin Figure 2This corresponds to Figure (2) in [19] and therebyagain validating our numerical scheme A little increase in thevelocity profile near the boundary layer is marked in Figure 3with the increase in the convective heat parameter becausethe fluid adjacent to the right surface of the plate becomeslighter by hot fluid and rises faster The boundary layer flowsdevelop adjacent to vertical surface and velocity reaches amaximum in the boundary layer It is evident from Figures4 and 5 that greater cooling of surface an increase in Gc andincrease in Gr result in an increase in the velocity It is dueto the fact that the increase in the values of Grashof numberand modified Grashof number has the tendency to increasethe thermal and mass buoyancy effect The increase is alsoevident due to the presence of source and chemical reactionparameters Furthermore the velocity increases rapidly andsuddenly falls near the boundary and then approaches thefar field boundary condition due to favorable buoyancy forcewith the increase of both Gr and Gc It can be seen that theincrease in the Prandtl number and Schmidt number leads toa fall in the velocity as shown in Figures 6 and 7
32 Temperature Profiles Figures 8ndash13 show the temperatureprofiles obtained by the numerical simulations for variousvalues of flow parameters Figure 8 clearly demonstrates thatthe temperature profiles increase with the increase of themagnetic field parameter which implies that the appliedmagnetic field tends to heat the fluid and thus reducesthe heat transfer from the wall Further it can be seenthat temperature profile increases due to increase of heatsource as well as chemical reaction parameter The thermalboundary layer thickness increases with an increase in the
Figure 8 Temperature profiles for different values of Ha 119878 and Krfor Gr = 01 Gc = 01 Sc = 062 Bi = 01 Nc = 001 and Pr = 072
plate surface convective heat parameter (Figure 9) and asimilar effect is also observed in Figure 12 with the increaseof Schmidt number It can be observed that the amplitudeof fluid temperature in presence of heat source and chemicalreacting substances is more in comparison to in absence ofthese parameters The steady state temperatures for differentGrashof number modified Grashof number internal heatsource and chemical reaction parameters are shown inFigures 10 and 11 The thermal boundary layer decreases withincreasing Grashof number and modified Grashof numberbut reverse effect is observed with the presence of chemicalreaction parameter This is also revealed in Figure 13 which
Figure 10 Temperature profiles for different values of Gc 119878 and Krfor Ha = 01 Gr = 01 Sc = 062 Bi = 01 Nc = 001 and Pr = 072
shows that thermal boundary layer thickness decreases asthe Prandtl number increases implying higher heat transferIt is due to that smaller values of Pr means increasing thethermal conductivity and therefore heat is able to diffuseaway from the plate more quickly than higher values of Prhence the rate of heat transfer is reduced It is noted thatowing to the presence of heat source effect (119878 gt 0) andchemical reaction parameter (Kr gt 0) the thermal stateof the fluid increases Hence the temperature of the fluidincreases within the boundary layers In the event that thestrength of the heat source and chemical reaction parametersare relatively large a remarkable change is observed in thetemperature profiles within the thermal boundary layer ascan be seen in Figure 13 Further the effect of heat generation
Figure 11 Temperature profiles for different values of Gr 119878 and KrforHa = 01 Gc = 01 119878c = 062 Bi = 01 Nc = 001 and Pr = 072
Sc = 062 119878 = 0 Kr = 0
Sc = 262 119878 = 0 Kr = 0
Sc = 062 119878 = 005 Kr = 005
Sc = 062 119878 = 005 Kr = 05
035
03
025
02
015
005
01
00 2 4 6 8 10
120578
120579(120578)
Figure 12 Temperature profiles for different values of Sc 119878 and Krfor Ha = 01 Gr = 01 Gc = 01 Bi = 01 Nc = 001 and Pr = 072
is more pronounced on temperature profiles for high Prandtlnumber fluids
33 Concentration Profile Figures 14ndash19 show the concen-tration profiles obtained by the numerical simulations forvarious values of nondimensional parameters Bi Ha GrGc Kr Pr Nc Sc and 119878 In Figure 14 the effect of anapplied magnetic field is found to increase the concentrationboundary layer However it is interesting to note that theconcentration profiles decrease with the increase of both heatsource and chemical reaction parameters (Figures 14 and 15)Figures 16 and 17 reveal the concentration variations with Grand Gc respectively It is due to the fact that an increase in thevalues of Grashof number and modified Grashof number hasthe tendency to increase the mass buoyancy effect This gives
Figure 14 Concentration profiles for different values of Ha 119878 andKr for Gr = 01 Gc = 01 Sc = 062 Bi = 01 Nc = 001 andPr = 072
rise to an increase in the induced flow and thereby decreasesconcentration Figure 18 depicts the effect of Schmidt numberon the concentration Like temperature the concentrationvalue is higher at the surface and falls exponentiallyThe con-centration decreases with an increase in Sc and the decrease ismore with the increase in concentration parameter Figure 19displays the effect of Pr on concentration profile against 120578withthe variation of source and chemical reaction parametersThe magnitude of concentration is higher at the plate andthen decays to zero asymptotically this is due to the fact thatthermal conductivity of fluid decreases with the increase of Pr
Figure 16 Concentration profiles for different values of Gc 119878 andKr for Ha = 01 Gr = 01 Sc = 062 Bi = 01 Nc = 001 andPr = 072
resulting a decrease in thermal boundary layer thickness andsource term further influences the decrease of concentration
4 Conclusions
The present numerical study has been carried out for heatand mass transfer of MHD flow over a moving vertical platein presence of heat source and chemical reaction along withconvective surface boundary conditionThe shootingmethodwith Runge-Kutta fourth-order iteration scheme has beenimplemented to solve the dimensionless velocity thermaland mass boundary layer equations It has been shownthat the magnitude of local skin friction and local Nusseltnumber increase whereas the plate surface temperature and
Figure 18 Concentration profiles for different values of Sc 119878 andKr for Ha = 01 Gr = 01 Gc = 01 Bi = 01 Nc = 001 andPr = 072
Sherwood number decreases with an increase in sourceparameter The increase in the strength of chemical reactingsubstances causes an increase in the magnitude of localskin friction the plate surface temperature and Sherwoodnumber but opposite behavior is seen for local Nusseltnumber The velocity profile decreases by increasing themagnetic parameter and even the increase is more prominentwith the increase in source and chemical reaction parameterThe thermal boundary layer thickness increases with theincrease of source chemical reaction parameter plate surfaceconvective heat parameter and Schmidt number while themass flux boundary layer thickness decreases Moreover thethermal boundary layer thickness the mass boundary layerand velocity decrease as the Prandtl number increases
[1] K Vajrevelu and J Nayfeh ldquoHydromagnetic convection at acone and a wedgerdquo International Communications in Heat andMass Transfer vol 19 pp 701ndash710 1992
[2] A J Chamkha ldquoHydromagnetic three-dimensional free con-vection on a vertical stretching surface with heat generation orabsorptionrdquo International Journal of Heat and Fluid Flow vol20 no 1 pp 84ndash92 1999
[3] D Moalem ldquoSteady state heat transfer within porous mediumwith temperature dependent heat generationrdquo InternationalJournal of Heat and Mass Transfer vol 19 no 5 pp 529ndash5371976
[4] M M Rahman and M A Sattar ldquoMagnetohydrodynamicconvective flow of a micropolar fluid past a continuouslymoving vertical porous plate in the presence of heat genera-tionabsorptionrdquo Journal of Heat Transfer vol 128 no 2 pp142ndash152 2006
[5] U N Das R Deka and V M Soundalgekar ldquoEffects of masstransfer on flowpast an impulsively started infinite vertical platewith constant heat flux and chemical reactionrdquo Forschung imIngenieurwesenEngineering Research vol 60 no 10 pp 284ndash287 1994
[6] S P Anjalidevi andRKandasamy ldquoEffects of chemical reactionheat and mass transfer on laminar flow along a semi infinitehorizontal platerdquoHeat andMass Transfer vol 35 no 6 pp 465ndash467 1999
[7] M A Seddeek A A Darwish andM S Abdelmeguid ldquoEffectsof chemical reaction and variable viscosity on hydromagneticmixed convection heat and mass transfer for Hiemenz flowthrough porous media with radiationrdquo Communications inNonlinear Science and Numerical Simulation vol 12 no 2 pp195ndash213 2007
[8] P M Patil and P S Kulkarni ldquoEffects of chemical reaction onfree convective flowof a polar fluid through a porousmedium inthe presence of internal heat generationrdquo International Journalof Thermal Sciences vol 47 no 8 pp 1043ndash1054 2008
[9] A M Salem and M Abd El-Aziz ldquoEffect of Hall currents andchemical reaction on hydromagnetic flow of a stretching ver-tical surface with internal heat generationabsorptionrdquo AppliedMathematical Modelling vol 32 no 7 pp 1236ndash1254 2008
[10] F S IbrahimAM Elaiw andAA Bakr ldquoEffect of the chemicalreaction and radiation absorption on the unsteady MHDfree convection flow past a semi infinite vertical permeablemoving plate with heat source and suctionrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 6 pp1056ndash1066 2008
[11] S K Parida M Acharya G C Dash and S Panda ldquoMHD heatand mass transfer in a rotating system with periodic suctionrdquoArabian Journal for Science and Engineering vol 36 no 6 pp1139ndash1151 2011
[12] R Rajeswari B Jothiram andV K Nelson ldquoChemical reactionheat and mass transfer on nonlinear MHD boundary layer flowthrough a vertical porous surface in the presence of suctionrdquoAppliedMathematical Sciences vol 3 no 49-52 pp 2469ndash24802009
[13] A Mahdy ldquoEffect of chemical reaction and heat generationor absorption on double-diffusive convection from a verticaltruncated cone in porous media with variable viscosityrdquo Inter-national Communications in Heat andMass Transfer vol 37 no5 pp 548ndash554 2010
[14] D Pal and B Talukdar ldquoPerturbation analysis of unsteadymagnetohydrodynamic convective heat and mass transfer in aboundary layer slip flow past a vertical permeable plate withthermal radiation and chemical reactionrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 7 pp1813ndash1830 2010
[15] O D Makinde and A Ogulu ldquoThe effect of thermal radiationon the heat and mass transfer flow of a variable viscosity fluidpast a vertical porous plate permeated by a transverse magneticfieldrdquo Chemical Engineering Communications vol 195 no 12pp 1575ndash1584 2008
[16] O D Makinde ldquoMHD mixed-convection interaction withthermal radiation and nth order chemical reaction past avertical porous plate embedded in a porous mediumrdquo ChemicalEngineering Communications vol 198 no 4 pp 590ndash608 2011
[17] 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
[18] P O Olanrewaju O T Arulogun and K Adebimpe ldquoInternalheat generation effect on thermal boundary layer with a con-vective surface boundary conditionrdquo American Journal of FluidDynamics vol 2 no 1 pp 1ndash4 2012
[19] ODMakinde ldquoOnMHDheat andmass transfer over amovingvertical plate with a convective surface boundary conditionrdquoCanadian Journal of Chemical Engineering vol 88 no 6 pp983ndash990 2010
[20] O D Makinde ldquoSimilarity solution of hydromagnetic heat andmass transfer over a vertical plate with a convective surfaceboundary conditionrdquo International Journal of Physical Sciencesvol 5 no 6 pp 700ndash710 2010
[21] O D Makinde ldquoSimilarity solution for natural convectionfrom a moving vertical plate with internal heat generation anda convective boundary conditionrdquo Thermal Science vol 15supplement 1 pp S137ndashS143 2011
[22] K Gangadhar N B Reddy and P K Kameswaran ldquoSimilaritysolution of hydromagnetic heat andmass transfer over a verticalplate with convective surface boundary condition and chemicalreactionrdquo International Journal of Nonlinear Science vol 3 no3 pp 298ndash307 2012
[23] NFM Noor S Abbabandy and I Hasim ldquoHeat and MassTransfer of thermophoretic MHD flow over an inclined radiateisothermal permeable surface in presence of heat source sinkrdquoInternational Journal of Heat and Mass Transfer vol 55 no 7pp 2122ndash2128 2012
[24] V Bisht M Kumar and Z Uddin ldquoEffect of variable thermalconductivity and chemical reaction on steadymixed convectionboundary layer flow with heat and mass transfer inside a conedue to a pointrdquo Journal of Applied Fluid Mechanics vol 4 no 4pp 59ndash63 2011
[25] A A Bakr ldquoEffects of chemical reaction on MHD free convec-tion andmass transfer flowof amicropolar fluidwith oscillatoryplate velocity and constant heat source in a rotating frame ofreferencerdquoCommunications inNonlinear Science andNumericalSimulation vol 16 no 2 pp 698ndash710 2011
The corresponding boundary conditions equation (5) forvelocity temperature and concentration fields in terms ofnondimensional variables are
119891 (0) = 0 1198911015840
(0) = 1
1205791015840
(0) = Bi119909[120579 (0) minus 1] 120601 (0) = 1
1198911015840
(infin) = 0 120579 (infin) = 0 120601 (infin) = 0
(11)
It is observed that in the absence of local source parameterand chemical reaction parameter that is for 119878
119909= 0 and
Kr119909= 0 (8) (9) and (10) together with boundary condition
(11) are the same as those obtained by Makinde [19] It isnoticed that the concentration equation (10) in presence ofthe chemical reaction parameter (Kr
119909) in the fluid yields
nonhomogeneous differential equationwhich is coupledwithmomentum equation (8) and in general difficult to solveanalytically In order to overcome this difficulty we solvethese equations numerically by fourth-order Runge-Kuttamethod in association with shooting technique Firstly theseequations together with associated boundary conditions arereduced to first-order differential equations Since equationsto be solved are the third order for the velocity and secondorder for the temperature and concentration the values of1198911015840 1205791015840 and 120601
1015840 are needed at 120578 = 0 Therefore the shootingmethod is used to solve this boundary value problem Thelocal skin friction coefficient the local Nusselt number thelocal Sherwood number and the plate surface temperatureare computed in terms of 11989110158401015840(0) minus1205791015840(0) minus1206011015840(0) and 120579(0)respectively It can be noted that the local parameters Ha
119909
Gr119909 Gc119909 Bi119909 119878119909 and Kr
119909in (8)ndash(10) are functions of 119909
and generate local similarity solution In order to have a truesimilarity solution we assume the following relation [19]
ℎ119891=
119886
radic119909 120590 =
119887
119909 120573 =
119888
119909
120573lowast
=119889
119909 119876
0=119890
119909 Kr1015840 = 119898
119909
(12)
where 119886 119887 119888 119889 119890 and 119898 are the constants with appropriatedimensions In view of relation (12) the parameters Ha
119909
Gr119909 Gc119909 Bi119909 119878119909 and Kr
119909are now independent of 119909 and
henceforth we drop the index ldquo119909rdquo for simplicity
3 Result Discussion
The numerical solutions of the boundary value problemfor system of ordinary differential equations were obtainedby Runge-Kutta method along with shooting techniqueSince the physical domain of the underlying problem isunbounded the computational domain is chosen sufficientlylarge in order to meet the far field boundary condition atinfinity Here the transverse distance is fixed to 10 and suitablymore than 10 depending upon the choice of the parametersTo demonstrate successful implementation of the numericalscheme the numerical results are compared to those obtainedby a previous published paper (see [19]) for the local skinfriction coefficient plate surface temperature and the local
1
08
06
04
02
00 2 4 6 8 10
Bi = 01 119878 = 0 Kr = 0
Bi = 10 119878 = 0 Kr = 0
Bi = 10 119878 = 01 Kr = 005
Bi = 10 119878 = 01 Kr = 05
119891998400
(120578)
120578
Figure 3 Velocity profiles for different values of Bi 119878 and Kr forHa = 01 Gr = 01 Gc = 01 Sc = 062 Nc = 001 and Pr = 072
Figure 4 Velocity profiles for different values of Gc 119878 and Kr forHa = 01 Gr = 01 Sc = 062 Bi = 01 Nc = 001 and Pr = 072
Sherwood and Nusselt numbers in Table 1 for the parametersembedded in absence of local chemical reaction parameter(Kr) and local heat source parameter (119878) Table 1 presents acomparison of 11989110158401015840(0) minus1205791015840(0) 120579(0) and minus120601
1015840
(0) between thepresent results and the results obtained by Makinde [19] forvarious values of Ha Gr Gc Bi Pr and Sc when 119878 = Kr =0 The results are found to be in excellent agreement It isimportant to note that the momentum equation is coupledwith heat and mass transfer equations and hence the Prandtlnumber Schmidt number chemical reaction parameter andsource term have an influence on skin friction in our present
International Journal of Chemical Engineering 5
Table 1 Comparison of 11989110158401015840(0) minus1205791015840(0) 120579(0) and minus1206011015840(0) for various values of Bi Ha Gr Gc Pr and Sc when Kr = 119878 = 0
Bi Gr Gc Ha Pr Sc Makinde [19] Present study11989110158401015840
Figure 5 Velocity profiles for different values of Gr 119878 and Kr forHa = 01 Gc = 01 Sc = 062 Bi = 01 Nc = 001 and Pr = 072
problem The influence of heat source and chemical reactionparameters on local skin friction local Nusselt number platesurface temperature and Sherwood number are highlightedin Tables 2 and 3 for various nondimensional flow param-eters It is clearly seen from Table 2 that the magnitude ofskin friction and Nusselt number increase whereas the platesurface temperature and Sherwood number decrease withthe increase of source parameter Furthermore increasingthe strength of chemical reacting substances is to increasethe local skin friction the plate surface temperature andSherwood number but opposite behavior is seen for localNusselt numberThe obvious observation fromTable 2 is thatthe fluid low Prandtl number increases the magnitude oflocal skin friction and local Nusselt number while decreasingthe plate surface temperature and local Sherwood numberAgain the fluid with low convective resistance (or externalresistance) decreases the magnitude of local skin frictionwhile the Nusselt number the plate surface temperatureand local Sherwood number increase It is also observed
Sc = 062 119878 = 0 Kr = 0
Sc = 262 119878 = 0 Kr = 0
Sc = 062 119878 = 005 Kr = 005
Sc = 062 119878 = 005 Kr = 05
1
08
06
04
02
00 2 4 6 8 10
119891998400 (120578)
120578
Figure 6 Velocity profiles for different values of Sc 119878 and Kr forHa = 01 Gr = 01 Gc = 01 Bi = 01 Nc = 001 and Pr = 072
Table 2 Computation of skin friction (11989110158401015840(0)) Nusselt number(minus1205791015840(0)) plate surface temperature (120579(0)) and Sherwood number(minus1206011015840(0)) for different values of Bi Pr 119878 andKrTheother parametersare Ha = 01 Gr = 01 Gc = 01 Sc = 062 and Nc = 001
from Table 3 that the increase of Schimdt number resultsin increase of the magnitude of the local skin friction butopposite behavior is marked in case of Nusselt number andthe plate surface temperature
6 International Journal of Chemical Engineering
Table 3 Computation of skin friction (11989110158401015840(0)) Nusselt number(minus1205791015840(0)) plate surface temperature (120579(0)) and Sherwood number(minus1206011015840(0)) for different values of Sc 119878 and Kr The other parametersare fixed at Bi = Gc = Gr = 01 Pr = 072 and Nc = 001
31 Velocity Profiles Figures 2ndash7 exhibit the velocity pro-files obtained by the numerical simulations for variousflow parameters involved in the problem The simulatedparameters are reported in the figure caption The effectsof magnetic parameter on the velocity field in presenceand absence of source and chemical reaction parameterare shown in Figure 2 It illustrates that the velocity profiledecreases with the increase of magnetic parameter in absenceof source and chemical reaction parameter because Lorentzforce acts against the flow if the magnetic field is applied inthe normal direction In presence of source and chemicalreaction parameter no such appreciable change is observedin Figure 2This corresponds to Figure (2) in [19] and therebyagain validating our numerical scheme A little increase in thevelocity profile near the boundary layer is marked in Figure 3with the increase in the convective heat parameter becausethe fluid adjacent to the right surface of the plate becomeslighter by hot fluid and rises faster The boundary layer flowsdevelop adjacent to vertical surface and velocity reaches amaximum in the boundary layer It is evident from Figures4 and 5 that greater cooling of surface an increase in Gc andincrease in Gr result in an increase in the velocity It is dueto the fact that the increase in the values of Grashof numberand modified Grashof number has the tendency to increasethe thermal and mass buoyancy effect The increase is alsoevident due to the presence of source and chemical reactionparameters Furthermore the velocity increases rapidly andsuddenly falls near the boundary and then approaches thefar field boundary condition due to favorable buoyancy forcewith the increase of both Gr and Gc It can be seen that theincrease in the Prandtl number and Schmidt number leads toa fall in the velocity as shown in Figures 6 and 7
32 Temperature Profiles Figures 8ndash13 show the temperatureprofiles obtained by the numerical simulations for variousvalues of flow parameters Figure 8 clearly demonstrates thatthe temperature profiles increase with the increase of themagnetic field parameter which implies that the appliedmagnetic field tends to heat the fluid and thus reducesthe heat transfer from the wall Further it can be seenthat temperature profile increases due to increase of heatsource as well as chemical reaction parameter The thermalboundary layer thickness increases with an increase in the
Figure 8 Temperature profiles for different values of Ha 119878 and Krfor Gr = 01 Gc = 01 Sc = 062 Bi = 01 Nc = 001 and Pr = 072
plate surface convective heat parameter (Figure 9) and asimilar effect is also observed in Figure 12 with the increaseof Schmidt number It can be observed that the amplitudeof fluid temperature in presence of heat source and chemicalreacting substances is more in comparison to in absence ofthese parameters The steady state temperatures for differentGrashof number modified Grashof number internal heatsource and chemical reaction parameters are shown inFigures 10 and 11 The thermal boundary layer decreases withincreasing Grashof number and modified Grashof numberbut reverse effect is observed with the presence of chemicalreaction parameter This is also revealed in Figure 13 which
Figure 10 Temperature profiles for different values of Gc 119878 and Krfor Ha = 01 Gr = 01 Sc = 062 Bi = 01 Nc = 001 and Pr = 072
shows that thermal boundary layer thickness decreases asthe Prandtl number increases implying higher heat transferIt is due to that smaller values of Pr means increasing thethermal conductivity and therefore heat is able to diffuseaway from the plate more quickly than higher values of Prhence the rate of heat transfer is reduced It is noted thatowing to the presence of heat source effect (119878 gt 0) andchemical reaction parameter (Kr gt 0) the thermal stateof the fluid increases Hence the temperature of the fluidincreases within the boundary layers In the event that thestrength of the heat source and chemical reaction parametersare relatively large a remarkable change is observed in thetemperature profiles within the thermal boundary layer ascan be seen in Figure 13 Further the effect of heat generation
Figure 11 Temperature profiles for different values of Gr 119878 and KrforHa = 01 Gc = 01 119878c = 062 Bi = 01 Nc = 001 and Pr = 072
Sc = 062 119878 = 0 Kr = 0
Sc = 262 119878 = 0 Kr = 0
Sc = 062 119878 = 005 Kr = 005
Sc = 062 119878 = 005 Kr = 05
035
03
025
02
015
005
01
00 2 4 6 8 10
120578
120579(120578)
Figure 12 Temperature profiles for different values of Sc 119878 and Krfor Ha = 01 Gr = 01 Gc = 01 Bi = 01 Nc = 001 and Pr = 072
is more pronounced on temperature profiles for high Prandtlnumber fluids
33 Concentration Profile Figures 14ndash19 show the concen-tration profiles obtained by the numerical simulations forvarious values of nondimensional parameters Bi Ha GrGc Kr Pr Nc Sc and 119878 In Figure 14 the effect of anapplied magnetic field is found to increase the concentrationboundary layer However it is interesting to note that theconcentration profiles decrease with the increase of both heatsource and chemical reaction parameters (Figures 14 and 15)Figures 16 and 17 reveal the concentration variations with Grand Gc respectively It is due to the fact that an increase in thevalues of Grashof number and modified Grashof number hasthe tendency to increase the mass buoyancy effect This gives
Figure 14 Concentration profiles for different values of Ha 119878 andKr for Gr = 01 Gc = 01 Sc = 062 Bi = 01 Nc = 001 andPr = 072
rise to an increase in the induced flow and thereby decreasesconcentration Figure 18 depicts the effect of Schmidt numberon the concentration Like temperature the concentrationvalue is higher at the surface and falls exponentiallyThe con-centration decreases with an increase in Sc and the decrease ismore with the increase in concentration parameter Figure 19displays the effect of Pr on concentration profile against 120578withthe variation of source and chemical reaction parametersThe magnitude of concentration is higher at the plate andthen decays to zero asymptotically this is due to the fact thatthermal conductivity of fluid decreases with the increase of Pr
Figure 16 Concentration profiles for different values of Gc 119878 andKr for Ha = 01 Gr = 01 Sc = 062 Bi = 01 Nc = 001 andPr = 072
resulting a decrease in thermal boundary layer thickness andsource term further influences the decrease of concentration
4 Conclusions
The present numerical study has been carried out for heatand mass transfer of MHD flow over a moving vertical platein presence of heat source and chemical reaction along withconvective surface boundary conditionThe shootingmethodwith Runge-Kutta fourth-order iteration scheme has beenimplemented to solve the dimensionless velocity thermaland mass boundary layer equations It has been shownthat the magnitude of local skin friction and local Nusseltnumber increase whereas the plate surface temperature and
Figure 18 Concentration profiles for different values of Sc 119878 andKr for Ha = 01 Gr = 01 Gc = 01 Bi = 01 Nc = 001 andPr = 072
Sherwood number decreases with an increase in sourceparameter The increase in the strength of chemical reactingsubstances causes an increase in the magnitude of localskin friction the plate surface temperature and Sherwoodnumber but opposite behavior is seen for local Nusseltnumber The velocity profile decreases by increasing themagnetic parameter and even the increase is more prominentwith the increase in source and chemical reaction parameterThe thermal boundary layer thickness increases with theincrease of source chemical reaction parameter plate surfaceconvective heat parameter and Schmidt number while themass flux boundary layer thickness decreases Moreover thethermal boundary layer thickness the mass boundary layerand velocity decrease as the Prandtl number increases
[1] K Vajrevelu and J Nayfeh ldquoHydromagnetic convection at acone and a wedgerdquo International Communications in Heat andMass Transfer vol 19 pp 701ndash710 1992
[2] A J Chamkha ldquoHydromagnetic three-dimensional free con-vection on a vertical stretching surface with heat generation orabsorptionrdquo International Journal of Heat and Fluid Flow vol20 no 1 pp 84ndash92 1999
[3] D Moalem ldquoSteady state heat transfer within porous mediumwith temperature dependent heat generationrdquo InternationalJournal of Heat and Mass Transfer vol 19 no 5 pp 529ndash5371976
[4] M M Rahman and M A Sattar ldquoMagnetohydrodynamicconvective flow of a micropolar fluid past a continuouslymoving vertical porous plate in the presence of heat genera-tionabsorptionrdquo Journal of Heat Transfer vol 128 no 2 pp142ndash152 2006
[5] U N Das R Deka and V M Soundalgekar ldquoEffects of masstransfer on flowpast an impulsively started infinite vertical platewith constant heat flux and chemical reactionrdquo Forschung imIngenieurwesenEngineering Research vol 60 no 10 pp 284ndash287 1994
[6] S P Anjalidevi andRKandasamy ldquoEffects of chemical reactionheat and mass transfer on laminar flow along a semi infinitehorizontal platerdquoHeat andMass Transfer vol 35 no 6 pp 465ndash467 1999
[7] M A Seddeek A A Darwish andM S Abdelmeguid ldquoEffectsof chemical reaction and variable viscosity on hydromagneticmixed convection heat and mass transfer for Hiemenz flowthrough porous media with radiationrdquo Communications inNonlinear Science and Numerical Simulation vol 12 no 2 pp195ndash213 2007
[8] P M Patil and P S Kulkarni ldquoEffects of chemical reaction onfree convective flowof a polar fluid through a porousmedium inthe presence of internal heat generationrdquo International Journalof Thermal Sciences vol 47 no 8 pp 1043ndash1054 2008
[9] A M Salem and M Abd El-Aziz ldquoEffect of Hall currents andchemical reaction on hydromagnetic flow of a stretching ver-tical surface with internal heat generationabsorptionrdquo AppliedMathematical Modelling vol 32 no 7 pp 1236ndash1254 2008
[10] F S IbrahimAM Elaiw andAA Bakr ldquoEffect of the chemicalreaction and radiation absorption on the unsteady MHDfree convection flow past a semi infinite vertical permeablemoving plate with heat source and suctionrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 6 pp1056ndash1066 2008
[11] S K Parida M Acharya G C Dash and S Panda ldquoMHD heatand mass transfer in a rotating system with periodic suctionrdquoArabian Journal for Science and Engineering vol 36 no 6 pp1139ndash1151 2011
[12] R Rajeswari B Jothiram andV K Nelson ldquoChemical reactionheat and mass transfer on nonlinear MHD boundary layer flowthrough a vertical porous surface in the presence of suctionrdquoAppliedMathematical Sciences vol 3 no 49-52 pp 2469ndash24802009
[13] A Mahdy ldquoEffect of chemical reaction and heat generationor absorption on double-diffusive convection from a verticaltruncated cone in porous media with variable viscosityrdquo Inter-national Communications in Heat andMass Transfer vol 37 no5 pp 548ndash554 2010
[14] D Pal and B Talukdar ldquoPerturbation analysis of unsteadymagnetohydrodynamic convective heat and mass transfer in aboundary layer slip flow past a vertical permeable plate withthermal radiation and chemical reactionrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 7 pp1813ndash1830 2010
[15] O D Makinde and A Ogulu ldquoThe effect of thermal radiationon the heat and mass transfer flow of a variable viscosity fluidpast a vertical porous plate permeated by a transverse magneticfieldrdquo Chemical Engineering Communications vol 195 no 12pp 1575ndash1584 2008
[16] O D Makinde ldquoMHD mixed-convection interaction withthermal radiation and nth order chemical reaction past avertical porous plate embedded in a porous mediumrdquo ChemicalEngineering Communications vol 198 no 4 pp 590ndash608 2011
[17] 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
[18] P O Olanrewaju O T Arulogun and K Adebimpe ldquoInternalheat generation effect on thermal boundary layer with a con-vective surface boundary conditionrdquo American Journal of FluidDynamics vol 2 no 1 pp 1ndash4 2012
[19] ODMakinde ldquoOnMHDheat andmass transfer over amovingvertical plate with a convective surface boundary conditionrdquoCanadian Journal of Chemical Engineering vol 88 no 6 pp983ndash990 2010
[20] O D Makinde ldquoSimilarity solution of hydromagnetic heat andmass transfer over a vertical plate with a convective surfaceboundary conditionrdquo International Journal of Physical Sciencesvol 5 no 6 pp 700ndash710 2010
[21] O D Makinde ldquoSimilarity solution for natural convectionfrom a moving vertical plate with internal heat generation anda convective boundary conditionrdquo Thermal Science vol 15supplement 1 pp S137ndashS143 2011
[22] K Gangadhar N B Reddy and P K Kameswaran ldquoSimilaritysolution of hydromagnetic heat andmass transfer over a verticalplate with convective surface boundary condition and chemicalreactionrdquo International Journal of Nonlinear Science vol 3 no3 pp 298ndash307 2012
[23] NFM Noor S Abbabandy and I Hasim ldquoHeat and MassTransfer of thermophoretic MHD flow over an inclined radiateisothermal permeable surface in presence of heat source sinkrdquoInternational Journal of Heat and Mass Transfer vol 55 no 7pp 2122ndash2128 2012
[24] V Bisht M Kumar and Z Uddin ldquoEffect of variable thermalconductivity and chemical reaction on steadymixed convectionboundary layer flow with heat and mass transfer inside a conedue to a pointrdquo Journal of Applied Fluid Mechanics vol 4 no 4pp 59ndash63 2011
[25] A A Bakr ldquoEffects of chemical reaction on MHD free convec-tion andmass transfer flowof amicropolar fluidwith oscillatoryplate velocity and constant heat source in a rotating frame ofreferencerdquoCommunications inNonlinear Science andNumericalSimulation vol 16 no 2 pp 698ndash710 2011
Table 1 Comparison of 11989110158401015840(0) minus1205791015840(0) 120579(0) and minus1206011015840(0) for various values of Bi Ha Gr Gc Pr and Sc when Kr = 119878 = 0
Bi Gr Gc Ha Pr Sc Makinde [19] Present study11989110158401015840
Figure 5 Velocity profiles for different values of Gr 119878 and Kr forHa = 01 Gc = 01 Sc = 062 Bi = 01 Nc = 001 and Pr = 072
problem The influence of heat source and chemical reactionparameters on local skin friction local Nusselt number platesurface temperature and Sherwood number are highlightedin Tables 2 and 3 for various nondimensional flow param-eters It is clearly seen from Table 2 that the magnitude ofskin friction and Nusselt number increase whereas the platesurface temperature and Sherwood number decrease withthe increase of source parameter Furthermore increasingthe strength of chemical reacting substances is to increasethe local skin friction the plate surface temperature andSherwood number but opposite behavior is seen for localNusselt numberThe obvious observation fromTable 2 is thatthe fluid low Prandtl number increases the magnitude oflocal skin friction and local Nusselt number while decreasingthe plate surface temperature and local Sherwood numberAgain the fluid with low convective resistance (or externalresistance) decreases the magnitude of local skin frictionwhile the Nusselt number the plate surface temperatureand local Sherwood number increase It is also observed
Sc = 062 119878 = 0 Kr = 0
Sc = 262 119878 = 0 Kr = 0
Sc = 062 119878 = 005 Kr = 005
Sc = 062 119878 = 005 Kr = 05
1
08
06
04
02
00 2 4 6 8 10
119891998400 (120578)
120578
Figure 6 Velocity profiles for different values of Sc 119878 and Kr forHa = 01 Gr = 01 Gc = 01 Bi = 01 Nc = 001 and Pr = 072
Table 2 Computation of skin friction (11989110158401015840(0)) Nusselt number(minus1205791015840(0)) plate surface temperature (120579(0)) and Sherwood number(minus1206011015840(0)) for different values of Bi Pr 119878 andKrTheother parametersare Ha = 01 Gr = 01 Gc = 01 Sc = 062 and Nc = 001
from Table 3 that the increase of Schimdt number resultsin increase of the magnitude of the local skin friction butopposite behavior is marked in case of Nusselt number andthe plate surface temperature
6 International Journal of Chemical Engineering
Table 3 Computation of skin friction (11989110158401015840(0)) Nusselt number(minus1205791015840(0)) plate surface temperature (120579(0)) and Sherwood number(minus1206011015840(0)) for different values of Sc 119878 and Kr The other parametersare fixed at Bi = Gc = Gr = 01 Pr = 072 and Nc = 001
31 Velocity Profiles Figures 2ndash7 exhibit the velocity pro-files obtained by the numerical simulations for variousflow parameters involved in the problem The simulatedparameters are reported in the figure caption The effectsof magnetic parameter on the velocity field in presenceand absence of source and chemical reaction parameterare shown in Figure 2 It illustrates that the velocity profiledecreases with the increase of magnetic parameter in absenceof source and chemical reaction parameter because Lorentzforce acts against the flow if the magnetic field is applied inthe normal direction In presence of source and chemicalreaction parameter no such appreciable change is observedin Figure 2This corresponds to Figure (2) in [19] and therebyagain validating our numerical scheme A little increase in thevelocity profile near the boundary layer is marked in Figure 3with the increase in the convective heat parameter becausethe fluid adjacent to the right surface of the plate becomeslighter by hot fluid and rises faster The boundary layer flowsdevelop adjacent to vertical surface and velocity reaches amaximum in the boundary layer It is evident from Figures4 and 5 that greater cooling of surface an increase in Gc andincrease in Gr result in an increase in the velocity It is dueto the fact that the increase in the values of Grashof numberand modified Grashof number has the tendency to increasethe thermal and mass buoyancy effect The increase is alsoevident due to the presence of source and chemical reactionparameters Furthermore the velocity increases rapidly andsuddenly falls near the boundary and then approaches thefar field boundary condition due to favorable buoyancy forcewith the increase of both Gr and Gc It can be seen that theincrease in the Prandtl number and Schmidt number leads toa fall in the velocity as shown in Figures 6 and 7
32 Temperature Profiles Figures 8ndash13 show the temperatureprofiles obtained by the numerical simulations for variousvalues of flow parameters Figure 8 clearly demonstrates thatthe temperature profiles increase with the increase of themagnetic field parameter which implies that the appliedmagnetic field tends to heat the fluid and thus reducesthe heat transfer from the wall Further it can be seenthat temperature profile increases due to increase of heatsource as well as chemical reaction parameter The thermalboundary layer thickness increases with an increase in the
Figure 8 Temperature profiles for different values of Ha 119878 and Krfor Gr = 01 Gc = 01 Sc = 062 Bi = 01 Nc = 001 and Pr = 072
plate surface convective heat parameter (Figure 9) and asimilar effect is also observed in Figure 12 with the increaseof Schmidt number It can be observed that the amplitudeof fluid temperature in presence of heat source and chemicalreacting substances is more in comparison to in absence ofthese parameters The steady state temperatures for differentGrashof number modified Grashof number internal heatsource and chemical reaction parameters are shown inFigures 10 and 11 The thermal boundary layer decreases withincreasing Grashof number and modified Grashof numberbut reverse effect is observed with the presence of chemicalreaction parameter This is also revealed in Figure 13 which
Figure 10 Temperature profiles for different values of Gc 119878 and Krfor Ha = 01 Gr = 01 Sc = 062 Bi = 01 Nc = 001 and Pr = 072
shows that thermal boundary layer thickness decreases asthe Prandtl number increases implying higher heat transferIt is due to that smaller values of Pr means increasing thethermal conductivity and therefore heat is able to diffuseaway from the plate more quickly than higher values of Prhence the rate of heat transfer is reduced It is noted thatowing to the presence of heat source effect (119878 gt 0) andchemical reaction parameter (Kr gt 0) the thermal stateof the fluid increases Hence the temperature of the fluidincreases within the boundary layers In the event that thestrength of the heat source and chemical reaction parametersare relatively large a remarkable change is observed in thetemperature profiles within the thermal boundary layer ascan be seen in Figure 13 Further the effect of heat generation
Figure 11 Temperature profiles for different values of Gr 119878 and KrforHa = 01 Gc = 01 119878c = 062 Bi = 01 Nc = 001 and Pr = 072
Sc = 062 119878 = 0 Kr = 0
Sc = 262 119878 = 0 Kr = 0
Sc = 062 119878 = 005 Kr = 005
Sc = 062 119878 = 005 Kr = 05
035
03
025
02
015
005
01
00 2 4 6 8 10
120578
120579(120578)
Figure 12 Temperature profiles for different values of Sc 119878 and Krfor Ha = 01 Gr = 01 Gc = 01 Bi = 01 Nc = 001 and Pr = 072
is more pronounced on temperature profiles for high Prandtlnumber fluids
33 Concentration Profile Figures 14ndash19 show the concen-tration profiles obtained by the numerical simulations forvarious values of nondimensional parameters Bi Ha GrGc Kr Pr Nc Sc and 119878 In Figure 14 the effect of anapplied magnetic field is found to increase the concentrationboundary layer However it is interesting to note that theconcentration profiles decrease with the increase of both heatsource and chemical reaction parameters (Figures 14 and 15)Figures 16 and 17 reveal the concentration variations with Grand Gc respectively It is due to the fact that an increase in thevalues of Grashof number and modified Grashof number hasthe tendency to increase the mass buoyancy effect This gives
Figure 14 Concentration profiles for different values of Ha 119878 andKr for Gr = 01 Gc = 01 Sc = 062 Bi = 01 Nc = 001 andPr = 072
rise to an increase in the induced flow and thereby decreasesconcentration Figure 18 depicts the effect of Schmidt numberon the concentration Like temperature the concentrationvalue is higher at the surface and falls exponentiallyThe con-centration decreases with an increase in Sc and the decrease ismore with the increase in concentration parameter Figure 19displays the effect of Pr on concentration profile against 120578withthe variation of source and chemical reaction parametersThe magnitude of concentration is higher at the plate andthen decays to zero asymptotically this is due to the fact thatthermal conductivity of fluid decreases with the increase of Pr
Figure 16 Concentration profiles for different values of Gc 119878 andKr for Ha = 01 Gr = 01 Sc = 062 Bi = 01 Nc = 001 andPr = 072
resulting a decrease in thermal boundary layer thickness andsource term further influences the decrease of concentration
4 Conclusions
The present numerical study has been carried out for heatand mass transfer of MHD flow over a moving vertical platein presence of heat source and chemical reaction along withconvective surface boundary conditionThe shootingmethodwith Runge-Kutta fourth-order iteration scheme has beenimplemented to solve the dimensionless velocity thermaland mass boundary layer equations It has been shownthat the magnitude of local skin friction and local Nusseltnumber increase whereas the plate surface temperature and
Figure 18 Concentration profiles for different values of Sc 119878 andKr for Ha = 01 Gr = 01 Gc = 01 Bi = 01 Nc = 001 andPr = 072
Sherwood number decreases with an increase in sourceparameter The increase in the strength of chemical reactingsubstances causes an increase in the magnitude of localskin friction the plate surface temperature and Sherwoodnumber but opposite behavior is seen for local Nusseltnumber The velocity profile decreases by increasing themagnetic parameter and even the increase is more prominentwith the increase in source and chemical reaction parameterThe thermal boundary layer thickness increases with theincrease of source chemical reaction parameter plate surfaceconvective heat parameter and Schmidt number while themass flux boundary layer thickness decreases Moreover thethermal boundary layer thickness the mass boundary layerand velocity decrease as the Prandtl number increases
[1] K Vajrevelu and J Nayfeh ldquoHydromagnetic convection at acone and a wedgerdquo International Communications in Heat andMass Transfer vol 19 pp 701ndash710 1992
[2] A J Chamkha ldquoHydromagnetic three-dimensional free con-vection on a vertical stretching surface with heat generation orabsorptionrdquo International Journal of Heat and Fluid Flow vol20 no 1 pp 84ndash92 1999
[3] D Moalem ldquoSteady state heat transfer within porous mediumwith temperature dependent heat generationrdquo InternationalJournal of Heat and Mass Transfer vol 19 no 5 pp 529ndash5371976
[4] M M Rahman and M A Sattar ldquoMagnetohydrodynamicconvective flow of a micropolar fluid past a continuouslymoving vertical porous plate in the presence of heat genera-tionabsorptionrdquo Journal of Heat Transfer vol 128 no 2 pp142ndash152 2006
[5] U N Das R Deka and V M Soundalgekar ldquoEffects of masstransfer on flowpast an impulsively started infinite vertical platewith constant heat flux and chemical reactionrdquo Forschung imIngenieurwesenEngineering Research vol 60 no 10 pp 284ndash287 1994
[6] S P Anjalidevi andRKandasamy ldquoEffects of chemical reactionheat and mass transfer on laminar flow along a semi infinitehorizontal platerdquoHeat andMass Transfer vol 35 no 6 pp 465ndash467 1999
[7] M A Seddeek A A Darwish andM S Abdelmeguid ldquoEffectsof chemical reaction and variable viscosity on hydromagneticmixed convection heat and mass transfer for Hiemenz flowthrough porous media with radiationrdquo Communications inNonlinear Science and Numerical Simulation vol 12 no 2 pp195ndash213 2007
[8] P M Patil and P S Kulkarni ldquoEffects of chemical reaction onfree convective flowof a polar fluid through a porousmedium inthe presence of internal heat generationrdquo International Journalof Thermal Sciences vol 47 no 8 pp 1043ndash1054 2008
[9] A M Salem and M Abd El-Aziz ldquoEffect of Hall currents andchemical reaction on hydromagnetic flow of a stretching ver-tical surface with internal heat generationabsorptionrdquo AppliedMathematical Modelling vol 32 no 7 pp 1236ndash1254 2008
[10] F S IbrahimAM Elaiw andAA Bakr ldquoEffect of the chemicalreaction and radiation absorption on the unsteady MHDfree convection flow past a semi infinite vertical permeablemoving plate with heat source and suctionrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 6 pp1056ndash1066 2008
[11] S K Parida M Acharya G C Dash and S Panda ldquoMHD heatand mass transfer in a rotating system with periodic suctionrdquoArabian Journal for Science and Engineering vol 36 no 6 pp1139ndash1151 2011
[12] R Rajeswari B Jothiram andV K Nelson ldquoChemical reactionheat and mass transfer on nonlinear MHD boundary layer flowthrough a vertical porous surface in the presence of suctionrdquoAppliedMathematical Sciences vol 3 no 49-52 pp 2469ndash24802009
[13] A Mahdy ldquoEffect of chemical reaction and heat generationor absorption on double-diffusive convection from a verticaltruncated cone in porous media with variable viscosityrdquo Inter-national Communications in Heat andMass Transfer vol 37 no5 pp 548ndash554 2010
[14] D Pal and B Talukdar ldquoPerturbation analysis of unsteadymagnetohydrodynamic convective heat and mass transfer in aboundary layer slip flow past a vertical permeable plate withthermal radiation and chemical reactionrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 7 pp1813ndash1830 2010
[15] O D Makinde and A Ogulu ldquoThe effect of thermal radiationon the heat and mass transfer flow of a variable viscosity fluidpast a vertical porous plate permeated by a transverse magneticfieldrdquo Chemical Engineering Communications vol 195 no 12pp 1575ndash1584 2008
[16] O D Makinde ldquoMHD mixed-convection interaction withthermal radiation and nth order chemical reaction past avertical porous plate embedded in a porous mediumrdquo ChemicalEngineering Communications vol 198 no 4 pp 590ndash608 2011
[17] 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
[18] P O Olanrewaju O T Arulogun and K Adebimpe ldquoInternalheat generation effect on thermal boundary layer with a con-vective surface boundary conditionrdquo American Journal of FluidDynamics vol 2 no 1 pp 1ndash4 2012
[19] ODMakinde ldquoOnMHDheat andmass transfer over amovingvertical plate with a convective surface boundary conditionrdquoCanadian Journal of Chemical Engineering vol 88 no 6 pp983ndash990 2010
[20] O D Makinde ldquoSimilarity solution of hydromagnetic heat andmass transfer over a vertical plate with a convective surfaceboundary conditionrdquo International Journal of Physical Sciencesvol 5 no 6 pp 700ndash710 2010
[21] O D Makinde ldquoSimilarity solution for natural convectionfrom a moving vertical plate with internal heat generation anda convective boundary conditionrdquo Thermal Science vol 15supplement 1 pp S137ndashS143 2011
[22] K Gangadhar N B Reddy and P K Kameswaran ldquoSimilaritysolution of hydromagnetic heat andmass transfer over a verticalplate with convective surface boundary condition and chemicalreactionrdquo International Journal of Nonlinear Science vol 3 no3 pp 298ndash307 2012
[23] NFM Noor S Abbabandy and I Hasim ldquoHeat and MassTransfer of thermophoretic MHD flow over an inclined radiateisothermal permeable surface in presence of heat source sinkrdquoInternational Journal of Heat and Mass Transfer vol 55 no 7pp 2122ndash2128 2012
[24] V Bisht M Kumar and Z Uddin ldquoEffect of variable thermalconductivity and chemical reaction on steadymixed convectionboundary layer flow with heat and mass transfer inside a conedue to a pointrdquo Journal of Applied Fluid Mechanics vol 4 no 4pp 59ndash63 2011
[25] A A Bakr ldquoEffects of chemical reaction on MHD free convec-tion andmass transfer flowof amicropolar fluidwith oscillatoryplate velocity and constant heat source in a rotating frame ofreferencerdquoCommunications inNonlinear Science andNumericalSimulation vol 16 no 2 pp 698ndash710 2011
Table 3 Computation of skin friction (11989110158401015840(0)) Nusselt number(minus1205791015840(0)) plate surface temperature (120579(0)) and Sherwood number(minus1206011015840(0)) for different values of Sc 119878 and Kr The other parametersare fixed at Bi = Gc = Gr = 01 Pr = 072 and Nc = 001
31 Velocity Profiles Figures 2ndash7 exhibit the velocity pro-files obtained by the numerical simulations for variousflow parameters involved in the problem The simulatedparameters are reported in the figure caption The effectsof magnetic parameter on the velocity field in presenceand absence of source and chemical reaction parameterare shown in Figure 2 It illustrates that the velocity profiledecreases with the increase of magnetic parameter in absenceof source and chemical reaction parameter because Lorentzforce acts against the flow if the magnetic field is applied inthe normal direction In presence of source and chemicalreaction parameter no such appreciable change is observedin Figure 2This corresponds to Figure (2) in [19] and therebyagain validating our numerical scheme A little increase in thevelocity profile near the boundary layer is marked in Figure 3with the increase in the convective heat parameter becausethe fluid adjacent to the right surface of the plate becomeslighter by hot fluid and rises faster The boundary layer flowsdevelop adjacent to vertical surface and velocity reaches amaximum in the boundary layer It is evident from Figures4 and 5 that greater cooling of surface an increase in Gc andincrease in Gr result in an increase in the velocity It is dueto the fact that the increase in the values of Grashof numberand modified Grashof number has the tendency to increasethe thermal and mass buoyancy effect The increase is alsoevident due to the presence of source and chemical reactionparameters Furthermore the velocity increases rapidly andsuddenly falls near the boundary and then approaches thefar field boundary condition due to favorable buoyancy forcewith the increase of both Gr and Gc It can be seen that theincrease in the Prandtl number and Schmidt number leads toa fall in the velocity as shown in Figures 6 and 7
32 Temperature Profiles Figures 8ndash13 show the temperatureprofiles obtained by the numerical simulations for variousvalues of flow parameters Figure 8 clearly demonstrates thatthe temperature profiles increase with the increase of themagnetic field parameter which implies that the appliedmagnetic field tends to heat the fluid and thus reducesthe heat transfer from the wall Further it can be seenthat temperature profile increases due to increase of heatsource as well as chemical reaction parameter The thermalboundary layer thickness increases with an increase in the
Figure 8 Temperature profiles for different values of Ha 119878 and Krfor Gr = 01 Gc = 01 Sc = 062 Bi = 01 Nc = 001 and Pr = 072
plate surface convective heat parameter (Figure 9) and asimilar effect is also observed in Figure 12 with the increaseof Schmidt number It can be observed that the amplitudeof fluid temperature in presence of heat source and chemicalreacting substances is more in comparison to in absence ofthese parameters The steady state temperatures for differentGrashof number modified Grashof number internal heatsource and chemical reaction parameters are shown inFigures 10 and 11 The thermal boundary layer decreases withincreasing Grashof number and modified Grashof numberbut reverse effect is observed with the presence of chemicalreaction parameter This is also revealed in Figure 13 which
Figure 10 Temperature profiles for different values of Gc 119878 and Krfor Ha = 01 Gr = 01 Sc = 062 Bi = 01 Nc = 001 and Pr = 072
shows that thermal boundary layer thickness decreases asthe Prandtl number increases implying higher heat transferIt is due to that smaller values of Pr means increasing thethermal conductivity and therefore heat is able to diffuseaway from the plate more quickly than higher values of Prhence the rate of heat transfer is reduced It is noted thatowing to the presence of heat source effect (119878 gt 0) andchemical reaction parameter (Kr gt 0) the thermal stateof the fluid increases Hence the temperature of the fluidincreases within the boundary layers In the event that thestrength of the heat source and chemical reaction parametersare relatively large a remarkable change is observed in thetemperature profiles within the thermal boundary layer ascan be seen in Figure 13 Further the effect of heat generation
Figure 11 Temperature profiles for different values of Gr 119878 and KrforHa = 01 Gc = 01 119878c = 062 Bi = 01 Nc = 001 and Pr = 072
Sc = 062 119878 = 0 Kr = 0
Sc = 262 119878 = 0 Kr = 0
Sc = 062 119878 = 005 Kr = 005
Sc = 062 119878 = 005 Kr = 05
035
03
025
02
015
005
01
00 2 4 6 8 10
120578
120579(120578)
Figure 12 Temperature profiles for different values of Sc 119878 and Krfor Ha = 01 Gr = 01 Gc = 01 Bi = 01 Nc = 001 and Pr = 072
is more pronounced on temperature profiles for high Prandtlnumber fluids
33 Concentration Profile Figures 14ndash19 show the concen-tration profiles obtained by the numerical simulations forvarious values of nondimensional parameters Bi Ha GrGc Kr Pr Nc Sc and 119878 In Figure 14 the effect of anapplied magnetic field is found to increase the concentrationboundary layer However it is interesting to note that theconcentration profiles decrease with the increase of both heatsource and chemical reaction parameters (Figures 14 and 15)Figures 16 and 17 reveal the concentration variations with Grand Gc respectively It is due to the fact that an increase in thevalues of Grashof number and modified Grashof number hasthe tendency to increase the mass buoyancy effect This gives
Figure 14 Concentration profiles for different values of Ha 119878 andKr for Gr = 01 Gc = 01 Sc = 062 Bi = 01 Nc = 001 andPr = 072
rise to an increase in the induced flow and thereby decreasesconcentration Figure 18 depicts the effect of Schmidt numberon the concentration Like temperature the concentrationvalue is higher at the surface and falls exponentiallyThe con-centration decreases with an increase in Sc and the decrease ismore with the increase in concentration parameter Figure 19displays the effect of Pr on concentration profile against 120578withthe variation of source and chemical reaction parametersThe magnitude of concentration is higher at the plate andthen decays to zero asymptotically this is due to the fact thatthermal conductivity of fluid decreases with the increase of Pr
Figure 16 Concentration profiles for different values of Gc 119878 andKr for Ha = 01 Gr = 01 Sc = 062 Bi = 01 Nc = 001 andPr = 072
resulting a decrease in thermal boundary layer thickness andsource term further influences the decrease of concentration
4 Conclusions
The present numerical study has been carried out for heatand mass transfer of MHD flow over a moving vertical platein presence of heat source and chemical reaction along withconvective surface boundary conditionThe shootingmethodwith Runge-Kutta fourth-order iteration scheme has beenimplemented to solve the dimensionless velocity thermaland mass boundary layer equations It has been shownthat the magnitude of local skin friction and local Nusseltnumber increase whereas the plate surface temperature and
Figure 18 Concentration profiles for different values of Sc 119878 andKr for Ha = 01 Gr = 01 Gc = 01 Bi = 01 Nc = 001 andPr = 072
Sherwood number decreases with an increase in sourceparameter The increase in the strength of chemical reactingsubstances causes an increase in the magnitude of localskin friction the plate surface temperature and Sherwoodnumber but opposite behavior is seen for local Nusseltnumber The velocity profile decreases by increasing themagnetic parameter and even the increase is more prominentwith the increase in source and chemical reaction parameterThe thermal boundary layer thickness increases with theincrease of source chemical reaction parameter plate surfaceconvective heat parameter and Schmidt number while themass flux boundary layer thickness decreases Moreover thethermal boundary layer thickness the mass boundary layerand velocity decrease as the Prandtl number increases
[1] K Vajrevelu and J Nayfeh ldquoHydromagnetic convection at acone and a wedgerdquo International Communications in Heat andMass Transfer vol 19 pp 701ndash710 1992
[2] A J Chamkha ldquoHydromagnetic three-dimensional free con-vection on a vertical stretching surface with heat generation orabsorptionrdquo International Journal of Heat and Fluid Flow vol20 no 1 pp 84ndash92 1999
[3] D Moalem ldquoSteady state heat transfer within porous mediumwith temperature dependent heat generationrdquo InternationalJournal of Heat and Mass Transfer vol 19 no 5 pp 529ndash5371976
[4] M M Rahman and M A Sattar ldquoMagnetohydrodynamicconvective flow of a micropolar fluid past a continuouslymoving vertical porous plate in the presence of heat genera-tionabsorptionrdquo Journal of Heat Transfer vol 128 no 2 pp142ndash152 2006
[5] U N Das R Deka and V M Soundalgekar ldquoEffects of masstransfer on flowpast an impulsively started infinite vertical platewith constant heat flux and chemical reactionrdquo Forschung imIngenieurwesenEngineering Research vol 60 no 10 pp 284ndash287 1994
[6] S P Anjalidevi andRKandasamy ldquoEffects of chemical reactionheat and mass transfer on laminar flow along a semi infinitehorizontal platerdquoHeat andMass Transfer vol 35 no 6 pp 465ndash467 1999
[7] M A Seddeek A A Darwish andM S Abdelmeguid ldquoEffectsof chemical reaction and variable viscosity on hydromagneticmixed convection heat and mass transfer for Hiemenz flowthrough porous media with radiationrdquo Communications inNonlinear Science and Numerical Simulation vol 12 no 2 pp195ndash213 2007
[8] P M Patil and P S Kulkarni ldquoEffects of chemical reaction onfree convective flowof a polar fluid through a porousmedium inthe presence of internal heat generationrdquo International Journalof Thermal Sciences vol 47 no 8 pp 1043ndash1054 2008
[9] A M Salem and M Abd El-Aziz ldquoEffect of Hall currents andchemical reaction on hydromagnetic flow of a stretching ver-tical surface with internal heat generationabsorptionrdquo AppliedMathematical Modelling vol 32 no 7 pp 1236ndash1254 2008
[10] F S IbrahimAM Elaiw andAA Bakr ldquoEffect of the chemicalreaction and radiation absorption on the unsteady MHDfree convection flow past a semi infinite vertical permeablemoving plate with heat source and suctionrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 6 pp1056ndash1066 2008
[11] S K Parida M Acharya G C Dash and S Panda ldquoMHD heatand mass transfer in a rotating system with periodic suctionrdquoArabian Journal for Science and Engineering vol 36 no 6 pp1139ndash1151 2011
[12] R Rajeswari B Jothiram andV K Nelson ldquoChemical reactionheat and mass transfer on nonlinear MHD boundary layer flowthrough a vertical porous surface in the presence of suctionrdquoAppliedMathematical Sciences vol 3 no 49-52 pp 2469ndash24802009
[13] A Mahdy ldquoEffect of chemical reaction and heat generationor absorption on double-diffusive convection from a verticaltruncated cone in porous media with variable viscosityrdquo Inter-national Communications in Heat andMass Transfer vol 37 no5 pp 548ndash554 2010
[14] D Pal and B Talukdar ldquoPerturbation analysis of unsteadymagnetohydrodynamic convective heat and mass transfer in aboundary layer slip flow past a vertical permeable plate withthermal radiation and chemical reactionrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 7 pp1813ndash1830 2010
[15] O D Makinde and A Ogulu ldquoThe effect of thermal radiationon the heat and mass transfer flow of a variable viscosity fluidpast a vertical porous plate permeated by a transverse magneticfieldrdquo Chemical Engineering Communications vol 195 no 12pp 1575ndash1584 2008
[16] O D Makinde ldquoMHD mixed-convection interaction withthermal radiation and nth order chemical reaction past avertical porous plate embedded in a porous mediumrdquo ChemicalEngineering Communications vol 198 no 4 pp 590ndash608 2011
[17] 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
[18] P O Olanrewaju O T Arulogun and K Adebimpe ldquoInternalheat generation effect on thermal boundary layer with a con-vective surface boundary conditionrdquo American Journal of FluidDynamics vol 2 no 1 pp 1ndash4 2012
[19] ODMakinde ldquoOnMHDheat andmass transfer over amovingvertical plate with a convective surface boundary conditionrdquoCanadian Journal of Chemical Engineering vol 88 no 6 pp983ndash990 2010
[20] O D Makinde ldquoSimilarity solution of hydromagnetic heat andmass transfer over a vertical plate with a convective surfaceboundary conditionrdquo International Journal of Physical Sciencesvol 5 no 6 pp 700ndash710 2010
[21] O D Makinde ldquoSimilarity solution for natural convectionfrom a moving vertical plate with internal heat generation anda convective boundary conditionrdquo Thermal Science vol 15supplement 1 pp S137ndashS143 2011
[22] K Gangadhar N B Reddy and P K Kameswaran ldquoSimilaritysolution of hydromagnetic heat andmass transfer over a verticalplate with convective surface boundary condition and chemicalreactionrdquo International Journal of Nonlinear Science vol 3 no3 pp 298ndash307 2012
[23] NFM Noor S Abbabandy and I Hasim ldquoHeat and MassTransfer of thermophoretic MHD flow over an inclined radiateisothermal permeable surface in presence of heat source sinkrdquoInternational Journal of Heat and Mass Transfer vol 55 no 7pp 2122ndash2128 2012
[24] V Bisht M Kumar and Z Uddin ldquoEffect of variable thermalconductivity and chemical reaction on steadymixed convectionboundary layer flow with heat and mass transfer inside a conedue to a pointrdquo Journal of Applied Fluid Mechanics vol 4 no 4pp 59ndash63 2011
[25] A A Bakr ldquoEffects of chemical reaction on MHD free convec-tion andmass transfer flowof amicropolar fluidwith oscillatoryplate velocity and constant heat source in a rotating frame ofreferencerdquoCommunications inNonlinear Science andNumericalSimulation vol 16 no 2 pp 698ndash710 2011
Figure 10 Temperature profiles for different values of Gc 119878 and Krfor Ha = 01 Gr = 01 Sc = 062 Bi = 01 Nc = 001 and Pr = 072
shows that thermal boundary layer thickness decreases asthe Prandtl number increases implying higher heat transferIt is due to that smaller values of Pr means increasing thethermal conductivity and therefore heat is able to diffuseaway from the plate more quickly than higher values of Prhence the rate of heat transfer is reduced It is noted thatowing to the presence of heat source effect (119878 gt 0) andchemical reaction parameter (Kr gt 0) the thermal stateof the fluid increases Hence the temperature of the fluidincreases within the boundary layers In the event that thestrength of the heat source and chemical reaction parametersare relatively large a remarkable change is observed in thetemperature profiles within the thermal boundary layer ascan be seen in Figure 13 Further the effect of heat generation
Figure 11 Temperature profiles for different values of Gr 119878 and KrforHa = 01 Gc = 01 119878c = 062 Bi = 01 Nc = 001 and Pr = 072
Sc = 062 119878 = 0 Kr = 0
Sc = 262 119878 = 0 Kr = 0
Sc = 062 119878 = 005 Kr = 005
Sc = 062 119878 = 005 Kr = 05
035
03
025
02
015
005
01
00 2 4 6 8 10
120578
120579(120578)
Figure 12 Temperature profiles for different values of Sc 119878 and Krfor Ha = 01 Gr = 01 Gc = 01 Bi = 01 Nc = 001 and Pr = 072
is more pronounced on temperature profiles for high Prandtlnumber fluids
33 Concentration Profile Figures 14ndash19 show the concen-tration profiles obtained by the numerical simulations forvarious values of nondimensional parameters Bi Ha GrGc Kr Pr Nc Sc and 119878 In Figure 14 the effect of anapplied magnetic field is found to increase the concentrationboundary layer However it is interesting to note that theconcentration profiles decrease with the increase of both heatsource and chemical reaction parameters (Figures 14 and 15)Figures 16 and 17 reveal the concentration variations with Grand Gc respectively It is due to the fact that an increase in thevalues of Grashof number and modified Grashof number hasthe tendency to increase the mass buoyancy effect This gives
Figure 14 Concentration profiles for different values of Ha 119878 andKr for Gr = 01 Gc = 01 Sc = 062 Bi = 01 Nc = 001 andPr = 072
rise to an increase in the induced flow and thereby decreasesconcentration Figure 18 depicts the effect of Schmidt numberon the concentration Like temperature the concentrationvalue is higher at the surface and falls exponentiallyThe con-centration decreases with an increase in Sc and the decrease ismore with the increase in concentration parameter Figure 19displays the effect of Pr on concentration profile against 120578withthe variation of source and chemical reaction parametersThe magnitude of concentration is higher at the plate andthen decays to zero asymptotically this is due to the fact thatthermal conductivity of fluid decreases with the increase of Pr
Figure 16 Concentration profiles for different values of Gc 119878 andKr for Ha = 01 Gr = 01 Sc = 062 Bi = 01 Nc = 001 andPr = 072
resulting a decrease in thermal boundary layer thickness andsource term further influences the decrease of concentration
4 Conclusions
The present numerical study has been carried out for heatand mass transfer of MHD flow over a moving vertical platein presence of heat source and chemical reaction along withconvective surface boundary conditionThe shootingmethodwith Runge-Kutta fourth-order iteration scheme has beenimplemented to solve the dimensionless velocity thermaland mass boundary layer equations It has been shownthat the magnitude of local skin friction and local Nusseltnumber increase whereas the plate surface temperature and
Figure 18 Concentration profiles for different values of Sc 119878 andKr for Ha = 01 Gr = 01 Gc = 01 Bi = 01 Nc = 001 andPr = 072
Sherwood number decreases with an increase in sourceparameter The increase in the strength of chemical reactingsubstances causes an increase in the magnitude of localskin friction the plate surface temperature and Sherwoodnumber but opposite behavior is seen for local Nusseltnumber The velocity profile decreases by increasing themagnetic parameter and even the increase is more prominentwith the increase in source and chemical reaction parameterThe thermal boundary layer thickness increases with theincrease of source chemical reaction parameter plate surfaceconvective heat parameter and Schmidt number while themass flux boundary layer thickness decreases Moreover thethermal boundary layer thickness the mass boundary layerand velocity decrease as the Prandtl number increases
[1] K Vajrevelu and J Nayfeh ldquoHydromagnetic convection at acone and a wedgerdquo International Communications in Heat andMass Transfer vol 19 pp 701ndash710 1992
[2] A J Chamkha ldquoHydromagnetic three-dimensional free con-vection on a vertical stretching surface with heat generation orabsorptionrdquo International Journal of Heat and Fluid Flow vol20 no 1 pp 84ndash92 1999
[3] D Moalem ldquoSteady state heat transfer within porous mediumwith temperature dependent heat generationrdquo InternationalJournal of Heat and Mass Transfer vol 19 no 5 pp 529ndash5371976
[4] M M Rahman and M A Sattar ldquoMagnetohydrodynamicconvective flow of a micropolar fluid past a continuouslymoving vertical porous plate in the presence of heat genera-tionabsorptionrdquo Journal of Heat Transfer vol 128 no 2 pp142ndash152 2006
[5] U N Das R Deka and V M Soundalgekar ldquoEffects of masstransfer on flowpast an impulsively started infinite vertical platewith constant heat flux and chemical reactionrdquo Forschung imIngenieurwesenEngineering Research vol 60 no 10 pp 284ndash287 1994
[6] S P Anjalidevi andRKandasamy ldquoEffects of chemical reactionheat and mass transfer on laminar flow along a semi infinitehorizontal platerdquoHeat andMass Transfer vol 35 no 6 pp 465ndash467 1999
[7] M A Seddeek A A Darwish andM S Abdelmeguid ldquoEffectsof chemical reaction and variable viscosity on hydromagneticmixed convection heat and mass transfer for Hiemenz flowthrough porous media with radiationrdquo Communications inNonlinear Science and Numerical Simulation vol 12 no 2 pp195ndash213 2007
[8] P M Patil and P S Kulkarni ldquoEffects of chemical reaction onfree convective flowof a polar fluid through a porousmedium inthe presence of internal heat generationrdquo International Journalof Thermal Sciences vol 47 no 8 pp 1043ndash1054 2008
[9] A M Salem and M Abd El-Aziz ldquoEffect of Hall currents andchemical reaction on hydromagnetic flow of a stretching ver-tical surface with internal heat generationabsorptionrdquo AppliedMathematical Modelling vol 32 no 7 pp 1236ndash1254 2008
[10] F S IbrahimAM Elaiw andAA Bakr ldquoEffect of the chemicalreaction and radiation absorption on the unsteady MHDfree convection flow past a semi infinite vertical permeablemoving plate with heat source and suctionrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 6 pp1056ndash1066 2008
[11] S K Parida M Acharya G C Dash and S Panda ldquoMHD heatand mass transfer in a rotating system with periodic suctionrdquoArabian Journal for Science and Engineering vol 36 no 6 pp1139ndash1151 2011
[12] R Rajeswari B Jothiram andV K Nelson ldquoChemical reactionheat and mass transfer on nonlinear MHD boundary layer flowthrough a vertical porous surface in the presence of suctionrdquoAppliedMathematical Sciences vol 3 no 49-52 pp 2469ndash24802009
[13] A Mahdy ldquoEffect of chemical reaction and heat generationor absorption on double-diffusive convection from a verticaltruncated cone in porous media with variable viscosityrdquo Inter-national Communications in Heat andMass Transfer vol 37 no5 pp 548ndash554 2010
[14] D Pal and B Talukdar ldquoPerturbation analysis of unsteadymagnetohydrodynamic convective heat and mass transfer in aboundary layer slip flow past a vertical permeable plate withthermal radiation and chemical reactionrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 7 pp1813ndash1830 2010
[15] O D Makinde and A Ogulu ldquoThe effect of thermal radiationon the heat and mass transfer flow of a variable viscosity fluidpast a vertical porous plate permeated by a transverse magneticfieldrdquo Chemical Engineering Communications vol 195 no 12pp 1575ndash1584 2008
[16] O D Makinde ldquoMHD mixed-convection interaction withthermal radiation and nth order chemical reaction past avertical porous plate embedded in a porous mediumrdquo ChemicalEngineering Communications vol 198 no 4 pp 590ndash608 2011
[17] 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
[18] P O Olanrewaju O T Arulogun and K Adebimpe ldquoInternalheat generation effect on thermal boundary layer with a con-vective surface boundary conditionrdquo American Journal of FluidDynamics vol 2 no 1 pp 1ndash4 2012
[19] ODMakinde ldquoOnMHDheat andmass transfer over amovingvertical plate with a convective surface boundary conditionrdquoCanadian Journal of Chemical Engineering vol 88 no 6 pp983ndash990 2010
[20] O D Makinde ldquoSimilarity solution of hydromagnetic heat andmass transfer over a vertical plate with a convective surfaceboundary conditionrdquo International Journal of Physical Sciencesvol 5 no 6 pp 700ndash710 2010
[21] O D Makinde ldquoSimilarity solution for natural convectionfrom a moving vertical plate with internal heat generation anda convective boundary conditionrdquo Thermal Science vol 15supplement 1 pp S137ndashS143 2011
[22] K Gangadhar N B Reddy and P K Kameswaran ldquoSimilaritysolution of hydromagnetic heat andmass transfer over a verticalplate with convective surface boundary condition and chemicalreactionrdquo International Journal of Nonlinear Science vol 3 no3 pp 298ndash307 2012
[23] NFM Noor S Abbabandy and I Hasim ldquoHeat and MassTransfer of thermophoretic MHD flow over an inclined radiateisothermal permeable surface in presence of heat source sinkrdquoInternational Journal of Heat and Mass Transfer vol 55 no 7pp 2122ndash2128 2012
[24] V Bisht M Kumar and Z Uddin ldquoEffect of variable thermalconductivity and chemical reaction on steadymixed convectionboundary layer flow with heat and mass transfer inside a conedue to a pointrdquo Journal of Applied Fluid Mechanics vol 4 no 4pp 59ndash63 2011
[25] A A Bakr ldquoEffects of chemical reaction on MHD free convec-tion andmass transfer flowof amicropolar fluidwith oscillatoryplate velocity and constant heat source in a rotating frame ofreferencerdquoCommunications inNonlinear Science andNumericalSimulation vol 16 no 2 pp 698ndash710 2011
Figure 14 Concentration profiles for different values of Ha 119878 andKr for Gr = 01 Gc = 01 Sc = 062 Bi = 01 Nc = 001 andPr = 072
rise to an increase in the induced flow and thereby decreasesconcentration Figure 18 depicts the effect of Schmidt numberon the concentration Like temperature the concentrationvalue is higher at the surface and falls exponentiallyThe con-centration decreases with an increase in Sc and the decrease ismore with the increase in concentration parameter Figure 19displays the effect of Pr on concentration profile against 120578withthe variation of source and chemical reaction parametersThe magnitude of concentration is higher at the plate andthen decays to zero asymptotically this is due to the fact thatthermal conductivity of fluid decreases with the increase of Pr
Figure 16 Concentration profiles for different values of Gc 119878 andKr for Ha = 01 Gr = 01 Sc = 062 Bi = 01 Nc = 001 andPr = 072
resulting a decrease in thermal boundary layer thickness andsource term further influences the decrease of concentration
4 Conclusions
The present numerical study has been carried out for heatand mass transfer of MHD flow over a moving vertical platein presence of heat source and chemical reaction along withconvective surface boundary conditionThe shootingmethodwith Runge-Kutta fourth-order iteration scheme has beenimplemented to solve the dimensionless velocity thermaland mass boundary layer equations It has been shownthat the magnitude of local skin friction and local Nusseltnumber increase whereas the plate surface temperature and
Figure 18 Concentration profiles for different values of Sc 119878 andKr for Ha = 01 Gr = 01 Gc = 01 Bi = 01 Nc = 001 andPr = 072
Sherwood number decreases with an increase in sourceparameter The increase in the strength of chemical reactingsubstances causes an increase in the magnitude of localskin friction the plate surface temperature and Sherwoodnumber but opposite behavior is seen for local Nusseltnumber The velocity profile decreases by increasing themagnetic parameter and even the increase is more prominentwith the increase in source and chemical reaction parameterThe thermal boundary layer thickness increases with theincrease of source chemical reaction parameter plate surfaceconvective heat parameter and Schmidt number while themass flux boundary layer thickness decreases Moreover thethermal boundary layer thickness the mass boundary layerand velocity decrease as the Prandtl number increases
[1] K Vajrevelu and J Nayfeh ldquoHydromagnetic convection at acone and a wedgerdquo International Communications in Heat andMass Transfer vol 19 pp 701ndash710 1992
[2] A J Chamkha ldquoHydromagnetic three-dimensional free con-vection on a vertical stretching surface with heat generation orabsorptionrdquo International Journal of Heat and Fluid Flow vol20 no 1 pp 84ndash92 1999
[3] D Moalem ldquoSteady state heat transfer within porous mediumwith temperature dependent heat generationrdquo InternationalJournal of Heat and Mass Transfer vol 19 no 5 pp 529ndash5371976
[4] M M Rahman and M A Sattar ldquoMagnetohydrodynamicconvective flow of a micropolar fluid past a continuouslymoving vertical porous plate in the presence of heat genera-tionabsorptionrdquo Journal of Heat Transfer vol 128 no 2 pp142ndash152 2006
[5] U N Das R Deka and V M Soundalgekar ldquoEffects of masstransfer on flowpast an impulsively started infinite vertical platewith constant heat flux and chemical reactionrdquo Forschung imIngenieurwesenEngineering Research vol 60 no 10 pp 284ndash287 1994
[6] S P Anjalidevi andRKandasamy ldquoEffects of chemical reactionheat and mass transfer on laminar flow along a semi infinitehorizontal platerdquoHeat andMass Transfer vol 35 no 6 pp 465ndash467 1999
[7] M A Seddeek A A Darwish andM S Abdelmeguid ldquoEffectsof chemical reaction and variable viscosity on hydromagneticmixed convection heat and mass transfer for Hiemenz flowthrough porous media with radiationrdquo Communications inNonlinear Science and Numerical Simulation vol 12 no 2 pp195ndash213 2007
[8] P M Patil and P S Kulkarni ldquoEffects of chemical reaction onfree convective flowof a polar fluid through a porousmedium inthe presence of internal heat generationrdquo International Journalof Thermal Sciences vol 47 no 8 pp 1043ndash1054 2008
[9] A M Salem and M Abd El-Aziz ldquoEffect of Hall currents andchemical reaction on hydromagnetic flow of a stretching ver-tical surface with internal heat generationabsorptionrdquo AppliedMathematical Modelling vol 32 no 7 pp 1236ndash1254 2008
[10] F S IbrahimAM Elaiw andAA Bakr ldquoEffect of the chemicalreaction and radiation absorption on the unsteady MHDfree convection flow past a semi infinite vertical permeablemoving plate with heat source and suctionrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 6 pp1056ndash1066 2008
[11] S K Parida M Acharya G C Dash and S Panda ldquoMHD heatand mass transfer in a rotating system with periodic suctionrdquoArabian Journal for Science and Engineering vol 36 no 6 pp1139ndash1151 2011
[12] R Rajeswari B Jothiram andV K Nelson ldquoChemical reactionheat and mass transfer on nonlinear MHD boundary layer flowthrough a vertical porous surface in the presence of suctionrdquoAppliedMathematical Sciences vol 3 no 49-52 pp 2469ndash24802009
[13] A Mahdy ldquoEffect of chemical reaction and heat generationor absorption on double-diffusive convection from a verticaltruncated cone in porous media with variable viscosityrdquo Inter-national Communications in Heat andMass Transfer vol 37 no5 pp 548ndash554 2010
[14] D Pal and B Talukdar ldquoPerturbation analysis of unsteadymagnetohydrodynamic convective heat and mass transfer in aboundary layer slip flow past a vertical permeable plate withthermal radiation and chemical reactionrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 7 pp1813ndash1830 2010
[15] O D Makinde and A Ogulu ldquoThe effect of thermal radiationon the heat and mass transfer flow of a variable viscosity fluidpast a vertical porous plate permeated by a transverse magneticfieldrdquo Chemical Engineering Communications vol 195 no 12pp 1575ndash1584 2008
[16] O D Makinde ldquoMHD mixed-convection interaction withthermal radiation and nth order chemical reaction past avertical porous plate embedded in a porous mediumrdquo ChemicalEngineering Communications vol 198 no 4 pp 590ndash608 2011
[17] 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
[18] P O Olanrewaju O T Arulogun and K Adebimpe ldquoInternalheat generation effect on thermal boundary layer with a con-vective surface boundary conditionrdquo American Journal of FluidDynamics vol 2 no 1 pp 1ndash4 2012
[19] ODMakinde ldquoOnMHDheat andmass transfer over amovingvertical plate with a convective surface boundary conditionrdquoCanadian Journal of Chemical Engineering vol 88 no 6 pp983ndash990 2010
[20] O D Makinde ldquoSimilarity solution of hydromagnetic heat andmass transfer over a vertical plate with a convective surfaceboundary conditionrdquo International Journal of Physical Sciencesvol 5 no 6 pp 700ndash710 2010
[21] O D Makinde ldquoSimilarity solution for natural convectionfrom a moving vertical plate with internal heat generation anda convective boundary conditionrdquo Thermal Science vol 15supplement 1 pp S137ndashS143 2011
[22] K Gangadhar N B Reddy and P K Kameswaran ldquoSimilaritysolution of hydromagnetic heat andmass transfer over a verticalplate with convective surface boundary condition and chemicalreactionrdquo International Journal of Nonlinear Science vol 3 no3 pp 298ndash307 2012
[23] NFM Noor S Abbabandy and I Hasim ldquoHeat and MassTransfer of thermophoretic MHD flow over an inclined radiateisothermal permeable surface in presence of heat source sinkrdquoInternational Journal of Heat and Mass Transfer vol 55 no 7pp 2122ndash2128 2012
[24] V Bisht M Kumar and Z Uddin ldquoEffect of variable thermalconductivity and chemical reaction on steadymixed convectionboundary layer flow with heat and mass transfer inside a conedue to a pointrdquo Journal of Applied Fluid Mechanics vol 4 no 4pp 59ndash63 2011
[25] A A Bakr ldquoEffects of chemical reaction on MHD free convec-tion andmass transfer flowof amicropolar fluidwith oscillatoryplate velocity and constant heat source in a rotating frame ofreferencerdquoCommunications inNonlinear Science andNumericalSimulation vol 16 no 2 pp 698ndash710 2011
Figure 18 Concentration profiles for different values of Sc 119878 andKr for Ha = 01 Gr = 01 Gc = 01 Bi = 01 Nc = 001 andPr = 072
Sherwood number decreases with an increase in sourceparameter The increase in the strength of chemical reactingsubstances causes an increase in the magnitude of localskin friction the plate surface temperature and Sherwoodnumber but opposite behavior is seen for local Nusseltnumber The velocity profile decreases by increasing themagnetic parameter and even the increase is more prominentwith the increase in source and chemical reaction parameterThe thermal boundary layer thickness increases with theincrease of source chemical reaction parameter plate surfaceconvective heat parameter and Schmidt number while themass flux boundary layer thickness decreases Moreover thethermal boundary layer thickness the mass boundary layerand velocity decrease as the Prandtl number increases
[1] K Vajrevelu and J Nayfeh ldquoHydromagnetic convection at acone and a wedgerdquo International Communications in Heat andMass Transfer vol 19 pp 701ndash710 1992
[2] A J Chamkha ldquoHydromagnetic three-dimensional free con-vection on a vertical stretching surface with heat generation orabsorptionrdquo International Journal of Heat and Fluid Flow vol20 no 1 pp 84ndash92 1999
[3] D Moalem ldquoSteady state heat transfer within porous mediumwith temperature dependent heat generationrdquo InternationalJournal of Heat and Mass Transfer vol 19 no 5 pp 529ndash5371976
[4] M M Rahman and M A Sattar ldquoMagnetohydrodynamicconvective flow of a micropolar fluid past a continuouslymoving vertical porous plate in the presence of heat genera-tionabsorptionrdquo Journal of Heat Transfer vol 128 no 2 pp142ndash152 2006
[5] U N Das R Deka and V M Soundalgekar ldquoEffects of masstransfer on flowpast an impulsively started infinite vertical platewith constant heat flux and chemical reactionrdquo Forschung imIngenieurwesenEngineering Research vol 60 no 10 pp 284ndash287 1994
[6] S P Anjalidevi andRKandasamy ldquoEffects of chemical reactionheat and mass transfer on laminar flow along a semi infinitehorizontal platerdquoHeat andMass Transfer vol 35 no 6 pp 465ndash467 1999
[7] M A Seddeek A A Darwish andM S Abdelmeguid ldquoEffectsof chemical reaction and variable viscosity on hydromagneticmixed convection heat and mass transfer for Hiemenz flowthrough porous media with radiationrdquo Communications inNonlinear Science and Numerical Simulation vol 12 no 2 pp195ndash213 2007
[8] P M Patil and P S Kulkarni ldquoEffects of chemical reaction onfree convective flowof a polar fluid through a porousmedium inthe presence of internal heat generationrdquo International Journalof Thermal Sciences vol 47 no 8 pp 1043ndash1054 2008
[9] A M Salem and M Abd El-Aziz ldquoEffect of Hall currents andchemical reaction on hydromagnetic flow of a stretching ver-tical surface with internal heat generationabsorptionrdquo AppliedMathematical Modelling vol 32 no 7 pp 1236ndash1254 2008
[10] F S IbrahimAM Elaiw andAA Bakr ldquoEffect of the chemicalreaction and radiation absorption on the unsteady MHDfree convection flow past a semi infinite vertical permeablemoving plate with heat source and suctionrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 6 pp1056ndash1066 2008
[11] S K Parida M Acharya G C Dash and S Panda ldquoMHD heatand mass transfer in a rotating system with periodic suctionrdquoArabian Journal for Science and Engineering vol 36 no 6 pp1139ndash1151 2011
[12] R Rajeswari B Jothiram andV K Nelson ldquoChemical reactionheat and mass transfer on nonlinear MHD boundary layer flowthrough a vertical porous surface in the presence of suctionrdquoAppliedMathematical Sciences vol 3 no 49-52 pp 2469ndash24802009
[13] A Mahdy ldquoEffect of chemical reaction and heat generationor absorption on double-diffusive convection from a verticaltruncated cone in porous media with variable viscosityrdquo Inter-national Communications in Heat andMass Transfer vol 37 no5 pp 548ndash554 2010
[14] D Pal and B Talukdar ldquoPerturbation analysis of unsteadymagnetohydrodynamic convective heat and mass transfer in aboundary layer slip flow past a vertical permeable plate withthermal radiation and chemical reactionrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 7 pp1813ndash1830 2010
[15] O D Makinde and A Ogulu ldquoThe effect of thermal radiationon the heat and mass transfer flow of a variable viscosity fluidpast a vertical porous plate permeated by a transverse magneticfieldrdquo Chemical Engineering Communications vol 195 no 12pp 1575ndash1584 2008
[16] O D Makinde ldquoMHD mixed-convection interaction withthermal radiation and nth order chemical reaction past avertical porous plate embedded in a porous mediumrdquo ChemicalEngineering Communications vol 198 no 4 pp 590ndash608 2011
[17] 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
[18] P O Olanrewaju O T Arulogun and K Adebimpe ldquoInternalheat generation effect on thermal boundary layer with a con-vective surface boundary conditionrdquo American Journal of FluidDynamics vol 2 no 1 pp 1ndash4 2012
[19] ODMakinde ldquoOnMHDheat andmass transfer over amovingvertical plate with a convective surface boundary conditionrdquoCanadian Journal of Chemical Engineering vol 88 no 6 pp983ndash990 2010
[20] O D Makinde ldquoSimilarity solution of hydromagnetic heat andmass transfer over a vertical plate with a convective surfaceboundary conditionrdquo International Journal of Physical Sciencesvol 5 no 6 pp 700ndash710 2010
[21] O D Makinde ldquoSimilarity solution for natural convectionfrom a moving vertical plate with internal heat generation anda convective boundary conditionrdquo Thermal Science vol 15supplement 1 pp S137ndashS143 2011
[22] K Gangadhar N B Reddy and P K Kameswaran ldquoSimilaritysolution of hydromagnetic heat andmass transfer over a verticalplate with convective surface boundary condition and chemicalreactionrdquo International Journal of Nonlinear Science vol 3 no3 pp 298ndash307 2012
[23] NFM Noor S Abbabandy and I Hasim ldquoHeat and MassTransfer of thermophoretic MHD flow over an inclined radiateisothermal permeable surface in presence of heat source sinkrdquoInternational Journal of Heat and Mass Transfer vol 55 no 7pp 2122ndash2128 2012
[24] V Bisht M Kumar and Z Uddin ldquoEffect of variable thermalconductivity and chemical reaction on steadymixed convectionboundary layer flow with heat and mass transfer inside a conedue to a pointrdquo Journal of Applied Fluid Mechanics vol 4 no 4pp 59ndash63 2011
[25] A A Bakr ldquoEffects of chemical reaction on MHD free convec-tion andmass transfer flowof amicropolar fluidwith oscillatoryplate velocity and constant heat source in a rotating frame ofreferencerdquoCommunications inNonlinear Science andNumericalSimulation vol 16 no 2 pp 698ndash710 2011
[1] K Vajrevelu and J Nayfeh ldquoHydromagnetic convection at acone and a wedgerdquo International Communications in Heat andMass Transfer vol 19 pp 701ndash710 1992
[2] A J Chamkha ldquoHydromagnetic three-dimensional free con-vection on a vertical stretching surface with heat generation orabsorptionrdquo International Journal of Heat and Fluid Flow vol20 no 1 pp 84ndash92 1999
[3] D Moalem ldquoSteady state heat transfer within porous mediumwith temperature dependent heat generationrdquo InternationalJournal of Heat and Mass Transfer vol 19 no 5 pp 529ndash5371976
[4] M M Rahman and M A Sattar ldquoMagnetohydrodynamicconvective flow of a micropolar fluid past a continuouslymoving vertical porous plate in the presence of heat genera-tionabsorptionrdquo Journal of Heat Transfer vol 128 no 2 pp142ndash152 2006
[5] U N Das R Deka and V M Soundalgekar ldquoEffects of masstransfer on flowpast an impulsively started infinite vertical platewith constant heat flux and chemical reactionrdquo Forschung imIngenieurwesenEngineering Research vol 60 no 10 pp 284ndash287 1994
[6] S P Anjalidevi andRKandasamy ldquoEffects of chemical reactionheat and mass transfer on laminar flow along a semi infinitehorizontal platerdquoHeat andMass Transfer vol 35 no 6 pp 465ndash467 1999
[7] M A Seddeek A A Darwish andM S Abdelmeguid ldquoEffectsof chemical reaction and variable viscosity on hydromagneticmixed convection heat and mass transfer for Hiemenz flowthrough porous media with radiationrdquo Communications inNonlinear Science and Numerical Simulation vol 12 no 2 pp195ndash213 2007
[8] P M Patil and P S Kulkarni ldquoEffects of chemical reaction onfree convective flowof a polar fluid through a porousmedium inthe presence of internal heat generationrdquo International Journalof Thermal Sciences vol 47 no 8 pp 1043ndash1054 2008
[9] A M Salem and M Abd El-Aziz ldquoEffect of Hall currents andchemical reaction on hydromagnetic flow of a stretching ver-tical surface with internal heat generationabsorptionrdquo AppliedMathematical Modelling vol 32 no 7 pp 1236ndash1254 2008
[10] F S IbrahimAM Elaiw andAA Bakr ldquoEffect of the chemicalreaction and radiation absorption on the unsteady MHDfree convection flow past a semi infinite vertical permeablemoving plate with heat source and suctionrdquo Communications inNonlinear Science and Numerical Simulation vol 13 no 6 pp1056ndash1066 2008
[11] S K Parida M Acharya G C Dash and S Panda ldquoMHD heatand mass transfer in a rotating system with periodic suctionrdquoArabian Journal for Science and Engineering vol 36 no 6 pp1139ndash1151 2011
[12] R Rajeswari B Jothiram andV K Nelson ldquoChemical reactionheat and mass transfer on nonlinear MHD boundary layer flowthrough a vertical porous surface in the presence of suctionrdquoAppliedMathematical Sciences vol 3 no 49-52 pp 2469ndash24802009
[13] A Mahdy ldquoEffect of chemical reaction and heat generationor absorption on double-diffusive convection from a verticaltruncated cone in porous media with variable viscosityrdquo Inter-national Communications in Heat andMass Transfer vol 37 no5 pp 548ndash554 2010
[14] D Pal and B Talukdar ldquoPerturbation analysis of unsteadymagnetohydrodynamic convective heat and mass transfer in aboundary layer slip flow past a vertical permeable plate withthermal radiation and chemical reactionrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 7 pp1813ndash1830 2010
[15] O D Makinde and A Ogulu ldquoThe effect of thermal radiationon the heat and mass transfer flow of a variable viscosity fluidpast a vertical porous plate permeated by a transverse magneticfieldrdquo Chemical Engineering Communications vol 195 no 12pp 1575ndash1584 2008
[16] O D Makinde ldquoMHD mixed-convection interaction withthermal radiation and nth order chemical reaction past avertical porous plate embedded in a porous mediumrdquo ChemicalEngineering Communications vol 198 no 4 pp 590ndash608 2011
[17] 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
[18] P O Olanrewaju O T Arulogun and K Adebimpe ldquoInternalheat generation effect on thermal boundary layer with a con-vective surface boundary conditionrdquo American Journal of FluidDynamics vol 2 no 1 pp 1ndash4 2012
[19] ODMakinde ldquoOnMHDheat andmass transfer over amovingvertical plate with a convective surface boundary conditionrdquoCanadian Journal of Chemical Engineering vol 88 no 6 pp983ndash990 2010
[20] O D Makinde ldquoSimilarity solution of hydromagnetic heat andmass transfer over a vertical plate with a convective surfaceboundary conditionrdquo International Journal of Physical Sciencesvol 5 no 6 pp 700ndash710 2010
[21] O D Makinde ldquoSimilarity solution for natural convectionfrom a moving vertical plate with internal heat generation anda convective boundary conditionrdquo Thermal Science vol 15supplement 1 pp S137ndashS143 2011
[22] K Gangadhar N B Reddy and P K Kameswaran ldquoSimilaritysolution of hydromagnetic heat andmass transfer over a verticalplate with convective surface boundary condition and chemicalreactionrdquo International Journal of Nonlinear Science vol 3 no3 pp 298ndash307 2012
[23] NFM Noor S Abbabandy and I Hasim ldquoHeat and MassTransfer of thermophoretic MHD flow over an inclined radiateisothermal permeable surface in presence of heat source sinkrdquoInternational Journal of Heat and Mass Transfer vol 55 no 7pp 2122ndash2128 2012
[24] V Bisht M Kumar and Z Uddin ldquoEffect of variable thermalconductivity and chemical reaction on steadymixed convectionboundary layer flow with heat and mass transfer inside a conedue to a pointrdquo Journal of Applied Fluid Mechanics vol 4 no 4pp 59ndash63 2011
[25] A A Bakr ldquoEffects of chemical reaction on MHD free convec-tion andmass transfer flowof amicropolar fluidwith oscillatoryplate velocity and constant heat source in a rotating frame ofreferencerdquoCommunications inNonlinear Science andNumericalSimulation vol 16 no 2 pp 698ndash710 2011
