Research ArticleRadiation Effects in Flow through Porous Medium overa Rotating Disk with Variable Fluid Properties
Shalini Jain and Shweta Bohra
Department of Mathematics amp Statistics Manipal University Jaipur Jaipur 303007 India
Correspondence should be addressed to Shalini Jain shalinijainjaipurmanipaledu
Received 30 June 2016 Revised 3 September 2016 Accepted 15 September 2016
Academic Editor Ali Cemal Benim
Copyright copy 2016 S Jain and S BohraThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The present study investigates the radiation effects in flow through porous medium over a permeable rotating disk with velocityslip and temperature jump Fluid properties density (120588) viscosity (120583) and thermal conductivity (120581) are taken to be dependenton temperature Particular case considering these fluid propertiesrsquo constant is also discussed The governing partial differentialequations are converted into nonlinear normal differential equation using similarity alterations Transformed system of equations issolved numerically by usingRunge-Kuttamethodwith shooting technique Effects of various parameters such as porosity parameter119870 suction parameter 119882119904 rotational Reynolds number Re Knudsen number Kn Prandtl number Pr radiation parameter 119873and relative temperature difference parameter 120576 on velocity profiles along radial tangential and axial direction and temperaturedistribution are investigated for both variable fluid properties and constant fluid properties Results obtained are analyzed anddepicted through graphs and table
1 Introduction
The study of rotating disk flows of electrically conducting flu-ids has practical applications in many areas such as rotatingmachinery lubrication computer storage devices oceanog-raphy viscometry and crystal growth processes In 1921Karman [1] was the first to investigate fluid flowdue to a rotat-ing disk He introduced similarity transformations to trans-form governing partial differential equations into ordinarydifferential equations Further Cochran [2] Benton [3] andTurkyilmazoglu [4] extended the work [1] and investigatedflow and heat transfer under different boundary conditionsChauhan and Gupta [5] investigated steady flow and heattransfer between two stationary naturally permeable disksHeat transfer from a rotating disk by convection has beeninvestigated theoretically under different physical and ther-mal conditions by Wagner [6] Millsaps and Pohlhausen[7] Kreith and Taylor [8] H-T Lin and L-K Lin [9] andVerma and Chauhan [10] Chauhan and Jain [11] studied flowbetween rotating disks they considered highly permeabledisk Turkyilmazoglu [12 13] investigated flow and heat
transfer of nanofluid due to rotating disk and on a radiallyshrinking rotating disk in the presence of a uniform verticalmagnetic field respectively The unsteady magnetohydrody-namic (MHD) squeezing flow between two parallel disks(which is filled with nanofluid) is considered by Azimiand Riazi [14] they have used Galerkin optimal homo-topy asymptotic method (GOHAM) to solve the problemSeveral researchers [15ndash17] have also investigated unsteadyfluid flow and heat transfer over permeable rotating diskrotating porous disk and infinite rotating disk under differentboundary conditions
Radiative effects have several applications in physics andengineering field Radiative heat transfer phenomena areused in nuclear reactors power generation system and hightemperature plasma on controlling heating factor in indus-tries and in liquid metal fluids Several researchers investi-gated the effects of radiation on convective flows Mansour[18] and Hossain et al [19] studied the effect of radiation onfree convection of fluid from a vertical plate and porous ver-tical plate respectively Raptis and Perdikis [20] investigatedthe MHD free convection flow in the presence of thermal
Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2016 Article ID 9671513 12 pageshttpdxdoiorg10115520169671513
2 Advances in Mathematical Physics
radiationThe investigation of the effect of radiation onmixedconvection flow of an optically dense viscous incompressiblefluid along a heated vertical flat plate with uniform freestream and uniform surface temperature has been done byHossain and Takhar [21] A Devi and R U Devi [22] studiedthermal radiation effect onMHD flow over a rotating infinitenonporous disk She also investigated porous rotating diskwith Hall effect
In recent years the slip flow regime has been widely stud-ied and researchers have been concentrating on the analysisof microscale in microelectromechanical systems (MEMS)associated with the embodiment of velocity slip and temper-ature jump Wang [23] examined the flow due to a stretch-ing boundary with partial slipmdashan exact solution of theNavier-Stokes equations Osalusi [24] studied the combinedhydromagnetic and slip flow of a steady laminar conductingviscous fluid in the presence of thermal radiation due to animpulsively started rotating porous disk with the variablefluid properties Khidir [25] investigated the effects of viscousdissipation and ohmic heating on steady MHD convectiveflow due to porous rotating disk taking into account thevariable fluid properties Sparrow et al [26] considered thefluid flow due to the rotation of a porous surfaced disk andemployed a set of linear slip flow conditions He observedthat a substantial reduction in torque occurred as a resultof surface slip Rashidi and Freidooni Mehr [27] investigatedeffects of velocity slip and temperature jump on the flow overa porous rotating disk The combined effects of temperatureand velocity jump on the heat transfer fluid flow and entropygeneration over a single rotating disk have been examined byArikoglu et al [28] Hayat et al [29] investigatedMHD steadyflow of viscous nanofluid due to a rotating disk with partialslip
In most of the research the fluid properties such asdensity (120588) viscosity (120583) and thermal conductivity (120581) areassumed to be constant However these properties remainunaltered if and only if temperature remains the same ordoes not change rapidly Therefore to predict flow behavioraccurately it is essential to consider variable fluid propertiesZakerullah and Ackroyd [30] analyzed the laminar naturalconvection boundary layer flow on a horizontal circular diskwith variable properties Herwig [31] and Herwig and Klemp[32] have extended the work [30] They have investigatedeffects of variable properties in a tube and concentric annulirespectively at constant heat flux FurtherMaleque and Sattar[33 34] have investigated laminar convective flow taking intoaccount the variable properties due to a porous rotating diskThey solved the problems numerically using Runge-Kuttamethod with shooting technique This work was extendedby Osalusi and Sibanda [35] They have studied flow in thepresence of magnetic field Rahman [36] made a study onthe slip flow with variable properties due to a porous rotatingdisk
Rashidi et al [37] obtained analytical solutions flow andheat transfer over a rotating disk in porous medium Hussainet al [38] obtained numerical solution of a disk rotating in
x
y
z
Ω
Tw
ru
w
Tinfin
pinfin
Figure 1 Coordinate system for the rotating disk flow
a viscous fluid Rashidi et al [39 40] analyzed entropy genera-tion in a MHD flow over a rotating porous disk with variablephysical properties They have also investigated fluid flowover a permeable rotating disk in the presence of Soret andDufour effect Alam et al [41] investigated thermophoreticdeposition of micron sized particles on flow due to rotatingdisk
The main goal of the present study is to investigateradiation effects in the steady flow over a rotating permeabledisk in porous medium with velocity slip and temperaturejump To predict the flow behavior accurately variable ther-mophysical properties are taken into consideration To thebest of the authorrsquos knowledge radiation effects of flow overrotating disk with velocity and temperature slip with variablethermal properties have not been studied yet The noveltyof present paper is to investigate flow and heat transferfor variable fluid properties with velocity slip and tempera-ture jump taken into consideration Also combined effectsfor both variable and constant fluid properties for variousphysical parameters on flow and heat transfer have beenobtained and depicted graphically which gives more insightabout fluid flow (Figure 1)
2 Mathematical Formulation
Consider a steady slip flow due to permeable rotating diskthrough porous medium Assume disk of 119903 radius is rotatingwith constant angular velocity Ω and placed at 119911 = 0 incylindrical polar coordinates (119903 120601 119911) where 119903 and 120601 are theradial and tangential axis respectively and 119911 is the verticalaxis
Let 119906 V and 119908 be the components of the fluid velocityin the direction of 119903 120601 and 119911 respectively 119901 and 119879 are thepressure and temperature of the fluidThe surface of the rotat-ing disk is considered at a uniform temperature 119879119908 Pressureand temperature for ambient fluid are 119901infin and 119879infin (119879119908 gt119879infin)
Advances in Mathematical Physics 3
Following [34 35 42] the fluid properties viscosity (120583)thermal conductivity (120581) and the density (120588) are taken asfunctions of temperature
120583 = 120583infin ( 119879119879infin)119886
120581 = 120581infin ( 119879119879infin)119887
120588 = 120588infin ( 119879119879infin)119888
(1)
where 119886 119887 and 119888 are arbitrary exponents and 120583infin 120581infin and 120588infinare the uniformviscosity thermal conductivity anddensity ofthe fluid Assume 119886 = 07 119887 = 083 and 119888 = minus10 (ideal gas)are the values of exponents for present investigation
The governing equations of continuity momentum andenergy for laminar incompressible flow in cylindrical coordi-nates are [35]
at 119911 997888rarr infin119906 997888rarr infinV 997888rarr infin119879 997888rarr 119879infin
(7)
where 120590V is the tangential momentum accommodation coef-ficient 120590119905 is the energy accommodation coefficient 120582 is meanfree path 120573 is the ratio of specific heats and 119879119904 is thetemperature of the fluid near to the disk surface
Rosseland approximation has been used for radiation 119902119903
where1205901 is the Stefan-Boltzmann constant and1198794 is themeanabsorption coefficient It is assumed that the temperaturedifferences within the flow are sufficiently small so that theterm1198794may be expressed as a linear function of temperatureThis is done by expanding 1198794 in a Taylor series about 119879infin andomitting the second- and higher-order terms leads to
1198794 cong 41198793infin119879 minus 31198794infin120597119902119903120597119911 = 120597120597119911 (minus4120590131198961
1205971198794120597119911 )
= 120597120597119911 (minus4120590131198961120597 (41198793infin119879 minus 31198794infin)120597119911 )
119901 minus 119901infin = minus120583infinΩ119875 (120578) 120579 (120578) = (119879 minus 119879infin)(119879119908 minus 119879infin)
(10)
where ]infin is uniformkinematic viscosity and119865119866119867 120579 and119875are nondimensional functions in terms of vertical coordinate120578 Substituting (10) in (2)ndash(6) we get the system of followingordinary differential equations
1198671015840 + 2119865 + 1198881205761198671205791015840 (1 + 120576120579)minus1 = 011986510158401015840 minus 119870119865 minus (1 + 120576120579)119888minus119886 [1198652 minus 1198662 + 1198671198651015840]+ 119886120576 (1 + 120576120579)minus1 12057910158401198651015840 = 0
11986610158401015840 minus 119870119866 minus (1 + 120576120579)119888minus119886 [1198671198661015840 + 2119865119866]+ 119886120576 (1 + 120576120579)minus1 12057910158401198661015840 = 0
where 119870 = ]infin119896Ω Pr = 120583infin119862119901120581infin and 119873 = 412059011198793infin1198961120581infinare the porosity parameter Prandtl number and radiationparameter respectively 120576 = Δ119879119879infin is the relative temper-ature difference parameter it is positive for a heated surfacenegative for a cooled surface and zero for the case of constantproperty
Kn = 120582119903 is the Knudsen number it is the ratio of meanfree path of fluid particle diameter For slip condition valueranges from 0001 to 01 For Kn lt 0001 no slip boundaryconditions are valid therefore velocity at the surface is zeroFor high Knudsen number high order continuum equation(Burnett equations) should be used [43] For present investi-gation slip regime of Knudsen number which lies in the range0001 lt Kn lt 01 has been considered
Re = Ω1199032]infin is the rotational Reynolds number forlaminar flow the value of the local Reynolds number is 18times 105 [44ndash46] For transition flow the local Reynolds numbervalues lie between 18 times 105 and 36 times 105 and for values higherthan 36 times 105 the flow becomes turbulent In this study thelaminar flow for local Reynolds number that lies in the range0 lt Re lt 10000 has been considered120574 = ((2minus120590V)120590V)KnradicRe 120574 is the slip factor depending onrotational Reynolds number and Knudsen number and mayvary from 0 to 12120593 = ((2 minus 120590119905)120590119905)(2120573(1 + 120573))(KnPr)radicRe 120593 is thetemperature jump factor varying from 0 to 12119882119904 = 1199080(Ω]infin)12 is the suction parameter which hasbeen taken as less than zero because suction of fluid is takingplace
The values of tangential momentum accommodationnumber (120590V) energy accommodation coefficient (120590119905) and thespecific heat ratio (120573) for air are considered as 09 09 and 14respectively Karniadakis et al [47]
3 Solution
The nonlinear coupled ordinary differential equations (11)with the boundary conditions (12) have been solved numer-ically applying fourth-order Runge-Kutta scheme togetherwith shooting method
The given boundary value problem is reduced to thefollowing system of initial value problem
Taking 120576 = 0 120583 = 120583infin (constant)Similarly at 120576 = 0 120588 = 120588infin and 120581 = 120581infinSubstitution of 120576 = 0 in (2)ndash(4) and (6) system of
equations is as follows
1198671015840 + 2119865 = 011986510158401015840 minus 1198671198651015840 minus 1198652 + 1198662 minus 119870119865 = 011986610158401015840 minus 1198671198661015840 minus 2119865119866 minus 119870119866 = 0(1 + 41198733 ) 12057910158401015840 minus Pr1198671205791015840 = 0
(16)
Equations (16) under the boundary condition (12) are trans-formed into the following system of initial value problems
Here 1199031 = 1198651015840(0) 1199032 = 1198661015840(0) and 1199033 = 1205791015840(0) are the initialguesses when fluid properties are variable and 1199034 = 1198651015840(0) 1199035 =1198661015840(0) and 1199036 = 1205791015840(0) are the initial guesses when fluid prop-erties are constant The essence of present numerical methodis to reduce the boundary value problem (BVP) into an initialvalue problem (IVP) Further shooting technique is used toguess 1199031 1199032 1199033 1199034 1199035 and 1199036 until the boundary conditionsare satisfied A number of iterations of Runge-Kutta fourth-order method has been performed to obtain final values ofthese guesses Initial guesses for different set of parametersare displayed in Table 1
In this problem the physical quantities of interest arelocal skin friction coefficients and the Nusselt number whichrepresents the wall shear stress and the rate of heat transferrespectively When variable fluid properties are taken intoconsideration the fluid near to the disk opposes rotation ofthe disk due to presence of tangential shear stressThereforeto maintain a steady rotation it is essential to have torque atthe shaft The skin frictions 119862119891119903 along radial direction and119862119891119905 along tangential direction at no slip condition are givenas
where Re = Ω1199032]infin is rotational Reynolds numberThus (24) shows that the radial and tangential skin
frictions coefficients are proportional to 1198651015840(0) and 1198661015840(0)respectively and Nusselt number is proportional to minus1205791015840(0)4 Result and Discussion
In this investigation Figures 2ndash6 show the effect of variousvalues of the physical parameters on the velocity (radialtangential and axial) and temperature distribution Resultsobtained for both constant fluid property and variable fluidproperty have been presented graphically
Figures 2(a)ndash2(d) show the effect of porosity parameter119870on all velocity components and temperature distribution It isobserved that radial tangential and axial velocity decreasesbut temperature increases by increasing the permeability forboth cases Whereas variation in porosity parameter has lesseffect on the temperature distribution radial velocity attains
Advances in Mathematical Physics 7
0 1 2 3 4 5 60
001002003004005006007008009
Freidoonimehr et al [40]Rashidi and Freidooni Mehr [27]
F(
)
K = = 0 ＋Ｈ = 0 ０Ｌ = 071
K = 0 = 02 ０Ｌ = 1K = = 0 ＋Ｈ = 005 ０Ｌ = 1
K = 05 = 02K = 05 = 0
K = 1 = 02K = 1 = 0
＋Ｈ = 005
０Ｌ = 1 ＋Ｈ = 005０Ｌ = 1 ＋Ｈ = 005
０Ｌ = 1 ＋Ｈ = 005０Ｌ = 1 ＋Ｈ = 005
(a)
0010203040506070809
1
0 1 2 3 4 5 6
G(
) K = = 0 ＋Ｈ = 0 ０Ｌ = 071K = 0 = 02
K = = 0 ＋Ｈ = 005 ０Ｌ = 1
K = 1 = 02K = 1 = 0
＋Ｈ = 005 ０Ｌ = 1
＋Ｈ = 005 ０Ｌ = 1 ＋Ｈ = 005 ０Ｌ = 1
Rashidi and Freidooni Mehr [27]Freidoonimehr et al [40]
(b)
0 1 2 3 4 5 6
H(
)
minus07
minus08
minus09
minus1
minus11
minus12
minus13
K = 10 = 02
K = 10 = 0K = 1 = 0
K = 0 = 02
K = = 0 ＋Ｈ = 005 ０Ｌ = 1K = 0 = 0
＋Ｈ = 005 ０Ｌ = 1
K = 1 = 02 ＋Ｈ = 005 ０Ｌ = 1 ＋Ｈ = 005 ０Ｌ = 1
＋Ｈ = 005 ０Ｌ = 1
＋Ｈ = 005 ０Ｌ = 1
＋Ｈ = 0 ０Ｌ = 071
Rashidi and Freidooni Mehr [27]Freidoonimehr et al [40]
(c)
0 2 4 6 8 10 12 140
0102030405060708
(
)
= 02 N = 1
= 0 N = 1
= 0 N = 0 K = 0K = 10 1 0
＋Ｈ = 005 ０Ｌ = 1
Rashidi and Freidooni Mehr [27]
(d)
Figure 2 Effect of variation in the porosity parameter on the (a) radial (b) tangential (c) axial and (d) temperature velocity profiles when119882119904 = minus1 and Re = 100
a maximum value close to the disk surface When we reducethe present problem into the literature available by takingpermeability parameter 119870 = 0 and radiation parameter119873 = 0 then the result obtained is exactly similar to thatof the Rashidi and Freidooni Mehr [27] and Freidoonimehret al [40] This validates the results obtained for presentinvestigation
Figures 3(a)ndash3(d) depict the effect of suction param-eter 119882119904 on the radial tangential axial and temperaturedistribution respectively Figures 3(a)ndash3(c) show that allvelocity components decrease as suction increases Physicalsignificance is that as suction increases adherence of thefluid with the wall increases and as a result boundary layerthickness decreases
Figure 3(d) depicts the effect of suction parameter ontemperature profile It is observed that as suction increasestemperature of fluid decreases because suction cools theboundary layer regime therefore suction is used for coolingthe flow in several engineering processes including MHDpower generators and nuclear energy processes
The effect of Reynolds number Re and Knudsen num-ber Kn on velocity and temperature distribution is plotted
in Figures 4 and 5 respectively Both the parameters areshowing the same effects on slip boundary conditions Itis observed that the increasing values of the Reynolds andKnudsen numbers decrease the fluid velocity componentsand temperature distribution The case when value of Kn liesbetween 0001 and 01 corresponds to slip at the surface of thedisk As slip increases the quantity of the fluid that can stickon the disk decreases Therefore circumferential velocity ofthe fluid reduces and causes reduction in centrifugal forceAs a result inward axial velocity decreases In other words asslip gets stronger flow of fluid drawn or pushed away alongthe velocity directions decreases the heat generation
Figure 6 shows the effect of Prandtl number on tem-perature profile As Prandtl number increases the thermalboundary layer thickness decreases Physical significance isthat Prandtl number precludes dispersal of heat in the fluid
Figure 7 depicts the variation of temperature profilewith radiation It is observed that as radiation increasestemperature of the fluid increases Also radiation parameterincreases temperature gradient near the surface of the disk
Figure 8 demonstrates the comparison of both constantand fluid properties on flow over a rotating disk with slip and
8 Advances in Mathematical Physics
0 1 2 3 4 5 6
0
0005
001
0015
002
0025F
()
Ws = minus1 minus2 minus4
= 0
= 02
(a)
0005
01015
02025
03035
04045
05
0 1 2 3 4 5 6
Ws = minus1 minus2 minus4
= 0
= 02
G(
)
(b)
0 1 2 3 4 5 6
= 0
= 02
Ws = minus1
Ws = minus2
Ws = minus4
H(
)
minus1
minus15
minus2
minus25
minus3
minus35
minus4
minus45
(c)
00102030405060708
Ws = minus1 minus2 minus4
= 0
= 02
10 1282 640 14
(
)
(d)
Figure 3 Effect of variation in the suction parameter on the (a) radial (b) tangential (c) axial and (d) temperature velocity profiles when119870 = 1 Kn = 005 Re = 100119873 = 1 and Pr = 1
Table 2 Comparison between the results of present study with the results reported by Kelson and Desseaux [48] Maleque and Sattar [33]and Alam et al [49] for the radial 1198651015840(0) and tangential minus1198661015840(0) skin friction coefficients for Pr = 071119872 = 0 120576 = 0 and Kn = 0
temperature jump The radial velocity achieves a maximumvalue near to the surface of the disk for all values of 120576 It is alsonoted that an increment in relative temperature differenceparameter causes shifting of maximum point from the diskThe velocity along the tangential direction increases asvalue of relative temperature difference parameter increaseswhereas the velocity along axial direction decreases Temper-ature increases as the value of relative temperature differenceparameter increasesThese results are similar to that obtainedby Rashidi et al [39]
Table 1 illustrates the effect of the parameters 120576 119870Kn Re Pr and 119873 on constant suction parameter on thenumerical values of (1198651015840(0) minus1198661015840(0)) and (minus1205791015840(0)) We observethat numerical value of 1198651015840(0) minus1198661015840(0) and minus1205791015840(0) decreases
with the increasing value of Kn Re and radiation parameter119873Tables 2 and 3 depict the comparison of skin friction coef-
ficients and rate of heat transfer between the results obtainedin the present investigation and the literature available [3348 49] It is observed that results obtained in present studyare very well in agreement with the existing results
5 Conclusions
In this study we have investigated radiation effect on velocityprofile for all components and temperature profile throughrotating disk in porous medium for variable fluid propertiesand in particular case for constant fluid properties also By
Advances in Mathematical Physics 9
0 1 2 3 4 5 6
= 0
= 02
Ws = minus1
２ = 1 10 100 1000 10000
00005
0010015
0020025
0030035
004
F(
)
(a)
= 0
= 02
5 63 41 20
Ws = minus1
２ = 1 10 100 1000 10000
0010203040506070809
1
G(
)
(b)
0 2 3 41 65
Ws = minus2 = 0
２ = 1
２ = 10
２ = 100
２ = 1000
２ = 10000
minus204minus2035minus203
minus2025minus202
minus2015minus201
minus2005minus2
minus1995minus199
H(
)
(c)
0 2 3 41 65
Ws = minus2 = 02 ２ = 1
２ = 10
２ = 100
２ = 1000
２ = 10000
minus2
minus195
minus19
minus185
minus18
minus175
minus17
H(
)
(d)
0 2 4 6 8 10 12 14
= 0
= 02
Ws = minus1
２ = 1 10 100 1000 10000
0010203040506070809
1
(
)
(e)
Figure 4 Effect of variation in the Reynolds number on the (a) radial (b) tangential (c) amp (d) axial and (e) temperature velocity profileswhen 119870 = 1 Kn = 005119873 = 1 and Pr = 1
Table 3 Comparison between the results of present study with the results reported by Kelson and Desseaux [48] Maleque and Sattar [33]and Alam et al [49] for rate of heat transfer minus1205791015840(0) for Pr = 071119872 = 0 120576 = 0 Kn = 0 and119873 = 0119882119904 Present Alam et al [49] Maleque and Sattar [33] Kelson and Desseaux [48]minus1205791015840(0) minus1205791015840(0) minus1205791015840(0) minus1205791015840(0)0 0326798372 032637889 032576 0325856minus2 1438764651 143876482 144212 1437782minus4 2842381877 284369011 284470 2842381minus5 3551223146 355222471 355411 3551223
10 Advances in Mathematical Physics
0 1 2 3 4 5 6
= 0
= 02
00005
0010015
0020025
0030035
004F
()
Ws = minus1
＋Ｈ = 0 002 005 01
(a)
0 1 2 3 4 5 6
= 0
= 02
0010203040506070809
1
G(
)
Ws = minus1
＋Ｈ = 0 002 005 01
(b)
0 1 2 3 4 5 6
= 0
= 02
minus2
minus195
minus19
minus185
minus18
minus175
minus17
H(
)
Ws = minus2
＋Ｈ = 005 002 0
(c)
= 0
= 02
0010203040506070809
1
(
)
104 6 8 12 1420
Ws = minus1
＋Ｈ = 0 002 005 01N = 0 ＋Ｈ = 005
(d)
Figure 5 Effect of variation in the Knudsen number on the (a) radial (b) tangential (c) axial and (d) temperature velocity profiles when119870 = 1 Re = 100119873 = 1 and Pr = 1
= 0
= 02
2 4 6 8 10 12 140
00102030405060708
(
)
N = 0 ０Ｌ = 4０Ｌ = 1 2 3 4
Figure 6 Effect of Prandtl number on temperature distributionwhen 119870 = 1119882119904 = minus1 Re = 100 Kn = 005 and119873 = 1
similarity transformation governing equations transformedinto nonlinear ordinary differential equations which aresolved numerically by usingRunge-Kuttamethodwith shoot-ing technique Based on the resulting solutions the followingconclusions can be drawn
= 0
= 02
2 4 6 8 10 12 140
N = 4 3 2 1 0
0010203040506070809
(
)
Figure 7 Effect of radiation parameter on temperature distributionwhen 119870 = 1119882119904 = minus1 Re = 100 Kn = 005 and Pr = 1
(i) The radial tangential and axial velocity profilesdecrease while the temperature increases with theincreasing values of porosity parameter
(ii) The increasing value of Reynolds and Knudsen num-ber decreases the fluid velocity components and
Advances in Mathematical Physics 11
1 2 3 4 5 60
= 0 02 04 10
0005
001
0015
002
0025F
()
(a)
1 2 54 60 3
0
01
02
03
04
05
06
G(
)
= 1 04 02 0
(b)
1 2 3 4 5 60
= 0
= 02
= 04
= 1
minus11
minus1
minus09
minus08
minus07
minus06
minus05
minus04
H(
)
(c)
00102030405060708
(
)
2 4 6 8 10 12 140
N = 0 = 02
= 1 04 02 0
(d)
Figure 8 Effect of variation in the relative temperature difference parameter on the (a) radial (b) tangential (c) axial and (d) temperaturevelocity profiles when119870 = 1119882119904 = minus1 Kn = 005 Re = 100119873 = 1 and Pr = 1
temperature and suction parameter also shows thesame effect
(iii) For the effect of the radiation parameter on the tem-perature distribution it is seen that the temperaturedistribution decreases with the increasing values ofradiation parameter and also it has been observed thatthe radial and tangential skin friction values decreasewith increase in the radiation parameter
Competing Interests
The authors declare that they have no competing interests
References
[1] Th V Karman ldquoUber laminare und turbulente reibunrdquo Zeits-chrift fur Angewandte Mathematik und Mechanik vol 1 no 4pp 233ndash252 1921
[2] W G Cochran ldquoThe flow due to a rotating diskrdquoMathematicalProceedings of the Cambridge Philosophical Society vol 30 no3 pp 365ndash375 1934
[3] E R Benton ldquoOn the flow due to a rotating diskrdquo Journal ofFluid Mechanics vol 24 no 4 pp 781ndash800 1966
[4] M Turkyilmazoglu ldquoPurely analytic solutions of the compress-ible boundary layer flow due to a porous rotating disk with heattransferrdquoPhysics of Fluids vol 21 no 10 Article ID 106104 2009
[5] D S Chauhan and S Gupta ldquoSteady flow and heat transferbetween two stationary naturally permeable disksrdquo Journal ofUltra Scientist of Physical Sciences vol 12 no 1 pp 45ndash52 2000
[6] C Wagner ldquoHeat transfer from a rotating disk to ambient airrdquoJournal of Applied Physics vol 19 no 9 pp 837ndash839 1948
[7] K Millsaps and K Pohlhausen ldquoHeat transfer by laminar flowfrom a rotating platerdquo Journal of the Aeronautical Sciences vol19 pp 120ndash126 1952
[8] F Kreith and J H Taylor ldquoHeat transfer from a rotating disk inturbulent flow no 1956rdquo ASME Paper 56-A-146 1956
[9] H-T Lin and L-K Lin ldquoHeat transfer from a rotating cone ordisk to fluids of any Prandtl numberrdquo International Communi-cations in Heat and Mass Transfer vol 14 no 3 pp 323ndash3321987
[10] P D Verma and D S Chauhan ldquoFlow between a torsionallyoscillating impermeable disc and a stationary naturally perme-able discrdquo Indian Journal of Pure and Applied Mathematics vol10 pp 1351ndash1361 1979
[11] D S Chauhan and S Jain ldquoSteady flow between highlypermeable rotating disksrdquo Indian Journal of Theoretical Physicsvol 52 no 1 pp 39ndash50 2004
[12] M Turkyilmazoglu ldquoNanofluid flow and heat transfer due to arotating diskrdquo Computers amp Fluids vol 94 pp 139ndash146 2014
[13] M Turkyilmazoglu ldquoMHD fluid flow and heat transfer due to ashrinking rotating diskrdquo Computers amp Fluids vol 90 pp 51ndash562014
[14] M Azimi and R Riazi ldquoHeat transfer analysis of GO-waternanofluid flow between two parallel disksrdquo Propulsion andPower Research vol 4 no 1 pp 23ndash30 2015
12 Advances in Mathematical Physics
[15] B R Sharma and H Konwar ldquoEffect of chemical reaction onmass transfer due to a permeable rotating heated diskrdquo Interna-tional Journal of Computer Applications vol 119 no 21 pp 5ndash92015
[16] H Shahmohamadi and M Mohammadpour ldquoA series solutionfor three-dimensional navier-stokes equations of flow near aninfinite rotating diskrdquo World Journal of Mechanics vol 4 pp117ndash127 2014
[17] S Srinivas A S Reddy T R Ramamohan and A K ShuklaldquoThermal-diffusion and diffusion-thermo effects onMHD flowof viscous fluid between expanding or contracting rotatingporous disks with viscous dissipationrdquo Journal of the EgyptianMathematical Society vol 24 no 1 pp 100ndash107 2016
[18] M A Mansour ldquoRadiative and free-convection effects on theoscillatory flow past a vertical platerdquo Astrophysics and SpaceScience vol 166 no 2 pp 269ndash275 1990
[19] M A Hossain M A Alim and D A S Rees ldquoThe effectof radiation on free convection from a porous vertical platerdquoInternational Journal of Heat and Mass Transfer vol 42 no 1pp 181ndash191 1999
[20] A Raptis and C Perdikis ldquoMHD free convection flow by thepresence of radiationrdquo International Journal of Magnetohydro-dynamics Plasma and Space Research vol 9 pp 237ndash252 2000
[21] M A Hossain and H S Takhar ldquoRadiation effect on mixedconvection along a vertical plate with uniform surface temper-aturerdquoHeat and Mass Transfer vol 31 no 4 pp 243ndash248 1996
[22] A Devi and R U Devi ldquoEffects of thermal radiation onhydromagnetic flow due to a porous rotating disk with halleffectrdquo Journal of Applied Fluid Mechanics vol 5 no 2 pp 1ndash7 2012
[23] C Y Wang ldquoFlow due to a stretching boundary with partialslipmdashan exact solution of the Navier-Stokes equationsrdquo Chemi-cal Engineering Science vol 57 no 17 pp 3745ndash3747 2002
[24] E Osalusi ldquoEffects of thermal radiation on MHD and slip flowover a porous rotating disk with variable propertiesrdquo RomanianJournal of Physics vol 52 no 3-4 pp 217ndash229 2007
[25] A A Khidir ldquoViscous dissipation Ohmic heating and radiationeffects on MHD flow past a rotating disk embedded in aporous medium with variable propertiesrdquo Arabian Journal ofMathematics vol 2 no 3 pp 263ndash277 2013
[26] E M Sparrow G S Beavers and L Y Hung ldquoFlow about aporous-surfaced rotating diskrdquo International Journal of Heatand Mass Transfer vol 14 no 7 pp 993ndash996 1971
[27] M M Rashidi and N Freidooni Mehr ldquoEffects of velocity slipand temperature jump on the entropy generation in magne-tohydrodynamic flow over a porous rotating diskrdquo Journal ofMechanical Engineering vol 1 no 3 2012
[28] A Arikoglu G Komurgoz I Ozkol and A Y Gunes ldquoCom-bined effects of temperature and velocity jump on the heattransfer fluid flow and entropy generation over a single rotatingdiskrdquo Journal of Heat Transfer vol 132 no 11 Article ID 1117032010
[29] THayatM RashidM Imtiaz andAAlsaedi ldquoMagnetohydro-dynamic (MHD) flow of Cu-water nanofluid due to a rotatingdisk with partial sliprdquo AIP Advances vol 5 no 6 Article ID067169 2015
[30] M Zakerullah and J A D Ackroyd ldquoLaminar natural convec-tion boundary-layers on Horizontal Circular disksrdquo Journal ofApplied Mathematics and Physics vol 30 pp 427ndash435 1979
[31] HHerwig ldquoThe effect of variable properties onmomentumandheat transfer in a tube with constant heat flux across the wallrdquoInternational Journal of Heat and Mass Transfer vol 28 no 2pp 423ndash431 1985
[32] H Herwig and K Klemp ldquoVariable property effects of fullydeveloped laminar flow in concentric annulirdquo Journal of HeatTransfer vol 110 no 2 pp 314ndash320 1988
[33] K AMaleque andMA Sattar ldquoSteady laminar convective flowwith variable properties due to a porous rotating diskrdquo Journalof Heat Transfer vol 127 no 12 pp 1406ndash1409 2005
[34] K A Maleque and M A Sattar ldquoThe effects of variable proper-ties and hall current on steady MHD laminar convective fluidflow due to a porous rotating diskrdquo International Journal of Heatand Mass Transfer vol 48 no 23-24 pp 4963ndash4972 2005
[35] E Osalusi and P Sibanda ldquoOn variable laminar convective flowproperties due to a porous rotating disk in a magnetic fieldrdquoRomanian Journal of Physics vol 9 no 10 pp 933ndash944 2006
[36] M M Rahman ldquoConvective Hydromagnetic slip flow withvariable properties due to a porous rotating diskrdquo The SultanQaboos University Journal for Science vol 15 pp 55ndash79 2010
[37] M M Rashidi S A Mohimanian Pour T Hayat and SObaidat ldquoAnalytic approximate solutions for steady flow over arotating disk in porousmediumwith heat transfer by homotopyanalysis methodrdquo Computers amp Fluids vol 54 no 1 pp 1ndash92012
[38] S Hussain F Ahmad M Shafique and S Hussain ldquoNumericalsolution for accelerated rotating disk in a viscous fluidrdquoAppliedMathematics vol 4 no 6 pp 899ndash902 2013
[39] M M Rashidi N Kavyani and S Abelman ldquoInvestigation ofentropy generation inMHDand slip flow over a rotating porousdisk with variable propertiesrdquo International Journal of Heat andMass Transfer vol 70 pp 892ndash917 2014
[40] N Freidoonimehr M M Rashidi S Abelman and G Loren-zini ldquoAnalytical modeling of MHD flow over a permeablerotating disk in the presence of Soret and Dufour effectsentropy analysisrdquo Entropy vol 18 no 5 article 131 2016
[41] M S Alam S M Chapal Hossain and M M Rahman ldquoTran-sient thermophoretic particle deposition on forced convectiveheat and mass transfer flow due to a rotating diskrdquo Ain ShamsEngineering Journal vol 7 no 1 pp 441ndash452 2016
[42] S Jayaraj ldquoThermophoresis in laminar flow over cold inclinedplates with variable propertiesrdquoHeat andMass Transfer vol 30no 3 pp 167ndash173 1995
[43] M M Rahman ldquoThermophoretic deposition of nanoparticlesdue to a permeable rotating disk effects of partial slip mag-netic field thermal radiation thermal-diffusion and diffusion-thermordquo International Journal of Mathematical ComputationalPhysical Electrical and Computer Engineering vol 7 no 5 2013
[44] I V Shevchuk Convective Heat and Mass Transfer in RotatingDisk Systems Springer Berlin Germany 2009
[45] C L Tien and D T Campbell ldquoHeat and mass transfer fromrotating conesrdquo Journal of FluidMechanics vol 17 no 1 pp 105ndash112 1963
[46] C J Elkins and J K Eaton ldquoHeat transfer in the rotatingdisk boundary layerrdquo Tech Rep TSD-103 Stanford UniversityDepartment of Mechanical EngineeringThermosciences Divi-sion Stanford Calif USA 1997
[47] G Karniadakis A Beskok and N Aluru Microflows Funda-mentals and Simulation Springer New York NY USA 2001
[48] N Kelson andA Desseaux ldquoNote on porous rotating disk flowrdquoAustralian amp New Zealand Industrial and Applied MathematicsJournal vol 42 pp 837ndash855 2000
[49] M Alam N Poddar M Rahman and K Vajravelu ldquoTransienthydromagnetic forced convective heat transfer slip flow due toa porous rotating disk with variable fluid propertiesrdquo AmericanJournal of Heat andMass Transfer vol 2 no 3 pp 165ndash189 2015
radiationThe investigation of the effect of radiation onmixedconvection flow of an optically dense viscous incompressiblefluid along a heated vertical flat plate with uniform freestream and uniform surface temperature has been done byHossain and Takhar [21] A Devi and R U Devi [22] studiedthermal radiation effect onMHD flow over a rotating infinitenonporous disk She also investigated porous rotating diskwith Hall effect
In recent years the slip flow regime has been widely stud-ied and researchers have been concentrating on the analysisof microscale in microelectromechanical systems (MEMS)associated with the embodiment of velocity slip and temper-ature jump Wang [23] examined the flow due to a stretch-ing boundary with partial slipmdashan exact solution of theNavier-Stokes equations Osalusi [24] studied the combinedhydromagnetic and slip flow of a steady laminar conductingviscous fluid in the presence of thermal radiation due to animpulsively started rotating porous disk with the variablefluid properties Khidir [25] investigated the effects of viscousdissipation and ohmic heating on steady MHD convectiveflow due to porous rotating disk taking into account thevariable fluid properties Sparrow et al [26] considered thefluid flow due to the rotation of a porous surfaced disk andemployed a set of linear slip flow conditions He observedthat a substantial reduction in torque occurred as a resultof surface slip Rashidi and Freidooni Mehr [27] investigatedeffects of velocity slip and temperature jump on the flow overa porous rotating disk The combined effects of temperatureand velocity jump on the heat transfer fluid flow and entropygeneration over a single rotating disk have been examined byArikoglu et al [28] Hayat et al [29] investigatedMHD steadyflow of viscous nanofluid due to a rotating disk with partialslip
In most of the research the fluid properties such asdensity (120588) viscosity (120583) and thermal conductivity (120581) areassumed to be constant However these properties remainunaltered if and only if temperature remains the same ordoes not change rapidly Therefore to predict flow behavioraccurately it is essential to consider variable fluid propertiesZakerullah and Ackroyd [30] analyzed the laminar naturalconvection boundary layer flow on a horizontal circular diskwith variable properties Herwig [31] and Herwig and Klemp[32] have extended the work [30] They have investigatedeffects of variable properties in a tube and concentric annulirespectively at constant heat flux FurtherMaleque and Sattar[33 34] have investigated laminar convective flow taking intoaccount the variable properties due to a porous rotating diskThey solved the problems numerically using Runge-Kuttamethod with shooting technique This work was extendedby Osalusi and Sibanda [35] They have studied flow in thepresence of magnetic field Rahman [36] made a study onthe slip flow with variable properties due to a porous rotatingdisk
Rashidi et al [37] obtained analytical solutions flow andheat transfer over a rotating disk in porous medium Hussainet al [38] obtained numerical solution of a disk rotating in
x
y
z
Ω
Tw
ru
w
Tinfin
pinfin
Figure 1 Coordinate system for the rotating disk flow
a viscous fluid Rashidi et al [39 40] analyzed entropy genera-tion in a MHD flow over a rotating porous disk with variablephysical properties They have also investigated fluid flowover a permeable rotating disk in the presence of Soret andDufour effect Alam et al [41] investigated thermophoreticdeposition of micron sized particles on flow due to rotatingdisk
The main goal of the present study is to investigateradiation effects in the steady flow over a rotating permeabledisk in porous medium with velocity slip and temperaturejump To predict the flow behavior accurately variable ther-mophysical properties are taken into consideration To thebest of the authorrsquos knowledge radiation effects of flow overrotating disk with velocity and temperature slip with variablethermal properties have not been studied yet The noveltyof present paper is to investigate flow and heat transferfor variable fluid properties with velocity slip and tempera-ture jump taken into consideration Also combined effectsfor both variable and constant fluid properties for variousphysical parameters on flow and heat transfer have beenobtained and depicted graphically which gives more insightabout fluid flow (Figure 1)
2 Mathematical Formulation
Consider a steady slip flow due to permeable rotating diskthrough porous medium Assume disk of 119903 radius is rotatingwith constant angular velocity Ω and placed at 119911 = 0 incylindrical polar coordinates (119903 120601 119911) where 119903 and 120601 are theradial and tangential axis respectively and 119911 is the verticalaxis
Let 119906 V and 119908 be the components of the fluid velocityin the direction of 119903 120601 and 119911 respectively 119901 and 119879 are thepressure and temperature of the fluidThe surface of the rotat-ing disk is considered at a uniform temperature 119879119908 Pressureand temperature for ambient fluid are 119901infin and 119879infin (119879119908 gt119879infin)
Advances in Mathematical Physics 3
Following [34 35 42] the fluid properties viscosity (120583)thermal conductivity (120581) and the density (120588) are taken asfunctions of temperature
120583 = 120583infin ( 119879119879infin)119886
120581 = 120581infin ( 119879119879infin)119887
120588 = 120588infin ( 119879119879infin)119888
(1)
where 119886 119887 and 119888 are arbitrary exponents and 120583infin 120581infin and 120588infinare the uniformviscosity thermal conductivity anddensity ofthe fluid Assume 119886 = 07 119887 = 083 and 119888 = minus10 (ideal gas)are the values of exponents for present investigation
The governing equations of continuity momentum andenergy for laminar incompressible flow in cylindrical coordi-nates are [35]
at 119911 997888rarr infin119906 997888rarr infinV 997888rarr infin119879 997888rarr 119879infin
(7)
where 120590V is the tangential momentum accommodation coef-ficient 120590119905 is the energy accommodation coefficient 120582 is meanfree path 120573 is the ratio of specific heats and 119879119904 is thetemperature of the fluid near to the disk surface
Rosseland approximation has been used for radiation 119902119903
where1205901 is the Stefan-Boltzmann constant and1198794 is themeanabsorption coefficient It is assumed that the temperaturedifferences within the flow are sufficiently small so that theterm1198794may be expressed as a linear function of temperatureThis is done by expanding 1198794 in a Taylor series about 119879infin andomitting the second- and higher-order terms leads to
1198794 cong 41198793infin119879 minus 31198794infin120597119902119903120597119911 = 120597120597119911 (minus4120590131198961
1205971198794120597119911 )
= 120597120597119911 (minus4120590131198961120597 (41198793infin119879 minus 31198794infin)120597119911 )
119901 minus 119901infin = minus120583infinΩ119875 (120578) 120579 (120578) = (119879 minus 119879infin)(119879119908 minus 119879infin)
(10)
where ]infin is uniformkinematic viscosity and119865119866119867 120579 and119875are nondimensional functions in terms of vertical coordinate120578 Substituting (10) in (2)ndash(6) we get the system of followingordinary differential equations
1198671015840 + 2119865 + 1198881205761198671205791015840 (1 + 120576120579)minus1 = 011986510158401015840 minus 119870119865 minus (1 + 120576120579)119888minus119886 [1198652 minus 1198662 + 1198671198651015840]+ 119886120576 (1 + 120576120579)minus1 12057910158401198651015840 = 0
11986610158401015840 minus 119870119866 minus (1 + 120576120579)119888minus119886 [1198671198661015840 + 2119865119866]+ 119886120576 (1 + 120576120579)minus1 12057910158401198661015840 = 0
where 119870 = ]infin119896Ω Pr = 120583infin119862119901120581infin and 119873 = 412059011198793infin1198961120581infinare the porosity parameter Prandtl number and radiationparameter respectively 120576 = Δ119879119879infin is the relative temper-ature difference parameter it is positive for a heated surfacenegative for a cooled surface and zero for the case of constantproperty
Kn = 120582119903 is the Knudsen number it is the ratio of meanfree path of fluid particle diameter For slip condition valueranges from 0001 to 01 For Kn lt 0001 no slip boundaryconditions are valid therefore velocity at the surface is zeroFor high Knudsen number high order continuum equation(Burnett equations) should be used [43] For present investi-gation slip regime of Knudsen number which lies in the range0001 lt Kn lt 01 has been considered
Re = Ω1199032]infin is the rotational Reynolds number forlaminar flow the value of the local Reynolds number is 18times 105 [44ndash46] For transition flow the local Reynolds numbervalues lie between 18 times 105 and 36 times 105 and for values higherthan 36 times 105 the flow becomes turbulent In this study thelaminar flow for local Reynolds number that lies in the range0 lt Re lt 10000 has been considered120574 = ((2minus120590V)120590V)KnradicRe 120574 is the slip factor depending onrotational Reynolds number and Knudsen number and mayvary from 0 to 12120593 = ((2 minus 120590119905)120590119905)(2120573(1 + 120573))(KnPr)radicRe 120593 is thetemperature jump factor varying from 0 to 12119882119904 = 1199080(Ω]infin)12 is the suction parameter which hasbeen taken as less than zero because suction of fluid is takingplace
The values of tangential momentum accommodationnumber (120590V) energy accommodation coefficient (120590119905) and thespecific heat ratio (120573) for air are considered as 09 09 and 14respectively Karniadakis et al [47]
3 Solution
The nonlinear coupled ordinary differential equations (11)with the boundary conditions (12) have been solved numer-ically applying fourth-order Runge-Kutta scheme togetherwith shooting method
The given boundary value problem is reduced to thefollowing system of initial value problem
Taking 120576 = 0 120583 = 120583infin (constant)Similarly at 120576 = 0 120588 = 120588infin and 120581 = 120581infinSubstitution of 120576 = 0 in (2)ndash(4) and (6) system of
equations is as follows
1198671015840 + 2119865 = 011986510158401015840 minus 1198671198651015840 minus 1198652 + 1198662 minus 119870119865 = 011986610158401015840 minus 1198671198661015840 minus 2119865119866 minus 119870119866 = 0(1 + 41198733 ) 12057910158401015840 minus Pr1198671205791015840 = 0
(16)
Equations (16) under the boundary condition (12) are trans-formed into the following system of initial value problems
Here 1199031 = 1198651015840(0) 1199032 = 1198661015840(0) and 1199033 = 1205791015840(0) are the initialguesses when fluid properties are variable and 1199034 = 1198651015840(0) 1199035 =1198661015840(0) and 1199036 = 1205791015840(0) are the initial guesses when fluid prop-erties are constant The essence of present numerical methodis to reduce the boundary value problem (BVP) into an initialvalue problem (IVP) Further shooting technique is used toguess 1199031 1199032 1199033 1199034 1199035 and 1199036 until the boundary conditionsare satisfied A number of iterations of Runge-Kutta fourth-order method has been performed to obtain final values ofthese guesses Initial guesses for different set of parametersare displayed in Table 1
In this problem the physical quantities of interest arelocal skin friction coefficients and the Nusselt number whichrepresents the wall shear stress and the rate of heat transferrespectively When variable fluid properties are taken intoconsideration the fluid near to the disk opposes rotation ofthe disk due to presence of tangential shear stressThereforeto maintain a steady rotation it is essential to have torque atthe shaft The skin frictions 119862119891119903 along radial direction and119862119891119905 along tangential direction at no slip condition are givenas
where Re = Ω1199032]infin is rotational Reynolds numberThus (24) shows that the radial and tangential skin
frictions coefficients are proportional to 1198651015840(0) and 1198661015840(0)respectively and Nusselt number is proportional to minus1205791015840(0)4 Result and Discussion
In this investigation Figures 2ndash6 show the effect of variousvalues of the physical parameters on the velocity (radialtangential and axial) and temperature distribution Resultsobtained for both constant fluid property and variable fluidproperty have been presented graphically
Figures 2(a)ndash2(d) show the effect of porosity parameter119870on all velocity components and temperature distribution It isobserved that radial tangential and axial velocity decreasesbut temperature increases by increasing the permeability forboth cases Whereas variation in porosity parameter has lesseffect on the temperature distribution radial velocity attains
Advances in Mathematical Physics 7
0 1 2 3 4 5 60
001002003004005006007008009
Freidoonimehr et al [40]Rashidi and Freidooni Mehr [27]
F(
)
K = = 0 ＋Ｈ = 0 ０Ｌ = 071
K = 0 = 02 ０Ｌ = 1K = = 0 ＋Ｈ = 005 ０Ｌ = 1
K = 05 = 02K = 05 = 0
K = 1 = 02K = 1 = 0
＋Ｈ = 005
０Ｌ = 1 ＋Ｈ = 005０Ｌ = 1 ＋Ｈ = 005
０Ｌ = 1 ＋Ｈ = 005０Ｌ = 1 ＋Ｈ = 005
(a)
0010203040506070809
1
0 1 2 3 4 5 6
G(
) K = = 0 ＋Ｈ = 0 ０Ｌ = 071K = 0 = 02
K = = 0 ＋Ｈ = 005 ０Ｌ = 1
K = 1 = 02K = 1 = 0
＋Ｈ = 005 ０Ｌ = 1
＋Ｈ = 005 ０Ｌ = 1 ＋Ｈ = 005 ０Ｌ = 1
Rashidi and Freidooni Mehr [27]Freidoonimehr et al [40]
(b)
0 1 2 3 4 5 6
H(
)
minus07
minus08
minus09
minus1
minus11
minus12
minus13
K = 10 = 02
K = 10 = 0K = 1 = 0
K = 0 = 02
K = = 0 ＋Ｈ = 005 ０Ｌ = 1K = 0 = 0
＋Ｈ = 005 ０Ｌ = 1
K = 1 = 02 ＋Ｈ = 005 ０Ｌ = 1 ＋Ｈ = 005 ０Ｌ = 1
＋Ｈ = 005 ０Ｌ = 1
＋Ｈ = 005 ０Ｌ = 1
＋Ｈ = 0 ０Ｌ = 071
Rashidi and Freidooni Mehr [27]Freidoonimehr et al [40]
(c)
0 2 4 6 8 10 12 140
0102030405060708
(
)
= 02 N = 1
= 0 N = 1
= 0 N = 0 K = 0K = 10 1 0
＋Ｈ = 005 ０Ｌ = 1
Rashidi and Freidooni Mehr [27]
(d)
Figure 2 Effect of variation in the porosity parameter on the (a) radial (b) tangential (c) axial and (d) temperature velocity profiles when119882119904 = minus1 and Re = 100
a maximum value close to the disk surface When we reducethe present problem into the literature available by takingpermeability parameter 119870 = 0 and radiation parameter119873 = 0 then the result obtained is exactly similar to thatof the Rashidi and Freidooni Mehr [27] and Freidoonimehret al [40] This validates the results obtained for presentinvestigation
Figures 3(a)ndash3(d) depict the effect of suction param-eter 119882119904 on the radial tangential axial and temperaturedistribution respectively Figures 3(a)ndash3(c) show that allvelocity components decrease as suction increases Physicalsignificance is that as suction increases adherence of thefluid with the wall increases and as a result boundary layerthickness decreases
Figure 3(d) depicts the effect of suction parameter ontemperature profile It is observed that as suction increasestemperature of fluid decreases because suction cools theboundary layer regime therefore suction is used for coolingthe flow in several engineering processes including MHDpower generators and nuclear energy processes
The effect of Reynolds number Re and Knudsen num-ber Kn on velocity and temperature distribution is plotted
in Figures 4 and 5 respectively Both the parameters areshowing the same effects on slip boundary conditions Itis observed that the increasing values of the Reynolds andKnudsen numbers decrease the fluid velocity componentsand temperature distribution The case when value of Kn liesbetween 0001 and 01 corresponds to slip at the surface of thedisk As slip increases the quantity of the fluid that can stickon the disk decreases Therefore circumferential velocity ofthe fluid reduces and causes reduction in centrifugal forceAs a result inward axial velocity decreases In other words asslip gets stronger flow of fluid drawn or pushed away alongthe velocity directions decreases the heat generation
Figure 6 shows the effect of Prandtl number on tem-perature profile As Prandtl number increases the thermalboundary layer thickness decreases Physical significance isthat Prandtl number precludes dispersal of heat in the fluid
Figure 7 depicts the variation of temperature profilewith radiation It is observed that as radiation increasestemperature of the fluid increases Also radiation parameterincreases temperature gradient near the surface of the disk
Figure 8 demonstrates the comparison of both constantand fluid properties on flow over a rotating disk with slip and
8 Advances in Mathematical Physics
0 1 2 3 4 5 6
0
0005
001
0015
002
0025F
()
Ws = minus1 minus2 minus4
= 0
= 02
(a)
0005
01015
02025
03035
04045
05
0 1 2 3 4 5 6
Ws = minus1 minus2 minus4
= 0
= 02
G(
)
(b)
0 1 2 3 4 5 6
= 0
= 02
Ws = minus1
Ws = minus2
Ws = minus4
H(
)
minus1
minus15
minus2
minus25
minus3
minus35
minus4
minus45
(c)
00102030405060708
Ws = minus1 minus2 minus4
= 0
= 02
10 1282 640 14
(
)
(d)
Figure 3 Effect of variation in the suction parameter on the (a) radial (b) tangential (c) axial and (d) temperature velocity profiles when119870 = 1 Kn = 005 Re = 100119873 = 1 and Pr = 1
Table 2 Comparison between the results of present study with the results reported by Kelson and Desseaux [48] Maleque and Sattar [33]and Alam et al [49] for the radial 1198651015840(0) and tangential minus1198661015840(0) skin friction coefficients for Pr = 071119872 = 0 120576 = 0 and Kn = 0
temperature jump The radial velocity achieves a maximumvalue near to the surface of the disk for all values of 120576 It is alsonoted that an increment in relative temperature differenceparameter causes shifting of maximum point from the diskThe velocity along the tangential direction increases asvalue of relative temperature difference parameter increaseswhereas the velocity along axial direction decreases Temper-ature increases as the value of relative temperature differenceparameter increasesThese results are similar to that obtainedby Rashidi et al [39]
Table 1 illustrates the effect of the parameters 120576 119870Kn Re Pr and 119873 on constant suction parameter on thenumerical values of (1198651015840(0) minus1198661015840(0)) and (minus1205791015840(0)) We observethat numerical value of 1198651015840(0) minus1198661015840(0) and minus1205791015840(0) decreases
with the increasing value of Kn Re and radiation parameter119873Tables 2 and 3 depict the comparison of skin friction coef-
ficients and rate of heat transfer between the results obtainedin the present investigation and the literature available [3348 49] It is observed that results obtained in present studyare very well in agreement with the existing results
5 Conclusions
In this study we have investigated radiation effect on velocityprofile for all components and temperature profile throughrotating disk in porous medium for variable fluid propertiesand in particular case for constant fluid properties also By
Advances in Mathematical Physics 9
0 1 2 3 4 5 6
= 0
= 02
Ws = minus1
２ = 1 10 100 1000 10000
00005
0010015
0020025
0030035
004
F(
)
(a)
= 0
= 02
5 63 41 20
Ws = minus1
２ = 1 10 100 1000 10000
0010203040506070809
1
G(
)
(b)
0 2 3 41 65
Ws = minus2 = 0
２ = 1
２ = 10
２ = 100
２ = 1000
２ = 10000
minus204minus2035minus203
minus2025minus202
minus2015minus201
minus2005minus2
minus1995minus199
H(
)
(c)
0 2 3 41 65
Ws = minus2 = 02 ２ = 1
２ = 10
２ = 100
２ = 1000
２ = 10000
minus2
minus195
minus19
minus185
minus18
minus175
minus17
H(
)
(d)
0 2 4 6 8 10 12 14
= 0
= 02
Ws = minus1
２ = 1 10 100 1000 10000
0010203040506070809
1
(
)
(e)
Figure 4 Effect of variation in the Reynolds number on the (a) radial (b) tangential (c) amp (d) axial and (e) temperature velocity profileswhen 119870 = 1 Kn = 005119873 = 1 and Pr = 1
Table 3 Comparison between the results of present study with the results reported by Kelson and Desseaux [48] Maleque and Sattar [33]and Alam et al [49] for rate of heat transfer minus1205791015840(0) for Pr = 071119872 = 0 120576 = 0 Kn = 0 and119873 = 0119882119904 Present Alam et al [49] Maleque and Sattar [33] Kelson and Desseaux [48]minus1205791015840(0) minus1205791015840(0) minus1205791015840(0) minus1205791015840(0)0 0326798372 032637889 032576 0325856minus2 1438764651 143876482 144212 1437782minus4 2842381877 284369011 284470 2842381minus5 3551223146 355222471 355411 3551223
10 Advances in Mathematical Physics
0 1 2 3 4 5 6
= 0
= 02
00005
0010015
0020025
0030035
004F
()
Ws = minus1
＋Ｈ = 0 002 005 01
(a)
0 1 2 3 4 5 6
= 0
= 02
0010203040506070809
1
G(
)
Ws = minus1
＋Ｈ = 0 002 005 01
(b)
0 1 2 3 4 5 6
= 0
= 02
minus2
minus195
minus19
minus185
minus18
minus175
minus17
H(
)
Ws = minus2
＋Ｈ = 005 002 0
(c)
= 0
= 02
0010203040506070809
1
(
)
104 6 8 12 1420
Ws = minus1
＋Ｈ = 0 002 005 01N = 0 ＋Ｈ = 005
(d)
Figure 5 Effect of variation in the Knudsen number on the (a) radial (b) tangential (c) axial and (d) temperature velocity profiles when119870 = 1 Re = 100119873 = 1 and Pr = 1
= 0
= 02
2 4 6 8 10 12 140
00102030405060708
(
)
N = 0 ０Ｌ = 4０Ｌ = 1 2 3 4
Figure 6 Effect of Prandtl number on temperature distributionwhen 119870 = 1119882119904 = minus1 Re = 100 Kn = 005 and119873 = 1
similarity transformation governing equations transformedinto nonlinear ordinary differential equations which aresolved numerically by usingRunge-Kuttamethodwith shoot-ing technique Based on the resulting solutions the followingconclusions can be drawn
= 0
= 02
2 4 6 8 10 12 140
N = 4 3 2 1 0
0010203040506070809
(
)
Figure 7 Effect of radiation parameter on temperature distributionwhen 119870 = 1119882119904 = minus1 Re = 100 Kn = 005 and Pr = 1
(i) The radial tangential and axial velocity profilesdecrease while the temperature increases with theincreasing values of porosity parameter
(ii) The increasing value of Reynolds and Knudsen num-ber decreases the fluid velocity components and
Advances in Mathematical Physics 11
1 2 3 4 5 60
= 0 02 04 10
0005
001
0015
002
0025F
()
(a)
1 2 54 60 3
0
01
02
03
04
05
06
G(
)
= 1 04 02 0
(b)
1 2 3 4 5 60
= 0
= 02
= 04
= 1
minus11
minus1
minus09
minus08
minus07
minus06
minus05
minus04
H(
)
(c)
00102030405060708
(
)
2 4 6 8 10 12 140
N = 0 = 02
= 1 04 02 0
(d)
Figure 8 Effect of variation in the relative temperature difference parameter on the (a) radial (b) tangential (c) axial and (d) temperaturevelocity profiles when119870 = 1119882119904 = minus1 Kn = 005 Re = 100119873 = 1 and Pr = 1
temperature and suction parameter also shows thesame effect
(iii) For the effect of the radiation parameter on the tem-perature distribution it is seen that the temperaturedistribution decreases with the increasing values ofradiation parameter and also it has been observed thatthe radial and tangential skin friction values decreasewith increase in the radiation parameter
Competing Interests
The authors declare that they have no competing interests
References
[1] Th V Karman ldquoUber laminare und turbulente reibunrdquo Zeits-chrift fur Angewandte Mathematik und Mechanik vol 1 no 4pp 233ndash252 1921
[2] W G Cochran ldquoThe flow due to a rotating diskrdquoMathematicalProceedings of the Cambridge Philosophical Society vol 30 no3 pp 365ndash375 1934
[3] E R Benton ldquoOn the flow due to a rotating diskrdquo Journal ofFluid Mechanics vol 24 no 4 pp 781ndash800 1966
[4] M Turkyilmazoglu ldquoPurely analytic solutions of the compress-ible boundary layer flow due to a porous rotating disk with heattransferrdquoPhysics of Fluids vol 21 no 10 Article ID 106104 2009
[5] D S Chauhan and S Gupta ldquoSteady flow and heat transferbetween two stationary naturally permeable disksrdquo Journal ofUltra Scientist of Physical Sciences vol 12 no 1 pp 45ndash52 2000
[6] C Wagner ldquoHeat transfer from a rotating disk to ambient airrdquoJournal of Applied Physics vol 19 no 9 pp 837ndash839 1948
[7] K Millsaps and K Pohlhausen ldquoHeat transfer by laminar flowfrom a rotating platerdquo Journal of the Aeronautical Sciences vol19 pp 120ndash126 1952
[8] F Kreith and J H Taylor ldquoHeat transfer from a rotating disk inturbulent flow no 1956rdquo ASME Paper 56-A-146 1956
[9] H-T Lin and L-K Lin ldquoHeat transfer from a rotating cone ordisk to fluids of any Prandtl numberrdquo International Communi-cations in Heat and Mass Transfer vol 14 no 3 pp 323ndash3321987
[10] P D Verma and D S Chauhan ldquoFlow between a torsionallyoscillating impermeable disc and a stationary naturally perme-able discrdquo Indian Journal of Pure and Applied Mathematics vol10 pp 1351ndash1361 1979
[11] D S Chauhan and S Jain ldquoSteady flow between highlypermeable rotating disksrdquo Indian Journal of Theoretical Physicsvol 52 no 1 pp 39ndash50 2004
[12] M Turkyilmazoglu ldquoNanofluid flow and heat transfer due to arotating diskrdquo Computers amp Fluids vol 94 pp 139ndash146 2014
[13] M Turkyilmazoglu ldquoMHD fluid flow and heat transfer due to ashrinking rotating diskrdquo Computers amp Fluids vol 90 pp 51ndash562014
[14] M Azimi and R Riazi ldquoHeat transfer analysis of GO-waternanofluid flow between two parallel disksrdquo Propulsion andPower Research vol 4 no 1 pp 23ndash30 2015
12 Advances in Mathematical Physics
[15] B R Sharma and H Konwar ldquoEffect of chemical reaction onmass transfer due to a permeable rotating heated diskrdquo Interna-tional Journal of Computer Applications vol 119 no 21 pp 5ndash92015
[16] H Shahmohamadi and M Mohammadpour ldquoA series solutionfor three-dimensional navier-stokes equations of flow near aninfinite rotating diskrdquo World Journal of Mechanics vol 4 pp117ndash127 2014
[17] S Srinivas A S Reddy T R Ramamohan and A K ShuklaldquoThermal-diffusion and diffusion-thermo effects onMHD flowof viscous fluid between expanding or contracting rotatingporous disks with viscous dissipationrdquo Journal of the EgyptianMathematical Society vol 24 no 1 pp 100ndash107 2016
[18] M A Mansour ldquoRadiative and free-convection effects on theoscillatory flow past a vertical platerdquo Astrophysics and SpaceScience vol 166 no 2 pp 269ndash275 1990
[19] M A Hossain M A Alim and D A S Rees ldquoThe effectof radiation on free convection from a porous vertical platerdquoInternational Journal of Heat and Mass Transfer vol 42 no 1pp 181ndash191 1999
[20] A Raptis and C Perdikis ldquoMHD free convection flow by thepresence of radiationrdquo International Journal of Magnetohydro-dynamics Plasma and Space Research vol 9 pp 237ndash252 2000
[21] M A Hossain and H S Takhar ldquoRadiation effect on mixedconvection along a vertical plate with uniform surface temper-aturerdquoHeat and Mass Transfer vol 31 no 4 pp 243ndash248 1996
[22] A Devi and R U Devi ldquoEffects of thermal radiation onhydromagnetic flow due to a porous rotating disk with halleffectrdquo Journal of Applied Fluid Mechanics vol 5 no 2 pp 1ndash7 2012
[23] C Y Wang ldquoFlow due to a stretching boundary with partialslipmdashan exact solution of the Navier-Stokes equationsrdquo Chemi-cal Engineering Science vol 57 no 17 pp 3745ndash3747 2002
[24] E Osalusi ldquoEffects of thermal radiation on MHD and slip flowover a porous rotating disk with variable propertiesrdquo RomanianJournal of Physics vol 52 no 3-4 pp 217ndash229 2007
[25] A A Khidir ldquoViscous dissipation Ohmic heating and radiationeffects on MHD flow past a rotating disk embedded in aporous medium with variable propertiesrdquo Arabian Journal ofMathematics vol 2 no 3 pp 263ndash277 2013
[26] E M Sparrow G S Beavers and L Y Hung ldquoFlow about aporous-surfaced rotating diskrdquo International Journal of Heatand Mass Transfer vol 14 no 7 pp 993ndash996 1971
[27] M M Rashidi and N Freidooni Mehr ldquoEffects of velocity slipand temperature jump on the entropy generation in magne-tohydrodynamic flow over a porous rotating diskrdquo Journal ofMechanical Engineering vol 1 no 3 2012
[28] A Arikoglu G Komurgoz I Ozkol and A Y Gunes ldquoCom-bined effects of temperature and velocity jump on the heattransfer fluid flow and entropy generation over a single rotatingdiskrdquo Journal of Heat Transfer vol 132 no 11 Article ID 1117032010
[29] THayatM RashidM Imtiaz andAAlsaedi ldquoMagnetohydro-dynamic (MHD) flow of Cu-water nanofluid due to a rotatingdisk with partial sliprdquo AIP Advances vol 5 no 6 Article ID067169 2015
[30] M Zakerullah and J A D Ackroyd ldquoLaminar natural convec-tion boundary-layers on Horizontal Circular disksrdquo Journal ofApplied Mathematics and Physics vol 30 pp 427ndash435 1979
[31] HHerwig ldquoThe effect of variable properties onmomentumandheat transfer in a tube with constant heat flux across the wallrdquoInternational Journal of Heat and Mass Transfer vol 28 no 2pp 423ndash431 1985
[32] H Herwig and K Klemp ldquoVariable property effects of fullydeveloped laminar flow in concentric annulirdquo Journal of HeatTransfer vol 110 no 2 pp 314ndash320 1988
[33] K AMaleque andMA Sattar ldquoSteady laminar convective flowwith variable properties due to a porous rotating diskrdquo Journalof Heat Transfer vol 127 no 12 pp 1406ndash1409 2005
[34] K A Maleque and M A Sattar ldquoThe effects of variable proper-ties and hall current on steady MHD laminar convective fluidflow due to a porous rotating diskrdquo International Journal of Heatand Mass Transfer vol 48 no 23-24 pp 4963ndash4972 2005
[35] E Osalusi and P Sibanda ldquoOn variable laminar convective flowproperties due to a porous rotating disk in a magnetic fieldrdquoRomanian Journal of Physics vol 9 no 10 pp 933ndash944 2006
[36] M M Rahman ldquoConvective Hydromagnetic slip flow withvariable properties due to a porous rotating diskrdquo The SultanQaboos University Journal for Science vol 15 pp 55ndash79 2010
[37] M M Rashidi S A Mohimanian Pour T Hayat and SObaidat ldquoAnalytic approximate solutions for steady flow over arotating disk in porousmediumwith heat transfer by homotopyanalysis methodrdquo Computers amp Fluids vol 54 no 1 pp 1ndash92012
[38] S Hussain F Ahmad M Shafique and S Hussain ldquoNumericalsolution for accelerated rotating disk in a viscous fluidrdquoAppliedMathematics vol 4 no 6 pp 899ndash902 2013
[39] M M Rashidi N Kavyani and S Abelman ldquoInvestigation ofentropy generation inMHDand slip flow over a rotating porousdisk with variable propertiesrdquo International Journal of Heat andMass Transfer vol 70 pp 892ndash917 2014
[40] N Freidoonimehr M M Rashidi S Abelman and G Loren-zini ldquoAnalytical modeling of MHD flow over a permeablerotating disk in the presence of Soret and Dufour effectsentropy analysisrdquo Entropy vol 18 no 5 article 131 2016
[41] M S Alam S M Chapal Hossain and M M Rahman ldquoTran-sient thermophoretic particle deposition on forced convectiveheat and mass transfer flow due to a rotating diskrdquo Ain ShamsEngineering Journal vol 7 no 1 pp 441ndash452 2016
[42] S Jayaraj ldquoThermophoresis in laminar flow over cold inclinedplates with variable propertiesrdquoHeat andMass Transfer vol 30no 3 pp 167ndash173 1995
[43] M M Rahman ldquoThermophoretic deposition of nanoparticlesdue to a permeable rotating disk effects of partial slip mag-netic field thermal radiation thermal-diffusion and diffusion-thermordquo International Journal of Mathematical ComputationalPhysical Electrical and Computer Engineering vol 7 no 5 2013
[44] I V Shevchuk Convective Heat and Mass Transfer in RotatingDisk Systems Springer Berlin Germany 2009
[45] C L Tien and D T Campbell ldquoHeat and mass transfer fromrotating conesrdquo Journal of FluidMechanics vol 17 no 1 pp 105ndash112 1963
[46] C J Elkins and J K Eaton ldquoHeat transfer in the rotatingdisk boundary layerrdquo Tech Rep TSD-103 Stanford UniversityDepartment of Mechanical EngineeringThermosciences Divi-sion Stanford Calif USA 1997
[47] G Karniadakis A Beskok and N Aluru Microflows Funda-mentals and Simulation Springer New York NY USA 2001
[48] N Kelson andA Desseaux ldquoNote on porous rotating disk flowrdquoAustralian amp New Zealand Industrial and Applied MathematicsJournal vol 42 pp 837ndash855 2000
[49] M Alam N Poddar M Rahman and K Vajravelu ldquoTransienthydromagnetic forced convective heat transfer slip flow due toa porous rotating disk with variable fluid propertiesrdquo AmericanJournal of Heat andMass Transfer vol 2 no 3 pp 165ndash189 2015
Following [34 35 42] the fluid properties viscosity (120583)thermal conductivity (120581) and the density (120588) are taken asfunctions of temperature
120583 = 120583infin ( 119879119879infin)119886
120581 = 120581infin ( 119879119879infin)119887
120588 = 120588infin ( 119879119879infin)119888
(1)
where 119886 119887 and 119888 are arbitrary exponents and 120583infin 120581infin and 120588infinare the uniformviscosity thermal conductivity anddensity ofthe fluid Assume 119886 = 07 119887 = 083 and 119888 = minus10 (ideal gas)are the values of exponents for present investigation
The governing equations of continuity momentum andenergy for laminar incompressible flow in cylindrical coordi-nates are [35]
at 119911 997888rarr infin119906 997888rarr infinV 997888rarr infin119879 997888rarr 119879infin
(7)
where 120590V is the tangential momentum accommodation coef-ficient 120590119905 is the energy accommodation coefficient 120582 is meanfree path 120573 is the ratio of specific heats and 119879119904 is thetemperature of the fluid near to the disk surface
Rosseland approximation has been used for radiation 119902119903
where1205901 is the Stefan-Boltzmann constant and1198794 is themeanabsorption coefficient It is assumed that the temperaturedifferences within the flow are sufficiently small so that theterm1198794may be expressed as a linear function of temperatureThis is done by expanding 1198794 in a Taylor series about 119879infin andomitting the second- and higher-order terms leads to
1198794 cong 41198793infin119879 minus 31198794infin120597119902119903120597119911 = 120597120597119911 (minus4120590131198961
1205971198794120597119911 )
= 120597120597119911 (minus4120590131198961120597 (41198793infin119879 minus 31198794infin)120597119911 )
119901 minus 119901infin = minus120583infinΩ119875 (120578) 120579 (120578) = (119879 minus 119879infin)(119879119908 minus 119879infin)
(10)
where ]infin is uniformkinematic viscosity and119865119866119867 120579 and119875are nondimensional functions in terms of vertical coordinate120578 Substituting (10) in (2)ndash(6) we get the system of followingordinary differential equations
1198671015840 + 2119865 + 1198881205761198671205791015840 (1 + 120576120579)minus1 = 011986510158401015840 minus 119870119865 minus (1 + 120576120579)119888minus119886 [1198652 minus 1198662 + 1198671198651015840]+ 119886120576 (1 + 120576120579)minus1 12057910158401198651015840 = 0
11986610158401015840 minus 119870119866 minus (1 + 120576120579)119888minus119886 [1198671198661015840 + 2119865119866]+ 119886120576 (1 + 120576120579)minus1 12057910158401198661015840 = 0
where 119870 = ]infin119896Ω Pr = 120583infin119862119901120581infin and 119873 = 412059011198793infin1198961120581infinare the porosity parameter Prandtl number and radiationparameter respectively 120576 = Δ119879119879infin is the relative temper-ature difference parameter it is positive for a heated surfacenegative for a cooled surface and zero for the case of constantproperty
Kn = 120582119903 is the Knudsen number it is the ratio of meanfree path of fluid particle diameter For slip condition valueranges from 0001 to 01 For Kn lt 0001 no slip boundaryconditions are valid therefore velocity at the surface is zeroFor high Knudsen number high order continuum equation(Burnett equations) should be used [43] For present investi-gation slip regime of Knudsen number which lies in the range0001 lt Kn lt 01 has been considered
Re = Ω1199032]infin is the rotational Reynolds number forlaminar flow the value of the local Reynolds number is 18times 105 [44ndash46] For transition flow the local Reynolds numbervalues lie between 18 times 105 and 36 times 105 and for values higherthan 36 times 105 the flow becomes turbulent In this study thelaminar flow for local Reynolds number that lies in the range0 lt Re lt 10000 has been considered120574 = ((2minus120590V)120590V)KnradicRe 120574 is the slip factor depending onrotational Reynolds number and Knudsen number and mayvary from 0 to 12120593 = ((2 minus 120590119905)120590119905)(2120573(1 + 120573))(KnPr)radicRe 120593 is thetemperature jump factor varying from 0 to 12119882119904 = 1199080(Ω]infin)12 is the suction parameter which hasbeen taken as less than zero because suction of fluid is takingplace
The values of tangential momentum accommodationnumber (120590V) energy accommodation coefficient (120590119905) and thespecific heat ratio (120573) for air are considered as 09 09 and 14respectively Karniadakis et al [47]
3 Solution
The nonlinear coupled ordinary differential equations (11)with the boundary conditions (12) have been solved numer-ically applying fourth-order Runge-Kutta scheme togetherwith shooting method
The given boundary value problem is reduced to thefollowing system of initial value problem
Taking 120576 = 0 120583 = 120583infin (constant)Similarly at 120576 = 0 120588 = 120588infin and 120581 = 120581infinSubstitution of 120576 = 0 in (2)ndash(4) and (6) system of
equations is as follows
1198671015840 + 2119865 = 011986510158401015840 minus 1198671198651015840 minus 1198652 + 1198662 minus 119870119865 = 011986610158401015840 minus 1198671198661015840 minus 2119865119866 minus 119870119866 = 0(1 + 41198733 ) 12057910158401015840 minus Pr1198671205791015840 = 0
(16)
Equations (16) under the boundary condition (12) are trans-formed into the following system of initial value problems
Here 1199031 = 1198651015840(0) 1199032 = 1198661015840(0) and 1199033 = 1205791015840(0) are the initialguesses when fluid properties are variable and 1199034 = 1198651015840(0) 1199035 =1198661015840(0) and 1199036 = 1205791015840(0) are the initial guesses when fluid prop-erties are constant The essence of present numerical methodis to reduce the boundary value problem (BVP) into an initialvalue problem (IVP) Further shooting technique is used toguess 1199031 1199032 1199033 1199034 1199035 and 1199036 until the boundary conditionsare satisfied A number of iterations of Runge-Kutta fourth-order method has been performed to obtain final values ofthese guesses Initial guesses for different set of parametersare displayed in Table 1
In this problem the physical quantities of interest arelocal skin friction coefficients and the Nusselt number whichrepresents the wall shear stress and the rate of heat transferrespectively When variable fluid properties are taken intoconsideration the fluid near to the disk opposes rotation ofthe disk due to presence of tangential shear stressThereforeto maintain a steady rotation it is essential to have torque atthe shaft The skin frictions 119862119891119903 along radial direction and119862119891119905 along tangential direction at no slip condition are givenas
where Re = Ω1199032]infin is rotational Reynolds numberThus (24) shows that the radial and tangential skin
frictions coefficients are proportional to 1198651015840(0) and 1198661015840(0)respectively and Nusselt number is proportional to minus1205791015840(0)4 Result and Discussion
In this investigation Figures 2ndash6 show the effect of variousvalues of the physical parameters on the velocity (radialtangential and axial) and temperature distribution Resultsobtained for both constant fluid property and variable fluidproperty have been presented graphically
Figures 2(a)ndash2(d) show the effect of porosity parameter119870on all velocity components and temperature distribution It isobserved that radial tangential and axial velocity decreasesbut temperature increases by increasing the permeability forboth cases Whereas variation in porosity parameter has lesseffect on the temperature distribution radial velocity attains
Advances in Mathematical Physics 7
0 1 2 3 4 5 60
001002003004005006007008009
Freidoonimehr et al [40]Rashidi and Freidooni Mehr [27]
F(
)
K = = 0 ＋Ｈ = 0 ０Ｌ = 071
K = 0 = 02 ０Ｌ = 1K = = 0 ＋Ｈ = 005 ０Ｌ = 1
K = 05 = 02K = 05 = 0
K = 1 = 02K = 1 = 0
＋Ｈ = 005
０Ｌ = 1 ＋Ｈ = 005０Ｌ = 1 ＋Ｈ = 005
０Ｌ = 1 ＋Ｈ = 005０Ｌ = 1 ＋Ｈ = 005
(a)
0010203040506070809
1
0 1 2 3 4 5 6
G(
) K = = 0 ＋Ｈ = 0 ０Ｌ = 071K = 0 = 02
K = = 0 ＋Ｈ = 005 ０Ｌ = 1
K = 1 = 02K = 1 = 0
＋Ｈ = 005 ０Ｌ = 1
＋Ｈ = 005 ０Ｌ = 1 ＋Ｈ = 005 ０Ｌ = 1
Rashidi and Freidooni Mehr [27]Freidoonimehr et al [40]
(b)
0 1 2 3 4 5 6
H(
)
minus07
minus08
minus09
minus1
minus11
minus12
minus13
K = 10 = 02
K = 10 = 0K = 1 = 0
K = 0 = 02
K = = 0 ＋Ｈ = 005 ０Ｌ = 1K = 0 = 0
＋Ｈ = 005 ０Ｌ = 1
K = 1 = 02 ＋Ｈ = 005 ０Ｌ = 1 ＋Ｈ = 005 ０Ｌ = 1
＋Ｈ = 005 ０Ｌ = 1
＋Ｈ = 005 ０Ｌ = 1
＋Ｈ = 0 ０Ｌ = 071
Rashidi and Freidooni Mehr [27]Freidoonimehr et al [40]
(c)
0 2 4 6 8 10 12 140
0102030405060708
(
)
= 02 N = 1
= 0 N = 1
= 0 N = 0 K = 0K = 10 1 0
＋Ｈ = 005 ０Ｌ = 1
Rashidi and Freidooni Mehr [27]
(d)
Figure 2 Effect of variation in the porosity parameter on the (a) radial (b) tangential (c) axial and (d) temperature velocity profiles when119882119904 = minus1 and Re = 100
a maximum value close to the disk surface When we reducethe present problem into the literature available by takingpermeability parameter 119870 = 0 and radiation parameter119873 = 0 then the result obtained is exactly similar to thatof the Rashidi and Freidooni Mehr [27] and Freidoonimehret al [40] This validates the results obtained for presentinvestigation
Figures 3(a)ndash3(d) depict the effect of suction param-eter 119882119904 on the radial tangential axial and temperaturedistribution respectively Figures 3(a)ndash3(c) show that allvelocity components decrease as suction increases Physicalsignificance is that as suction increases adherence of thefluid with the wall increases and as a result boundary layerthickness decreases
Figure 3(d) depicts the effect of suction parameter ontemperature profile It is observed that as suction increasestemperature of fluid decreases because suction cools theboundary layer regime therefore suction is used for coolingthe flow in several engineering processes including MHDpower generators and nuclear energy processes
The effect of Reynolds number Re and Knudsen num-ber Kn on velocity and temperature distribution is plotted
in Figures 4 and 5 respectively Both the parameters areshowing the same effects on slip boundary conditions Itis observed that the increasing values of the Reynolds andKnudsen numbers decrease the fluid velocity componentsand temperature distribution The case when value of Kn liesbetween 0001 and 01 corresponds to slip at the surface of thedisk As slip increases the quantity of the fluid that can stickon the disk decreases Therefore circumferential velocity ofthe fluid reduces and causes reduction in centrifugal forceAs a result inward axial velocity decreases In other words asslip gets stronger flow of fluid drawn or pushed away alongthe velocity directions decreases the heat generation
Figure 6 shows the effect of Prandtl number on tem-perature profile As Prandtl number increases the thermalboundary layer thickness decreases Physical significance isthat Prandtl number precludes dispersal of heat in the fluid
Figure 7 depicts the variation of temperature profilewith radiation It is observed that as radiation increasestemperature of the fluid increases Also radiation parameterincreases temperature gradient near the surface of the disk
Figure 8 demonstrates the comparison of both constantand fluid properties on flow over a rotating disk with slip and
8 Advances in Mathematical Physics
0 1 2 3 4 5 6
0
0005
001
0015
002
0025F
()
Ws = minus1 minus2 minus4
= 0
= 02
(a)
0005
01015
02025
03035
04045
05
0 1 2 3 4 5 6
Ws = minus1 minus2 minus4
= 0
= 02
G(
)
(b)
0 1 2 3 4 5 6
= 0
= 02
Ws = minus1
Ws = minus2
Ws = minus4
H(
)
minus1
minus15
minus2
minus25
minus3
minus35
minus4
minus45
(c)
00102030405060708
Ws = minus1 minus2 minus4
= 0
= 02
10 1282 640 14
(
)
(d)
Figure 3 Effect of variation in the suction parameter on the (a) radial (b) tangential (c) axial and (d) temperature velocity profiles when119870 = 1 Kn = 005 Re = 100119873 = 1 and Pr = 1
Table 2 Comparison between the results of present study with the results reported by Kelson and Desseaux [48] Maleque and Sattar [33]and Alam et al [49] for the radial 1198651015840(0) and tangential minus1198661015840(0) skin friction coefficients for Pr = 071119872 = 0 120576 = 0 and Kn = 0
temperature jump The radial velocity achieves a maximumvalue near to the surface of the disk for all values of 120576 It is alsonoted that an increment in relative temperature differenceparameter causes shifting of maximum point from the diskThe velocity along the tangential direction increases asvalue of relative temperature difference parameter increaseswhereas the velocity along axial direction decreases Temper-ature increases as the value of relative temperature differenceparameter increasesThese results are similar to that obtainedby Rashidi et al [39]
Table 1 illustrates the effect of the parameters 120576 119870Kn Re Pr and 119873 on constant suction parameter on thenumerical values of (1198651015840(0) minus1198661015840(0)) and (minus1205791015840(0)) We observethat numerical value of 1198651015840(0) minus1198661015840(0) and minus1205791015840(0) decreases
with the increasing value of Kn Re and radiation parameter119873Tables 2 and 3 depict the comparison of skin friction coef-
ficients and rate of heat transfer between the results obtainedin the present investigation and the literature available [3348 49] It is observed that results obtained in present studyare very well in agreement with the existing results
5 Conclusions
In this study we have investigated radiation effect on velocityprofile for all components and temperature profile throughrotating disk in porous medium for variable fluid propertiesand in particular case for constant fluid properties also By
Advances in Mathematical Physics 9
0 1 2 3 4 5 6
= 0
= 02
Ws = minus1
２ = 1 10 100 1000 10000
00005
0010015
0020025
0030035
004
F(
)
(a)
= 0
= 02
5 63 41 20
Ws = minus1
２ = 1 10 100 1000 10000
0010203040506070809
1
G(
)
(b)
0 2 3 41 65
Ws = minus2 = 0
２ = 1
２ = 10
２ = 100
２ = 1000
２ = 10000
minus204minus2035minus203
minus2025minus202
minus2015minus201
minus2005minus2
minus1995minus199
H(
)
(c)
0 2 3 41 65
Ws = minus2 = 02 ２ = 1
２ = 10
２ = 100
２ = 1000
２ = 10000
minus2
minus195
minus19
minus185
minus18
minus175
minus17
H(
)
(d)
0 2 4 6 8 10 12 14
= 0
= 02
Ws = minus1
２ = 1 10 100 1000 10000
0010203040506070809
1
(
)
(e)
Figure 4 Effect of variation in the Reynolds number on the (a) radial (b) tangential (c) amp (d) axial and (e) temperature velocity profileswhen 119870 = 1 Kn = 005119873 = 1 and Pr = 1
Table 3 Comparison between the results of present study with the results reported by Kelson and Desseaux [48] Maleque and Sattar [33]and Alam et al [49] for rate of heat transfer minus1205791015840(0) for Pr = 071119872 = 0 120576 = 0 Kn = 0 and119873 = 0119882119904 Present Alam et al [49] Maleque and Sattar [33] Kelson and Desseaux [48]minus1205791015840(0) minus1205791015840(0) minus1205791015840(0) minus1205791015840(0)0 0326798372 032637889 032576 0325856minus2 1438764651 143876482 144212 1437782minus4 2842381877 284369011 284470 2842381minus5 3551223146 355222471 355411 3551223
10 Advances in Mathematical Physics
0 1 2 3 4 5 6
= 0
= 02
00005
0010015
0020025
0030035
004F
()
Ws = minus1
＋Ｈ = 0 002 005 01
(a)
0 1 2 3 4 5 6
= 0
= 02
0010203040506070809
1
G(
)
Ws = minus1
＋Ｈ = 0 002 005 01
(b)
0 1 2 3 4 5 6
= 0
= 02
minus2
minus195
minus19
minus185
minus18
minus175
minus17
H(
)
Ws = minus2
＋Ｈ = 005 002 0
(c)
= 0
= 02
0010203040506070809
1
(
)
104 6 8 12 1420
Ws = minus1
＋Ｈ = 0 002 005 01N = 0 ＋Ｈ = 005
(d)
Figure 5 Effect of variation in the Knudsen number on the (a) radial (b) tangential (c) axial and (d) temperature velocity profiles when119870 = 1 Re = 100119873 = 1 and Pr = 1
= 0
= 02
2 4 6 8 10 12 140
00102030405060708
(
)
N = 0 ０Ｌ = 4０Ｌ = 1 2 3 4
Figure 6 Effect of Prandtl number on temperature distributionwhen 119870 = 1119882119904 = minus1 Re = 100 Kn = 005 and119873 = 1
similarity transformation governing equations transformedinto nonlinear ordinary differential equations which aresolved numerically by usingRunge-Kuttamethodwith shoot-ing technique Based on the resulting solutions the followingconclusions can be drawn
= 0
= 02
2 4 6 8 10 12 140
N = 4 3 2 1 0
0010203040506070809
(
)
Figure 7 Effect of radiation parameter on temperature distributionwhen 119870 = 1119882119904 = minus1 Re = 100 Kn = 005 and Pr = 1
(i) The radial tangential and axial velocity profilesdecrease while the temperature increases with theincreasing values of porosity parameter
(ii) The increasing value of Reynolds and Knudsen num-ber decreases the fluid velocity components and
Advances in Mathematical Physics 11
1 2 3 4 5 60
= 0 02 04 10
0005
001
0015
002
0025F
()
(a)
1 2 54 60 3
0
01
02
03
04
05
06
G(
)
= 1 04 02 0
(b)
1 2 3 4 5 60
= 0
= 02
= 04
= 1
minus11
minus1
minus09
minus08
minus07
minus06
minus05
minus04
H(
)
(c)
00102030405060708
(
)
2 4 6 8 10 12 140
N = 0 = 02
= 1 04 02 0
(d)
Figure 8 Effect of variation in the relative temperature difference parameter on the (a) radial (b) tangential (c) axial and (d) temperaturevelocity profiles when119870 = 1119882119904 = minus1 Kn = 005 Re = 100119873 = 1 and Pr = 1
temperature and suction parameter also shows thesame effect
(iii) For the effect of the radiation parameter on the tem-perature distribution it is seen that the temperaturedistribution decreases with the increasing values ofradiation parameter and also it has been observed thatthe radial and tangential skin friction values decreasewith increase in the radiation parameter
Competing Interests
The authors declare that they have no competing interests
References
[1] Th V Karman ldquoUber laminare und turbulente reibunrdquo Zeits-chrift fur Angewandte Mathematik und Mechanik vol 1 no 4pp 233ndash252 1921
[2] W G Cochran ldquoThe flow due to a rotating diskrdquoMathematicalProceedings of the Cambridge Philosophical Society vol 30 no3 pp 365ndash375 1934
[3] E R Benton ldquoOn the flow due to a rotating diskrdquo Journal ofFluid Mechanics vol 24 no 4 pp 781ndash800 1966
[4] M Turkyilmazoglu ldquoPurely analytic solutions of the compress-ible boundary layer flow due to a porous rotating disk with heattransferrdquoPhysics of Fluids vol 21 no 10 Article ID 106104 2009
[5] D S Chauhan and S Gupta ldquoSteady flow and heat transferbetween two stationary naturally permeable disksrdquo Journal ofUltra Scientist of Physical Sciences vol 12 no 1 pp 45ndash52 2000
[6] C Wagner ldquoHeat transfer from a rotating disk to ambient airrdquoJournal of Applied Physics vol 19 no 9 pp 837ndash839 1948
[7] K Millsaps and K Pohlhausen ldquoHeat transfer by laminar flowfrom a rotating platerdquo Journal of the Aeronautical Sciences vol19 pp 120ndash126 1952
[8] F Kreith and J H Taylor ldquoHeat transfer from a rotating disk inturbulent flow no 1956rdquo ASME Paper 56-A-146 1956
[9] H-T Lin and L-K Lin ldquoHeat transfer from a rotating cone ordisk to fluids of any Prandtl numberrdquo International Communi-cations in Heat and Mass Transfer vol 14 no 3 pp 323ndash3321987
[10] P D Verma and D S Chauhan ldquoFlow between a torsionallyoscillating impermeable disc and a stationary naturally perme-able discrdquo Indian Journal of Pure and Applied Mathematics vol10 pp 1351ndash1361 1979
[11] D S Chauhan and S Jain ldquoSteady flow between highlypermeable rotating disksrdquo Indian Journal of Theoretical Physicsvol 52 no 1 pp 39ndash50 2004
[12] M Turkyilmazoglu ldquoNanofluid flow and heat transfer due to arotating diskrdquo Computers amp Fluids vol 94 pp 139ndash146 2014
[13] M Turkyilmazoglu ldquoMHD fluid flow and heat transfer due to ashrinking rotating diskrdquo Computers amp Fluids vol 90 pp 51ndash562014
[14] M Azimi and R Riazi ldquoHeat transfer analysis of GO-waternanofluid flow between two parallel disksrdquo Propulsion andPower Research vol 4 no 1 pp 23ndash30 2015
12 Advances in Mathematical Physics
[15] B R Sharma and H Konwar ldquoEffect of chemical reaction onmass transfer due to a permeable rotating heated diskrdquo Interna-tional Journal of Computer Applications vol 119 no 21 pp 5ndash92015
[16] H Shahmohamadi and M Mohammadpour ldquoA series solutionfor three-dimensional navier-stokes equations of flow near aninfinite rotating diskrdquo World Journal of Mechanics vol 4 pp117ndash127 2014
[17] S Srinivas A S Reddy T R Ramamohan and A K ShuklaldquoThermal-diffusion and diffusion-thermo effects onMHD flowof viscous fluid between expanding or contracting rotatingporous disks with viscous dissipationrdquo Journal of the EgyptianMathematical Society vol 24 no 1 pp 100ndash107 2016
[18] M A Mansour ldquoRadiative and free-convection effects on theoscillatory flow past a vertical platerdquo Astrophysics and SpaceScience vol 166 no 2 pp 269ndash275 1990
[19] M A Hossain M A Alim and D A S Rees ldquoThe effectof radiation on free convection from a porous vertical platerdquoInternational Journal of Heat and Mass Transfer vol 42 no 1pp 181ndash191 1999
[20] A Raptis and C Perdikis ldquoMHD free convection flow by thepresence of radiationrdquo International Journal of Magnetohydro-dynamics Plasma and Space Research vol 9 pp 237ndash252 2000
[21] M A Hossain and H S Takhar ldquoRadiation effect on mixedconvection along a vertical plate with uniform surface temper-aturerdquoHeat and Mass Transfer vol 31 no 4 pp 243ndash248 1996
[22] A Devi and R U Devi ldquoEffects of thermal radiation onhydromagnetic flow due to a porous rotating disk with halleffectrdquo Journal of Applied Fluid Mechanics vol 5 no 2 pp 1ndash7 2012
[23] C Y Wang ldquoFlow due to a stretching boundary with partialslipmdashan exact solution of the Navier-Stokes equationsrdquo Chemi-cal Engineering Science vol 57 no 17 pp 3745ndash3747 2002
[24] E Osalusi ldquoEffects of thermal radiation on MHD and slip flowover a porous rotating disk with variable propertiesrdquo RomanianJournal of Physics vol 52 no 3-4 pp 217ndash229 2007
[25] A A Khidir ldquoViscous dissipation Ohmic heating and radiationeffects on MHD flow past a rotating disk embedded in aporous medium with variable propertiesrdquo Arabian Journal ofMathematics vol 2 no 3 pp 263ndash277 2013
[26] E M Sparrow G S Beavers and L Y Hung ldquoFlow about aporous-surfaced rotating diskrdquo International Journal of Heatand Mass Transfer vol 14 no 7 pp 993ndash996 1971
[27] M M Rashidi and N Freidooni Mehr ldquoEffects of velocity slipand temperature jump on the entropy generation in magne-tohydrodynamic flow over a porous rotating diskrdquo Journal ofMechanical Engineering vol 1 no 3 2012
[28] A Arikoglu G Komurgoz I Ozkol and A Y Gunes ldquoCom-bined effects of temperature and velocity jump on the heattransfer fluid flow and entropy generation over a single rotatingdiskrdquo Journal of Heat Transfer vol 132 no 11 Article ID 1117032010
[29] THayatM RashidM Imtiaz andAAlsaedi ldquoMagnetohydro-dynamic (MHD) flow of Cu-water nanofluid due to a rotatingdisk with partial sliprdquo AIP Advances vol 5 no 6 Article ID067169 2015
[30] M Zakerullah and J A D Ackroyd ldquoLaminar natural convec-tion boundary-layers on Horizontal Circular disksrdquo Journal ofApplied Mathematics and Physics vol 30 pp 427ndash435 1979
[31] HHerwig ldquoThe effect of variable properties onmomentumandheat transfer in a tube with constant heat flux across the wallrdquoInternational Journal of Heat and Mass Transfer vol 28 no 2pp 423ndash431 1985
[32] H Herwig and K Klemp ldquoVariable property effects of fullydeveloped laminar flow in concentric annulirdquo Journal of HeatTransfer vol 110 no 2 pp 314ndash320 1988
[33] K AMaleque andMA Sattar ldquoSteady laminar convective flowwith variable properties due to a porous rotating diskrdquo Journalof Heat Transfer vol 127 no 12 pp 1406ndash1409 2005
[34] K A Maleque and M A Sattar ldquoThe effects of variable proper-ties and hall current on steady MHD laminar convective fluidflow due to a porous rotating diskrdquo International Journal of Heatand Mass Transfer vol 48 no 23-24 pp 4963ndash4972 2005
[35] E Osalusi and P Sibanda ldquoOn variable laminar convective flowproperties due to a porous rotating disk in a magnetic fieldrdquoRomanian Journal of Physics vol 9 no 10 pp 933ndash944 2006
[36] M M Rahman ldquoConvective Hydromagnetic slip flow withvariable properties due to a porous rotating diskrdquo The SultanQaboos University Journal for Science vol 15 pp 55ndash79 2010
[37] M M Rashidi S A Mohimanian Pour T Hayat and SObaidat ldquoAnalytic approximate solutions for steady flow over arotating disk in porousmediumwith heat transfer by homotopyanalysis methodrdquo Computers amp Fluids vol 54 no 1 pp 1ndash92012
[38] S Hussain F Ahmad M Shafique and S Hussain ldquoNumericalsolution for accelerated rotating disk in a viscous fluidrdquoAppliedMathematics vol 4 no 6 pp 899ndash902 2013
[39] M M Rashidi N Kavyani and S Abelman ldquoInvestigation ofentropy generation inMHDand slip flow over a rotating porousdisk with variable propertiesrdquo International Journal of Heat andMass Transfer vol 70 pp 892ndash917 2014
[40] N Freidoonimehr M M Rashidi S Abelman and G Loren-zini ldquoAnalytical modeling of MHD flow over a permeablerotating disk in the presence of Soret and Dufour effectsentropy analysisrdquo Entropy vol 18 no 5 article 131 2016
[41] M S Alam S M Chapal Hossain and M M Rahman ldquoTran-sient thermophoretic particle deposition on forced convectiveheat and mass transfer flow due to a rotating diskrdquo Ain ShamsEngineering Journal vol 7 no 1 pp 441ndash452 2016
[42] S Jayaraj ldquoThermophoresis in laminar flow over cold inclinedplates with variable propertiesrdquoHeat andMass Transfer vol 30no 3 pp 167ndash173 1995
[43] M M Rahman ldquoThermophoretic deposition of nanoparticlesdue to a permeable rotating disk effects of partial slip mag-netic field thermal radiation thermal-diffusion and diffusion-thermordquo International Journal of Mathematical ComputationalPhysical Electrical and Computer Engineering vol 7 no 5 2013
[44] I V Shevchuk Convective Heat and Mass Transfer in RotatingDisk Systems Springer Berlin Germany 2009
[45] C L Tien and D T Campbell ldquoHeat and mass transfer fromrotating conesrdquo Journal of FluidMechanics vol 17 no 1 pp 105ndash112 1963
[46] C J Elkins and J K Eaton ldquoHeat transfer in the rotatingdisk boundary layerrdquo Tech Rep TSD-103 Stanford UniversityDepartment of Mechanical EngineeringThermosciences Divi-sion Stanford Calif USA 1997
[47] G Karniadakis A Beskok and N Aluru Microflows Funda-mentals and Simulation Springer New York NY USA 2001
[48] N Kelson andA Desseaux ldquoNote on porous rotating disk flowrdquoAustralian amp New Zealand Industrial and Applied MathematicsJournal vol 42 pp 837ndash855 2000
[49] M Alam N Poddar M Rahman and K Vajravelu ldquoTransienthydromagnetic forced convective heat transfer slip flow due toa porous rotating disk with variable fluid propertiesrdquo AmericanJournal of Heat andMass Transfer vol 2 no 3 pp 165ndash189 2015
119901 minus 119901infin = minus120583infinΩ119875 (120578) 120579 (120578) = (119879 minus 119879infin)(119879119908 minus 119879infin)
(10)
where ]infin is uniformkinematic viscosity and119865119866119867 120579 and119875are nondimensional functions in terms of vertical coordinate120578 Substituting (10) in (2)ndash(6) we get the system of followingordinary differential equations
1198671015840 + 2119865 + 1198881205761198671205791015840 (1 + 120576120579)minus1 = 011986510158401015840 minus 119870119865 minus (1 + 120576120579)119888minus119886 [1198652 minus 1198662 + 1198671198651015840]+ 119886120576 (1 + 120576120579)minus1 12057910158401198651015840 = 0
11986610158401015840 minus 119870119866 minus (1 + 120576120579)119888minus119886 [1198671198661015840 + 2119865119866]+ 119886120576 (1 + 120576120579)minus1 12057910158401198661015840 = 0
where 119870 = ]infin119896Ω Pr = 120583infin119862119901120581infin and 119873 = 412059011198793infin1198961120581infinare the porosity parameter Prandtl number and radiationparameter respectively 120576 = Δ119879119879infin is the relative temper-ature difference parameter it is positive for a heated surfacenegative for a cooled surface and zero for the case of constantproperty
Kn = 120582119903 is the Knudsen number it is the ratio of meanfree path of fluid particle diameter For slip condition valueranges from 0001 to 01 For Kn lt 0001 no slip boundaryconditions are valid therefore velocity at the surface is zeroFor high Knudsen number high order continuum equation(Burnett equations) should be used [43] For present investi-gation slip regime of Knudsen number which lies in the range0001 lt Kn lt 01 has been considered
Re = Ω1199032]infin is the rotational Reynolds number forlaminar flow the value of the local Reynolds number is 18times 105 [44ndash46] For transition flow the local Reynolds numbervalues lie between 18 times 105 and 36 times 105 and for values higherthan 36 times 105 the flow becomes turbulent In this study thelaminar flow for local Reynolds number that lies in the range0 lt Re lt 10000 has been considered120574 = ((2minus120590V)120590V)KnradicRe 120574 is the slip factor depending onrotational Reynolds number and Knudsen number and mayvary from 0 to 12120593 = ((2 minus 120590119905)120590119905)(2120573(1 + 120573))(KnPr)radicRe 120593 is thetemperature jump factor varying from 0 to 12119882119904 = 1199080(Ω]infin)12 is the suction parameter which hasbeen taken as less than zero because suction of fluid is takingplace
The values of tangential momentum accommodationnumber (120590V) energy accommodation coefficient (120590119905) and thespecific heat ratio (120573) for air are considered as 09 09 and 14respectively Karniadakis et al [47]
3 Solution
The nonlinear coupled ordinary differential equations (11)with the boundary conditions (12) have been solved numer-ically applying fourth-order Runge-Kutta scheme togetherwith shooting method
The given boundary value problem is reduced to thefollowing system of initial value problem
Taking 120576 = 0 120583 = 120583infin (constant)Similarly at 120576 = 0 120588 = 120588infin and 120581 = 120581infinSubstitution of 120576 = 0 in (2)ndash(4) and (6) system of
equations is as follows
1198671015840 + 2119865 = 011986510158401015840 minus 1198671198651015840 minus 1198652 + 1198662 minus 119870119865 = 011986610158401015840 minus 1198671198661015840 minus 2119865119866 minus 119870119866 = 0(1 + 41198733 ) 12057910158401015840 minus Pr1198671205791015840 = 0
(16)
Equations (16) under the boundary condition (12) are trans-formed into the following system of initial value problems
Here 1199031 = 1198651015840(0) 1199032 = 1198661015840(0) and 1199033 = 1205791015840(0) are the initialguesses when fluid properties are variable and 1199034 = 1198651015840(0) 1199035 =1198661015840(0) and 1199036 = 1205791015840(0) are the initial guesses when fluid prop-erties are constant The essence of present numerical methodis to reduce the boundary value problem (BVP) into an initialvalue problem (IVP) Further shooting technique is used toguess 1199031 1199032 1199033 1199034 1199035 and 1199036 until the boundary conditionsare satisfied A number of iterations of Runge-Kutta fourth-order method has been performed to obtain final values ofthese guesses Initial guesses for different set of parametersare displayed in Table 1
In this problem the physical quantities of interest arelocal skin friction coefficients and the Nusselt number whichrepresents the wall shear stress and the rate of heat transferrespectively When variable fluid properties are taken intoconsideration the fluid near to the disk opposes rotation ofthe disk due to presence of tangential shear stressThereforeto maintain a steady rotation it is essential to have torque atthe shaft The skin frictions 119862119891119903 along radial direction and119862119891119905 along tangential direction at no slip condition are givenas
where Re = Ω1199032]infin is rotational Reynolds numberThus (24) shows that the radial and tangential skin
frictions coefficients are proportional to 1198651015840(0) and 1198661015840(0)respectively and Nusselt number is proportional to minus1205791015840(0)4 Result and Discussion
In this investigation Figures 2ndash6 show the effect of variousvalues of the physical parameters on the velocity (radialtangential and axial) and temperature distribution Resultsobtained for both constant fluid property and variable fluidproperty have been presented graphically
Figures 2(a)ndash2(d) show the effect of porosity parameter119870on all velocity components and temperature distribution It isobserved that radial tangential and axial velocity decreasesbut temperature increases by increasing the permeability forboth cases Whereas variation in porosity parameter has lesseffect on the temperature distribution radial velocity attains
Advances in Mathematical Physics 7
0 1 2 3 4 5 60
001002003004005006007008009
Freidoonimehr et al [40]Rashidi and Freidooni Mehr [27]
F(
)
K = = 0 ＋Ｈ = 0 ０Ｌ = 071
K = 0 = 02 ０Ｌ = 1K = = 0 ＋Ｈ = 005 ０Ｌ = 1
K = 05 = 02K = 05 = 0
K = 1 = 02K = 1 = 0
＋Ｈ = 005
０Ｌ = 1 ＋Ｈ = 005０Ｌ = 1 ＋Ｈ = 005
０Ｌ = 1 ＋Ｈ = 005０Ｌ = 1 ＋Ｈ = 005
(a)
0010203040506070809
1
0 1 2 3 4 5 6
G(
) K = = 0 ＋Ｈ = 0 ０Ｌ = 071K = 0 = 02
K = = 0 ＋Ｈ = 005 ０Ｌ = 1
K = 1 = 02K = 1 = 0
＋Ｈ = 005 ０Ｌ = 1
＋Ｈ = 005 ０Ｌ = 1 ＋Ｈ = 005 ０Ｌ = 1
Rashidi and Freidooni Mehr [27]Freidoonimehr et al [40]
(b)
0 1 2 3 4 5 6
H(
)
minus07
minus08
minus09
minus1
minus11
minus12
minus13
K = 10 = 02
K = 10 = 0K = 1 = 0
K = 0 = 02
K = = 0 ＋Ｈ = 005 ０Ｌ = 1K = 0 = 0
＋Ｈ = 005 ０Ｌ = 1
K = 1 = 02 ＋Ｈ = 005 ０Ｌ = 1 ＋Ｈ = 005 ０Ｌ = 1
＋Ｈ = 005 ０Ｌ = 1
＋Ｈ = 005 ０Ｌ = 1
＋Ｈ = 0 ０Ｌ = 071
Rashidi and Freidooni Mehr [27]Freidoonimehr et al [40]
(c)
0 2 4 6 8 10 12 140
0102030405060708
(
)
= 02 N = 1
= 0 N = 1
= 0 N = 0 K = 0K = 10 1 0
＋Ｈ = 005 ０Ｌ = 1
Rashidi and Freidooni Mehr [27]
(d)
Figure 2 Effect of variation in the porosity parameter on the (a) radial (b) tangential (c) axial and (d) temperature velocity profiles when119882119904 = minus1 and Re = 100
a maximum value close to the disk surface When we reducethe present problem into the literature available by takingpermeability parameter 119870 = 0 and radiation parameter119873 = 0 then the result obtained is exactly similar to thatof the Rashidi and Freidooni Mehr [27] and Freidoonimehret al [40] This validates the results obtained for presentinvestigation
Figures 3(a)ndash3(d) depict the effect of suction param-eter 119882119904 on the radial tangential axial and temperaturedistribution respectively Figures 3(a)ndash3(c) show that allvelocity components decrease as suction increases Physicalsignificance is that as suction increases adherence of thefluid with the wall increases and as a result boundary layerthickness decreases
Figure 3(d) depicts the effect of suction parameter ontemperature profile It is observed that as suction increasestemperature of fluid decreases because suction cools theboundary layer regime therefore suction is used for coolingthe flow in several engineering processes including MHDpower generators and nuclear energy processes
The effect of Reynolds number Re and Knudsen num-ber Kn on velocity and temperature distribution is plotted
in Figures 4 and 5 respectively Both the parameters areshowing the same effects on slip boundary conditions Itis observed that the increasing values of the Reynolds andKnudsen numbers decrease the fluid velocity componentsand temperature distribution The case when value of Kn liesbetween 0001 and 01 corresponds to slip at the surface of thedisk As slip increases the quantity of the fluid that can stickon the disk decreases Therefore circumferential velocity ofthe fluid reduces and causes reduction in centrifugal forceAs a result inward axial velocity decreases In other words asslip gets stronger flow of fluid drawn or pushed away alongthe velocity directions decreases the heat generation
Figure 6 shows the effect of Prandtl number on tem-perature profile As Prandtl number increases the thermalboundary layer thickness decreases Physical significance isthat Prandtl number precludes dispersal of heat in the fluid
Figure 7 depicts the variation of temperature profilewith radiation It is observed that as radiation increasestemperature of the fluid increases Also radiation parameterincreases temperature gradient near the surface of the disk
Figure 8 demonstrates the comparison of both constantand fluid properties on flow over a rotating disk with slip and
8 Advances in Mathematical Physics
0 1 2 3 4 5 6
0
0005
001
0015
002
0025F
()
Ws = minus1 minus2 minus4
= 0
= 02
(a)
0005
01015
02025
03035
04045
05
0 1 2 3 4 5 6
Ws = minus1 minus2 minus4
= 0
= 02
G(
)
(b)
0 1 2 3 4 5 6
= 0
= 02
Ws = minus1
Ws = minus2
Ws = minus4
H(
)
minus1
minus15
minus2
minus25
minus3
minus35
minus4
minus45
(c)
00102030405060708
Ws = minus1 minus2 minus4
= 0
= 02
10 1282 640 14
(
)
(d)
Figure 3 Effect of variation in the suction parameter on the (a) radial (b) tangential (c) axial and (d) temperature velocity profiles when119870 = 1 Kn = 005 Re = 100119873 = 1 and Pr = 1
Table 2 Comparison between the results of present study with the results reported by Kelson and Desseaux [48] Maleque and Sattar [33]and Alam et al [49] for the radial 1198651015840(0) and tangential minus1198661015840(0) skin friction coefficients for Pr = 071119872 = 0 120576 = 0 and Kn = 0
temperature jump The radial velocity achieves a maximumvalue near to the surface of the disk for all values of 120576 It is alsonoted that an increment in relative temperature differenceparameter causes shifting of maximum point from the diskThe velocity along the tangential direction increases asvalue of relative temperature difference parameter increaseswhereas the velocity along axial direction decreases Temper-ature increases as the value of relative temperature differenceparameter increasesThese results are similar to that obtainedby Rashidi et al [39]
Table 1 illustrates the effect of the parameters 120576 119870Kn Re Pr and 119873 on constant suction parameter on thenumerical values of (1198651015840(0) minus1198661015840(0)) and (minus1205791015840(0)) We observethat numerical value of 1198651015840(0) minus1198661015840(0) and minus1205791015840(0) decreases
with the increasing value of Kn Re and radiation parameter119873Tables 2 and 3 depict the comparison of skin friction coef-
ficients and rate of heat transfer between the results obtainedin the present investigation and the literature available [3348 49] It is observed that results obtained in present studyare very well in agreement with the existing results
5 Conclusions
In this study we have investigated radiation effect on velocityprofile for all components and temperature profile throughrotating disk in porous medium for variable fluid propertiesand in particular case for constant fluid properties also By
Advances in Mathematical Physics 9
0 1 2 3 4 5 6
= 0
= 02
Ws = minus1
２ = 1 10 100 1000 10000
00005
0010015
0020025
0030035
004
F(
)
(a)
= 0
= 02
5 63 41 20
Ws = minus1
２ = 1 10 100 1000 10000
0010203040506070809
1
G(
)
(b)
0 2 3 41 65
Ws = minus2 = 0
２ = 1
２ = 10
２ = 100
２ = 1000
２ = 10000
minus204minus2035minus203
minus2025minus202
minus2015minus201
minus2005minus2
minus1995minus199
H(
)
(c)
0 2 3 41 65
Ws = minus2 = 02 ２ = 1
２ = 10
２ = 100
２ = 1000
２ = 10000
minus2
minus195
minus19
minus185
minus18
minus175
minus17
H(
)
(d)
0 2 4 6 8 10 12 14
= 0
= 02
Ws = minus1
２ = 1 10 100 1000 10000
0010203040506070809
1
(
)
(e)
Figure 4 Effect of variation in the Reynolds number on the (a) radial (b) tangential (c) amp (d) axial and (e) temperature velocity profileswhen 119870 = 1 Kn = 005119873 = 1 and Pr = 1
Table 3 Comparison between the results of present study with the results reported by Kelson and Desseaux [48] Maleque and Sattar [33]and Alam et al [49] for rate of heat transfer minus1205791015840(0) for Pr = 071119872 = 0 120576 = 0 Kn = 0 and119873 = 0119882119904 Present Alam et al [49] Maleque and Sattar [33] Kelson and Desseaux [48]minus1205791015840(0) minus1205791015840(0) minus1205791015840(0) minus1205791015840(0)0 0326798372 032637889 032576 0325856minus2 1438764651 143876482 144212 1437782minus4 2842381877 284369011 284470 2842381minus5 3551223146 355222471 355411 3551223
10 Advances in Mathematical Physics
0 1 2 3 4 5 6
= 0
= 02
00005
0010015
0020025
0030035
004F
()
Ws = minus1
＋Ｈ = 0 002 005 01
(a)
0 1 2 3 4 5 6
= 0
= 02
0010203040506070809
1
G(
)
Ws = minus1
＋Ｈ = 0 002 005 01
(b)
0 1 2 3 4 5 6
= 0
= 02
minus2
minus195
minus19
minus185
minus18
minus175
minus17
H(
)
Ws = minus2
＋Ｈ = 005 002 0
(c)
= 0
= 02
0010203040506070809
1
(
)
104 6 8 12 1420
Ws = minus1
＋Ｈ = 0 002 005 01N = 0 ＋Ｈ = 005
(d)
Figure 5 Effect of variation in the Knudsen number on the (a) radial (b) tangential (c) axial and (d) temperature velocity profiles when119870 = 1 Re = 100119873 = 1 and Pr = 1
= 0
= 02
2 4 6 8 10 12 140
00102030405060708
(
)
N = 0 ０Ｌ = 4０Ｌ = 1 2 3 4
Figure 6 Effect of Prandtl number on temperature distributionwhen 119870 = 1119882119904 = minus1 Re = 100 Kn = 005 and119873 = 1
similarity transformation governing equations transformedinto nonlinear ordinary differential equations which aresolved numerically by usingRunge-Kuttamethodwith shoot-ing technique Based on the resulting solutions the followingconclusions can be drawn
= 0
= 02
2 4 6 8 10 12 140
N = 4 3 2 1 0
0010203040506070809
(
)
Figure 7 Effect of radiation parameter on temperature distributionwhen 119870 = 1119882119904 = minus1 Re = 100 Kn = 005 and Pr = 1
(i) The radial tangential and axial velocity profilesdecrease while the temperature increases with theincreasing values of porosity parameter
(ii) The increasing value of Reynolds and Knudsen num-ber decreases the fluid velocity components and
Advances in Mathematical Physics 11
1 2 3 4 5 60
= 0 02 04 10
0005
001
0015
002
0025F
()
(a)
1 2 54 60 3
0
01
02
03
04
05
06
G(
)
= 1 04 02 0
(b)
1 2 3 4 5 60
= 0
= 02
= 04
= 1
minus11
minus1
minus09
minus08
minus07
minus06
minus05
minus04
H(
)
(c)
00102030405060708
(
)
2 4 6 8 10 12 140
N = 0 = 02
= 1 04 02 0
(d)
Figure 8 Effect of variation in the relative temperature difference parameter on the (a) radial (b) tangential (c) axial and (d) temperaturevelocity profiles when119870 = 1119882119904 = minus1 Kn = 005 Re = 100119873 = 1 and Pr = 1
temperature and suction parameter also shows thesame effect
(iii) For the effect of the radiation parameter on the tem-perature distribution it is seen that the temperaturedistribution decreases with the increasing values ofradiation parameter and also it has been observed thatthe radial and tangential skin friction values decreasewith increase in the radiation parameter
Competing Interests
The authors declare that they have no competing interests
References
[1] Th V Karman ldquoUber laminare und turbulente reibunrdquo Zeits-chrift fur Angewandte Mathematik und Mechanik vol 1 no 4pp 233ndash252 1921
[2] W G Cochran ldquoThe flow due to a rotating diskrdquoMathematicalProceedings of the Cambridge Philosophical Society vol 30 no3 pp 365ndash375 1934
[3] E R Benton ldquoOn the flow due to a rotating diskrdquo Journal ofFluid Mechanics vol 24 no 4 pp 781ndash800 1966
[4] M Turkyilmazoglu ldquoPurely analytic solutions of the compress-ible boundary layer flow due to a porous rotating disk with heattransferrdquoPhysics of Fluids vol 21 no 10 Article ID 106104 2009
[5] D S Chauhan and S Gupta ldquoSteady flow and heat transferbetween two stationary naturally permeable disksrdquo Journal ofUltra Scientist of Physical Sciences vol 12 no 1 pp 45ndash52 2000
[6] C Wagner ldquoHeat transfer from a rotating disk to ambient airrdquoJournal of Applied Physics vol 19 no 9 pp 837ndash839 1948
[7] K Millsaps and K Pohlhausen ldquoHeat transfer by laminar flowfrom a rotating platerdquo Journal of the Aeronautical Sciences vol19 pp 120ndash126 1952
[8] F Kreith and J H Taylor ldquoHeat transfer from a rotating disk inturbulent flow no 1956rdquo ASME Paper 56-A-146 1956
[9] H-T Lin and L-K Lin ldquoHeat transfer from a rotating cone ordisk to fluids of any Prandtl numberrdquo International Communi-cations in Heat and Mass Transfer vol 14 no 3 pp 323ndash3321987
[10] P D Verma and D S Chauhan ldquoFlow between a torsionallyoscillating impermeable disc and a stationary naturally perme-able discrdquo Indian Journal of Pure and Applied Mathematics vol10 pp 1351ndash1361 1979
[11] D S Chauhan and S Jain ldquoSteady flow between highlypermeable rotating disksrdquo Indian Journal of Theoretical Physicsvol 52 no 1 pp 39ndash50 2004
[12] M Turkyilmazoglu ldquoNanofluid flow and heat transfer due to arotating diskrdquo Computers amp Fluids vol 94 pp 139ndash146 2014
[13] M Turkyilmazoglu ldquoMHD fluid flow and heat transfer due to ashrinking rotating diskrdquo Computers amp Fluids vol 90 pp 51ndash562014
[14] M Azimi and R Riazi ldquoHeat transfer analysis of GO-waternanofluid flow between two parallel disksrdquo Propulsion andPower Research vol 4 no 1 pp 23ndash30 2015
12 Advances in Mathematical Physics
[15] B R Sharma and H Konwar ldquoEffect of chemical reaction onmass transfer due to a permeable rotating heated diskrdquo Interna-tional Journal of Computer Applications vol 119 no 21 pp 5ndash92015
[16] H Shahmohamadi and M Mohammadpour ldquoA series solutionfor three-dimensional navier-stokes equations of flow near aninfinite rotating diskrdquo World Journal of Mechanics vol 4 pp117ndash127 2014
[17] S Srinivas A S Reddy T R Ramamohan and A K ShuklaldquoThermal-diffusion and diffusion-thermo effects onMHD flowof viscous fluid between expanding or contracting rotatingporous disks with viscous dissipationrdquo Journal of the EgyptianMathematical Society vol 24 no 1 pp 100ndash107 2016
[18] M A Mansour ldquoRadiative and free-convection effects on theoscillatory flow past a vertical platerdquo Astrophysics and SpaceScience vol 166 no 2 pp 269ndash275 1990
[19] M A Hossain M A Alim and D A S Rees ldquoThe effectof radiation on free convection from a porous vertical platerdquoInternational Journal of Heat and Mass Transfer vol 42 no 1pp 181ndash191 1999
[20] A Raptis and C Perdikis ldquoMHD free convection flow by thepresence of radiationrdquo International Journal of Magnetohydro-dynamics Plasma and Space Research vol 9 pp 237ndash252 2000
[21] M A Hossain and H S Takhar ldquoRadiation effect on mixedconvection along a vertical plate with uniform surface temper-aturerdquoHeat and Mass Transfer vol 31 no 4 pp 243ndash248 1996
[22] A Devi and R U Devi ldquoEffects of thermal radiation onhydromagnetic flow due to a porous rotating disk with halleffectrdquo Journal of Applied Fluid Mechanics vol 5 no 2 pp 1ndash7 2012
[23] C Y Wang ldquoFlow due to a stretching boundary with partialslipmdashan exact solution of the Navier-Stokes equationsrdquo Chemi-cal Engineering Science vol 57 no 17 pp 3745ndash3747 2002
[24] E Osalusi ldquoEffects of thermal radiation on MHD and slip flowover a porous rotating disk with variable propertiesrdquo RomanianJournal of Physics vol 52 no 3-4 pp 217ndash229 2007
[25] A A Khidir ldquoViscous dissipation Ohmic heating and radiationeffects on MHD flow past a rotating disk embedded in aporous medium with variable propertiesrdquo Arabian Journal ofMathematics vol 2 no 3 pp 263ndash277 2013
[26] E M Sparrow G S Beavers and L Y Hung ldquoFlow about aporous-surfaced rotating diskrdquo International Journal of Heatand Mass Transfer vol 14 no 7 pp 993ndash996 1971
[27] M M Rashidi and N Freidooni Mehr ldquoEffects of velocity slipand temperature jump on the entropy generation in magne-tohydrodynamic flow over a porous rotating diskrdquo Journal ofMechanical Engineering vol 1 no 3 2012
[28] A Arikoglu G Komurgoz I Ozkol and A Y Gunes ldquoCom-bined effects of temperature and velocity jump on the heattransfer fluid flow and entropy generation over a single rotatingdiskrdquo Journal of Heat Transfer vol 132 no 11 Article ID 1117032010
[29] THayatM RashidM Imtiaz andAAlsaedi ldquoMagnetohydro-dynamic (MHD) flow of Cu-water nanofluid due to a rotatingdisk with partial sliprdquo AIP Advances vol 5 no 6 Article ID067169 2015
[30] M Zakerullah and J A D Ackroyd ldquoLaminar natural convec-tion boundary-layers on Horizontal Circular disksrdquo Journal ofApplied Mathematics and Physics vol 30 pp 427ndash435 1979
[31] HHerwig ldquoThe effect of variable properties onmomentumandheat transfer in a tube with constant heat flux across the wallrdquoInternational Journal of Heat and Mass Transfer vol 28 no 2pp 423ndash431 1985
[32] H Herwig and K Klemp ldquoVariable property effects of fullydeveloped laminar flow in concentric annulirdquo Journal of HeatTransfer vol 110 no 2 pp 314ndash320 1988
[33] K AMaleque andMA Sattar ldquoSteady laminar convective flowwith variable properties due to a porous rotating diskrdquo Journalof Heat Transfer vol 127 no 12 pp 1406ndash1409 2005
[34] K A Maleque and M A Sattar ldquoThe effects of variable proper-ties and hall current on steady MHD laminar convective fluidflow due to a porous rotating diskrdquo International Journal of Heatand Mass Transfer vol 48 no 23-24 pp 4963ndash4972 2005
[35] E Osalusi and P Sibanda ldquoOn variable laminar convective flowproperties due to a porous rotating disk in a magnetic fieldrdquoRomanian Journal of Physics vol 9 no 10 pp 933ndash944 2006
[36] M M Rahman ldquoConvective Hydromagnetic slip flow withvariable properties due to a porous rotating diskrdquo The SultanQaboos University Journal for Science vol 15 pp 55ndash79 2010
[37] M M Rashidi S A Mohimanian Pour T Hayat and SObaidat ldquoAnalytic approximate solutions for steady flow over arotating disk in porousmediumwith heat transfer by homotopyanalysis methodrdquo Computers amp Fluids vol 54 no 1 pp 1ndash92012
[38] S Hussain F Ahmad M Shafique and S Hussain ldquoNumericalsolution for accelerated rotating disk in a viscous fluidrdquoAppliedMathematics vol 4 no 6 pp 899ndash902 2013
[39] M M Rashidi N Kavyani and S Abelman ldquoInvestigation ofentropy generation inMHDand slip flow over a rotating porousdisk with variable propertiesrdquo International Journal of Heat andMass Transfer vol 70 pp 892ndash917 2014
[40] N Freidoonimehr M M Rashidi S Abelman and G Loren-zini ldquoAnalytical modeling of MHD flow over a permeablerotating disk in the presence of Soret and Dufour effectsentropy analysisrdquo Entropy vol 18 no 5 article 131 2016
[41] M S Alam S M Chapal Hossain and M M Rahman ldquoTran-sient thermophoretic particle deposition on forced convectiveheat and mass transfer flow due to a rotating diskrdquo Ain ShamsEngineering Journal vol 7 no 1 pp 441ndash452 2016
[42] S Jayaraj ldquoThermophoresis in laminar flow over cold inclinedplates with variable propertiesrdquoHeat andMass Transfer vol 30no 3 pp 167ndash173 1995
[43] M M Rahman ldquoThermophoretic deposition of nanoparticlesdue to a permeable rotating disk effects of partial slip mag-netic field thermal radiation thermal-diffusion and diffusion-thermordquo International Journal of Mathematical ComputationalPhysical Electrical and Computer Engineering vol 7 no 5 2013
[44] I V Shevchuk Convective Heat and Mass Transfer in RotatingDisk Systems Springer Berlin Germany 2009
[45] C L Tien and D T Campbell ldquoHeat and mass transfer fromrotating conesrdquo Journal of FluidMechanics vol 17 no 1 pp 105ndash112 1963
[46] C J Elkins and J K Eaton ldquoHeat transfer in the rotatingdisk boundary layerrdquo Tech Rep TSD-103 Stanford UniversityDepartment of Mechanical EngineeringThermosciences Divi-sion Stanford Calif USA 1997
[47] G Karniadakis A Beskok and N Aluru Microflows Funda-mentals and Simulation Springer New York NY USA 2001
[48] N Kelson andA Desseaux ldquoNote on porous rotating disk flowrdquoAustralian amp New Zealand Industrial and Applied MathematicsJournal vol 42 pp 837ndash855 2000
[49] M Alam N Poddar M Rahman and K Vajravelu ldquoTransienthydromagnetic forced convective heat transfer slip flow due toa porous rotating disk with variable fluid propertiesrdquo AmericanJournal of Heat andMass Transfer vol 2 no 3 pp 165ndash189 2015
Taking 120576 = 0 120583 = 120583infin (constant)Similarly at 120576 = 0 120588 = 120588infin and 120581 = 120581infinSubstitution of 120576 = 0 in (2)ndash(4) and (6) system of
equations is as follows
1198671015840 + 2119865 = 011986510158401015840 minus 1198671198651015840 minus 1198652 + 1198662 minus 119870119865 = 011986610158401015840 minus 1198671198661015840 minus 2119865119866 minus 119870119866 = 0(1 + 41198733 ) 12057910158401015840 minus Pr1198671205791015840 = 0
(16)
Equations (16) under the boundary condition (12) are trans-formed into the following system of initial value problems
Here 1199031 = 1198651015840(0) 1199032 = 1198661015840(0) and 1199033 = 1205791015840(0) are the initialguesses when fluid properties are variable and 1199034 = 1198651015840(0) 1199035 =1198661015840(0) and 1199036 = 1205791015840(0) are the initial guesses when fluid prop-erties are constant The essence of present numerical methodis to reduce the boundary value problem (BVP) into an initialvalue problem (IVP) Further shooting technique is used toguess 1199031 1199032 1199033 1199034 1199035 and 1199036 until the boundary conditionsare satisfied A number of iterations of Runge-Kutta fourth-order method has been performed to obtain final values ofthese guesses Initial guesses for different set of parametersare displayed in Table 1
In this problem the physical quantities of interest arelocal skin friction coefficients and the Nusselt number whichrepresents the wall shear stress and the rate of heat transferrespectively When variable fluid properties are taken intoconsideration the fluid near to the disk opposes rotation ofthe disk due to presence of tangential shear stressThereforeto maintain a steady rotation it is essential to have torque atthe shaft The skin frictions 119862119891119903 along radial direction and119862119891119905 along tangential direction at no slip condition are givenas
where Re = Ω1199032]infin is rotational Reynolds numberThus (24) shows that the radial and tangential skin
frictions coefficients are proportional to 1198651015840(0) and 1198661015840(0)respectively and Nusselt number is proportional to minus1205791015840(0)4 Result and Discussion
In this investigation Figures 2ndash6 show the effect of variousvalues of the physical parameters on the velocity (radialtangential and axial) and temperature distribution Resultsobtained for both constant fluid property and variable fluidproperty have been presented graphically
Figures 2(a)ndash2(d) show the effect of porosity parameter119870on all velocity components and temperature distribution It isobserved that radial tangential and axial velocity decreasesbut temperature increases by increasing the permeability forboth cases Whereas variation in porosity parameter has lesseffect on the temperature distribution radial velocity attains
Advances in Mathematical Physics 7
0 1 2 3 4 5 60
001002003004005006007008009
Freidoonimehr et al [40]Rashidi and Freidooni Mehr [27]
F(
)
K = = 0 ＋Ｈ = 0 ０Ｌ = 071
K = 0 = 02 ０Ｌ = 1K = = 0 ＋Ｈ = 005 ０Ｌ = 1
K = 05 = 02K = 05 = 0
K = 1 = 02K = 1 = 0
＋Ｈ = 005
０Ｌ = 1 ＋Ｈ = 005０Ｌ = 1 ＋Ｈ = 005
０Ｌ = 1 ＋Ｈ = 005０Ｌ = 1 ＋Ｈ = 005
(a)
0010203040506070809
1
0 1 2 3 4 5 6
G(
) K = = 0 ＋Ｈ = 0 ０Ｌ = 071K = 0 = 02
K = = 0 ＋Ｈ = 005 ０Ｌ = 1
K = 1 = 02K = 1 = 0
＋Ｈ = 005 ０Ｌ = 1
＋Ｈ = 005 ０Ｌ = 1 ＋Ｈ = 005 ０Ｌ = 1
Rashidi and Freidooni Mehr [27]Freidoonimehr et al [40]
(b)
0 1 2 3 4 5 6
H(
)
minus07
minus08
minus09
minus1
minus11
minus12
minus13
K = 10 = 02
K = 10 = 0K = 1 = 0
K = 0 = 02
K = = 0 ＋Ｈ = 005 ０Ｌ = 1K = 0 = 0
＋Ｈ = 005 ０Ｌ = 1
K = 1 = 02 ＋Ｈ = 005 ０Ｌ = 1 ＋Ｈ = 005 ０Ｌ = 1
＋Ｈ = 005 ０Ｌ = 1
＋Ｈ = 005 ０Ｌ = 1
＋Ｈ = 0 ０Ｌ = 071
Rashidi and Freidooni Mehr [27]Freidoonimehr et al [40]
(c)
0 2 4 6 8 10 12 140
0102030405060708
(
)
= 02 N = 1
= 0 N = 1
= 0 N = 0 K = 0K = 10 1 0
＋Ｈ = 005 ０Ｌ = 1
Rashidi and Freidooni Mehr [27]
(d)
Figure 2 Effect of variation in the porosity parameter on the (a) radial (b) tangential (c) axial and (d) temperature velocity profiles when119882119904 = minus1 and Re = 100
a maximum value close to the disk surface When we reducethe present problem into the literature available by takingpermeability parameter 119870 = 0 and radiation parameter119873 = 0 then the result obtained is exactly similar to thatof the Rashidi and Freidooni Mehr [27] and Freidoonimehret al [40] This validates the results obtained for presentinvestigation
Figures 3(a)ndash3(d) depict the effect of suction param-eter 119882119904 on the radial tangential axial and temperaturedistribution respectively Figures 3(a)ndash3(c) show that allvelocity components decrease as suction increases Physicalsignificance is that as suction increases adherence of thefluid with the wall increases and as a result boundary layerthickness decreases
Figure 3(d) depicts the effect of suction parameter ontemperature profile It is observed that as suction increasestemperature of fluid decreases because suction cools theboundary layer regime therefore suction is used for coolingthe flow in several engineering processes including MHDpower generators and nuclear energy processes
The effect of Reynolds number Re and Knudsen num-ber Kn on velocity and temperature distribution is plotted
in Figures 4 and 5 respectively Both the parameters areshowing the same effects on slip boundary conditions Itis observed that the increasing values of the Reynolds andKnudsen numbers decrease the fluid velocity componentsand temperature distribution The case when value of Kn liesbetween 0001 and 01 corresponds to slip at the surface of thedisk As slip increases the quantity of the fluid that can stickon the disk decreases Therefore circumferential velocity ofthe fluid reduces and causes reduction in centrifugal forceAs a result inward axial velocity decreases In other words asslip gets stronger flow of fluid drawn or pushed away alongthe velocity directions decreases the heat generation
Figure 6 shows the effect of Prandtl number on tem-perature profile As Prandtl number increases the thermalboundary layer thickness decreases Physical significance isthat Prandtl number precludes dispersal of heat in the fluid
Figure 7 depicts the variation of temperature profilewith radiation It is observed that as radiation increasestemperature of the fluid increases Also radiation parameterincreases temperature gradient near the surface of the disk
Figure 8 demonstrates the comparison of both constantand fluid properties on flow over a rotating disk with slip and
8 Advances in Mathematical Physics
0 1 2 3 4 5 6
0
0005
001
0015
002
0025F
()
Ws = minus1 minus2 minus4
= 0
= 02
(a)
0005
01015
02025
03035
04045
05
0 1 2 3 4 5 6
Ws = minus1 minus2 minus4
= 0
= 02
G(
)
(b)
0 1 2 3 4 5 6
= 0
= 02
Ws = minus1
Ws = minus2
Ws = minus4
H(
)
minus1
minus15
minus2
minus25
minus3
minus35
minus4
minus45
(c)
00102030405060708
Ws = minus1 minus2 minus4
= 0
= 02
10 1282 640 14
(
)
(d)
Figure 3 Effect of variation in the suction parameter on the (a) radial (b) tangential (c) axial and (d) temperature velocity profiles when119870 = 1 Kn = 005 Re = 100119873 = 1 and Pr = 1
Table 2 Comparison between the results of present study with the results reported by Kelson and Desseaux [48] Maleque and Sattar [33]and Alam et al [49] for the radial 1198651015840(0) and tangential minus1198661015840(0) skin friction coefficients for Pr = 071119872 = 0 120576 = 0 and Kn = 0
temperature jump The radial velocity achieves a maximumvalue near to the surface of the disk for all values of 120576 It is alsonoted that an increment in relative temperature differenceparameter causes shifting of maximum point from the diskThe velocity along the tangential direction increases asvalue of relative temperature difference parameter increaseswhereas the velocity along axial direction decreases Temper-ature increases as the value of relative temperature differenceparameter increasesThese results are similar to that obtainedby Rashidi et al [39]
Table 1 illustrates the effect of the parameters 120576 119870Kn Re Pr and 119873 on constant suction parameter on thenumerical values of (1198651015840(0) minus1198661015840(0)) and (minus1205791015840(0)) We observethat numerical value of 1198651015840(0) minus1198661015840(0) and minus1205791015840(0) decreases
with the increasing value of Kn Re and radiation parameter119873Tables 2 and 3 depict the comparison of skin friction coef-
ficients and rate of heat transfer between the results obtainedin the present investigation and the literature available [3348 49] It is observed that results obtained in present studyare very well in agreement with the existing results
5 Conclusions
In this study we have investigated radiation effect on velocityprofile for all components and temperature profile throughrotating disk in porous medium for variable fluid propertiesand in particular case for constant fluid properties also By
Advances in Mathematical Physics 9
0 1 2 3 4 5 6
= 0
= 02
Ws = minus1
２ = 1 10 100 1000 10000
00005
0010015
0020025
0030035
004
F(
)
(a)
= 0
= 02
5 63 41 20
Ws = minus1
２ = 1 10 100 1000 10000
0010203040506070809
1
G(
)
(b)
0 2 3 41 65
Ws = minus2 = 0
２ = 1
２ = 10
２ = 100
２ = 1000
２ = 10000
minus204minus2035minus203
minus2025minus202
minus2015minus201
minus2005minus2
minus1995minus199
H(
)
(c)
0 2 3 41 65
Ws = minus2 = 02 ２ = 1
２ = 10
２ = 100
２ = 1000
２ = 10000
minus2
minus195
minus19
minus185
minus18
minus175
minus17
H(
)
(d)
0 2 4 6 8 10 12 14
= 0
= 02
Ws = minus1
２ = 1 10 100 1000 10000
0010203040506070809
1
(
)
(e)
Figure 4 Effect of variation in the Reynolds number on the (a) radial (b) tangential (c) amp (d) axial and (e) temperature velocity profileswhen 119870 = 1 Kn = 005119873 = 1 and Pr = 1
Table 3 Comparison between the results of present study with the results reported by Kelson and Desseaux [48] Maleque and Sattar [33]and Alam et al [49] for rate of heat transfer minus1205791015840(0) for Pr = 071119872 = 0 120576 = 0 Kn = 0 and119873 = 0119882119904 Present Alam et al [49] Maleque and Sattar [33] Kelson and Desseaux [48]minus1205791015840(0) minus1205791015840(0) minus1205791015840(0) minus1205791015840(0)0 0326798372 032637889 032576 0325856minus2 1438764651 143876482 144212 1437782minus4 2842381877 284369011 284470 2842381minus5 3551223146 355222471 355411 3551223
10 Advances in Mathematical Physics
0 1 2 3 4 5 6
= 0
= 02
00005
0010015
0020025
0030035
004F
()
Ws = minus1
＋Ｈ = 0 002 005 01
(a)
0 1 2 3 4 5 6
= 0
= 02
0010203040506070809
1
G(
)
Ws = minus1
＋Ｈ = 0 002 005 01
(b)
0 1 2 3 4 5 6
= 0
= 02
minus2
minus195
minus19
minus185
minus18
minus175
minus17
H(
)
Ws = minus2
＋Ｈ = 005 002 0
(c)
= 0
= 02
0010203040506070809
1
(
)
104 6 8 12 1420
Ws = minus1
＋Ｈ = 0 002 005 01N = 0 ＋Ｈ = 005
(d)
Figure 5 Effect of variation in the Knudsen number on the (a) radial (b) tangential (c) axial and (d) temperature velocity profiles when119870 = 1 Re = 100119873 = 1 and Pr = 1
= 0
= 02
2 4 6 8 10 12 140
00102030405060708
(
)
N = 0 ０Ｌ = 4０Ｌ = 1 2 3 4
Figure 6 Effect of Prandtl number on temperature distributionwhen 119870 = 1119882119904 = minus1 Re = 100 Kn = 005 and119873 = 1
similarity transformation governing equations transformedinto nonlinear ordinary differential equations which aresolved numerically by usingRunge-Kuttamethodwith shoot-ing technique Based on the resulting solutions the followingconclusions can be drawn
= 0
= 02
2 4 6 8 10 12 140
N = 4 3 2 1 0
0010203040506070809
(
)
Figure 7 Effect of radiation parameter on temperature distributionwhen 119870 = 1119882119904 = minus1 Re = 100 Kn = 005 and Pr = 1
(i) The radial tangential and axial velocity profilesdecrease while the temperature increases with theincreasing values of porosity parameter
(ii) The increasing value of Reynolds and Knudsen num-ber decreases the fluid velocity components and
Advances in Mathematical Physics 11
1 2 3 4 5 60
= 0 02 04 10
0005
001
0015
002
0025F
()
(a)
1 2 54 60 3
0
01
02
03
04
05
06
G(
)
= 1 04 02 0
(b)
1 2 3 4 5 60
= 0
= 02
= 04
= 1
minus11
minus1
minus09
minus08
minus07
minus06
minus05
minus04
H(
)
(c)
00102030405060708
(
)
2 4 6 8 10 12 140
N = 0 = 02
= 1 04 02 0
(d)
Figure 8 Effect of variation in the relative temperature difference parameter on the (a) radial (b) tangential (c) axial and (d) temperaturevelocity profiles when119870 = 1119882119904 = minus1 Kn = 005 Re = 100119873 = 1 and Pr = 1
temperature and suction parameter also shows thesame effect
(iii) For the effect of the radiation parameter on the tem-perature distribution it is seen that the temperaturedistribution decreases with the increasing values ofradiation parameter and also it has been observed thatthe radial and tangential skin friction values decreasewith increase in the radiation parameter
Competing Interests
The authors declare that they have no competing interests
References
[1] Th V Karman ldquoUber laminare und turbulente reibunrdquo Zeits-chrift fur Angewandte Mathematik und Mechanik vol 1 no 4pp 233ndash252 1921
[2] W G Cochran ldquoThe flow due to a rotating diskrdquoMathematicalProceedings of the Cambridge Philosophical Society vol 30 no3 pp 365ndash375 1934
[3] E R Benton ldquoOn the flow due to a rotating diskrdquo Journal ofFluid Mechanics vol 24 no 4 pp 781ndash800 1966
[4] M Turkyilmazoglu ldquoPurely analytic solutions of the compress-ible boundary layer flow due to a porous rotating disk with heattransferrdquoPhysics of Fluids vol 21 no 10 Article ID 106104 2009
[5] D S Chauhan and S Gupta ldquoSteady flow and heat transferbetween two stationary naturally permeable disksrdquo Journal ofUltra Scientist of Physical Sciences vol 12 no 1 pp 45ndash52 2000
[6] C Wagner ldquoHeat transfer from a rotating disk to ambient airrdquoJournal of Applied Physics vol 19 no 9 pp 837ndash839 1948
[7] K Millsaps and K Pohlhausen ldquoHeat transfer by laminar flowfrom a rotating platerdquo Journal of the Aeronautical Sciences vol19 pp 120ndash126 1952
[8] F Kreith and J H Taylor ldquoHeat transfer from a rotating disk inturbulent flow no 1956rdquo ASME Paper 56-A-146 1956
[9] H-T Lin and L-K Lin ldquoHeat transfer from a rotating cone ordisk to fluids of any Prandtl numberrdquo International Communi-cations in Heat and Mass Transfer vol 14 no 3 pp 323ndash3321987
[10] P D Verma and D S Chauhan ldquoFlow between a torsionallyoscillating impermeable disc and a stationary naturally perme-able discrdquo Indian Journal of Pure and Applied Mathematics vol10 pp 1351ndash1361 1979
[11] D S Chauhan and S Jain ldquoSteady flow between highlypermeable rotating disksrdquo Indian Journal of Theoretical Physicsvol 52 no 1 pp 39ndash50 2004
[12] M Turkyilmazoglu ldquoNanofluid flow and heat transfer due to arotating diskrdquo Computers amp Fluids vol 94 pp 139ndash146 2014
[13] M Turkyilmazoglu ldquoMHD fluid flow and heat transfer due to ashrinking rotating diskrdquo Computers amp Fluids vol 90 pp 51ndash562014
[14] M Azimi and R Riazi ldquoHeat transfer analysis of GO-waternanofluid flow between two parallel disksrdquo Propulsion andPower Research vol 4 no 1 pp 23ndash30 2015
12 Advances in Mathematical Physics
[15] B R Sharma and H Konwar ldquoEffect of chemical reaction onmass transfer due to a permeable rotating heated diskrdquo Interna-tional Journal of Computer Applications vol 119 no 21 pp 5ndash92015
[16] H Shahmohamadi and M Mohammadpour ldquoA series solutionfor three-dimensional navier-stokes equations of flow near aninfinite rotating diskrdquo World Journal of Mechanics vol 4 pp117ndash127 2014
[17] S Srinivas A S Reddy T R Ramamohan and A K ShuklaldquoThermal-diffusion and diffusion-thermo effects onMHD flowof viscous fluid between expanding or contracting rotatingporous disks with viscous dissipationrdquo Journal of the EgyptianMathematical Society vol 24 no 1 pp 100ndash107 2016
[18] M A Mansour ldquoRadiative and free-convection effects on theoscillatory flow past a vertical platerdquo Astrophysics and SpaceScience vol 166 no 2 pp 269ndash275 1990
[19] M A Hossain M A Alim and D A S Rees ldquoThe effectof radiation on free convection from a porous vertical platerdquoInternational Journal of Heat and Mass Transfer vol 42 no 1pp 181ndash191 1999
[20] A Raptis and C Perdikis ldquoMHD free convection flow by thepresence of radiationrdquo International Journal of Magnetohydro-dynamics Plasma and Space Research vol 9 pp 237ndash252 2000
[21] M A Hossain and H S Takhar ldquoRadiation effect on mixedconvection along a vertical plate with uniform surface temper-aturerdquoHeat and Mass Transfer vol 31 no 4 pp 243ndash248 1996
[22] A Devi and R U Devi ldquoEffects of thermal radiation onhydromagnetic flow due to a porous rotating disk with halleffectrdquo Journal of Applied Fluid Mechanics vol 5 no 2 pp 1ndash7 2012
[23] C Y Wang ldquoFlow due to a stretching boundary with partialslipmdashan exact solution of the Navier-Stokes equationsrdquo Chemi-cal Engineering Science vol 57 no 17 pp 3745ndash3747 2002
[24] E Osalusi ldquoEffects of thermal radiation on MHD and slip flowover a porous rotating disk with variable propertiesrdquo RomanianJournal of Physics vol 52 no 3-4 pp 217ndash229 2007
[25] A A Khidir ldquoViscous dissipation Ohmic heating and radiationeffects on MHD flow past a rotating disk embedded in aporous medium with variable propertiesrdquo Arabian Journal ofMathematics vol 2 no 3 pp 263ndash277 2013
[26] E M Sparrow G S Beavers and L Y Hung ldquoFlow about aporous-surfaced rotating diskrdquo International Journal of Heatand Mass Transfer vol 14 no 7 pp 993ndash996 1971
[27] M M Rashidi and N Freidooni Mehr ldquoEffects of velocity slipand temperature jump on the entropy generation in magne-tohydrodynamic flow over a porous rotating diskrdquo Journal ofMechanical Engineering vol 1 no 3 2012
[28] A Arikoglu G Komurgoz I Ozkol and A Y Gunes ldquoCom-bined effects of temperature and velocity jump on the heattransfer fluid flow and entropy generation over a single rotatingdiskrdquo Journal of Heat Transfer vol 132 no 11 Article ID 1117032010
[29] THayatM RashidM Imtiaz andAAlsaedi ldquoMagnetohydro-dynamic (MHD) flow of Cu-water nanofluid due to a rotatingdisk with partial sliprdquo AIP Advances vol 5 no 6 Article ID067169 2015
[30] M Zakerullah and J A D Ackroyd ldquoLaminar natural convec-tion boundary-layers on Horizontal Circular disksrdquo Journal ofApplied Mathematics and Physics vol 30 pp 427ndash435 1979
[31] HHerwig ldquoThe effect of variable properties onmomentumandheat transfer in a tube with constant heat flux across the wallrdquoInternational Journal of Heat and Mass Transfer vol 28 no 2pp 423ndash431 1985
[32] H Herwig and K Klemp ldquoVariable property effects of fullydeveloped laminar flow in concentric annulirdquo Journal of HeatTransfer vol 110 no 2 pp 314ndash320 1988
[33] K AMaleque andMA Sattar ldquoSteady laminar convective flowwith variable properties due to a porous rotating diskrdquo Journalof Heat Transfer vol 127 no 12 pp 1406ndash1409 2005
[34] K A Maleque and M A Sattar ldquoThe effects of variable proper-ties and hall current on steady MHD laminar convective fluidflow due to a porous rotating diskrdquo International Journal of Heatand Mass Transfer vol 48 no 23-24 pp 4963ndash4972 2005
[35] E Osalusi and P Sibanda ldquoOn variable laminar convective flowproperties due to a porous rotating disk in a magnetic fieldrdquoRomanian Journal of Physics vol 9 no 10 pp 933ndash944 2006
[36] M M Rahman ldquoConvective Hydromagnetic slip flow withvariable properties due to a porous rotating diskrdquo The SultanQaboos University Journal for Science vol 15 pp 55ndash79 2010
[37] M M Rashidi S A Mohimanian Pour T Hayat and SObaidat ldquoAnalytic approximate solutions for steady flow over arotating disk in porousmediumwith heat transfer by homotopyanalysis methodrdquo Computers amp Fluids vol 54 no 1 pp 1ndash92012
[38] S Hussain F Ahmad M Shafique and S Hussain ldquoNumericalsolution for accelerated rotating disk in a viscous fluidrdquoAppliedMathematics vol 4 no 6 pp 899ndash902 2013
[39] M M Rashidi N Kavyani and S Abelman ldquoInvestigation ofentropy generation inMHDand slip flow over a rotating porousdisk with variable propertiesrdquo International Journal of Heat andMass Transfer vol 70 pp 892ndash917 2014
[40] N Freidoonimehr M M Rashidi S Abelman and G Loren-zini ldquoAnalytical modeling of MHD flow over a permeablerotating disk in the presence of Soret and Dufour effectsentropy analysisrdquo Entropy vol 18 no 5 article 131 2016
[41] M S Alam S M Chapal Hossain and M M Rahman ldquoTran-sient thermophoretic particle deposition on forced convectiveheat and mass transfer flow due to a rotating diskrdquo Ain ShamsEngineering Journal vol 7 no 1 pp 441ndash452 2016
[42] S Jayaraj ldquoThermophoresis in laminar flow over cold inclinedplates with variable propertiesrdquoHeat andMass Transfer vol 30no 3 pp 167ndash173 1995
[43] M M Rahman ldquoThermophoretic deposition of nanoparticlesdue to a permeable rotating disk effects of partial slip mag-netic field thermal radiation thermal-diffusion and diffusion-thermordquo International Journal of Mathematical ComputationalPhysical Electrical and Computer Engineering vol 7 no 5 2013
[44] I V Shevchuk Convective Heat and Mass Transfer in RotatingDisk Systems Springer Berlin Germany 2009
[45] C L Tien and D T Campbell ldquoHeat and mass transfer fromrotating conesrdquo Journal of FluidMechanics vol 17 no 1 pp 105ndash112 1963
[46] C J Elkins and J K Eaton ldquoHeat transfer in the rotatingdisk boundary layerrdquo Tech Rep TSD-103 Stanford UniversityDepartment of Mechanical EngineeringThermosciences Divi-sion Stanford Calif USA 1997
[47] G Karniadakis A Beskok and N Aluru Microflows Funda-mentals and Simulation Springer New York NY USA 2001
[48] N Kelson andA Desseaux ldquoNote on porous rotating disk flowrdquoAustralian amp New Zealand Industrial and Applied MathematicsJournal vol 42 pp 837ndash855 2000
[49] M Alam N Poddar M Rahman and K Vajravelu ldquoTransienthydromagnetic forced convective heat transfer slip flow due toa porous rotating disk with variable fluid propertiesrdquo AmericanJournal of Heat andMass Transfer vol 2 no 3 pp 165ndash189 2015
where Re = Ω1199032]infin is rotational Reynolds numberThus (24) shows that the radial and tangential skin
frictions coefficients are proportional to 1198651015840(0) and 1198661015840(0)respectively and Nusselt number is proportional to minus1205791015840(0)4 Result and Discussion
In this investigation Figures 2ndash6 show the effect of variousvalues of the physical parameters on the velocity (radialtangential and axial) and temperature distribution Resultsobtained for both constant fluid property and variable fluidproperty have been presented graphically
Figures 2(a)ndash2(d) show the effect of porosity parameter119870on all velocity components and temperature distribution It isobserved that radial tangential and axial velocity decreasesbut temperature increases by increasing the permeability forboth cases Whereas variation in porosity parameter has lesseffect on the temperature distribution radial velocity attains
Advances in Mathematical Physics 7
0 1 2 3 4 5 60
001002003004005006007008009
Freidoonimehr et al [40]Rashidi and Freidooni Mehr [27]
F(
)
K = = 0 ＋Ｈ = 0 ０Ｌ = 071
K = 0 = 02 ０Ｌ = 1K = = 0 ＋Ｈ = 005 ０Ｌ = 1
K = 05 = 02K = 05 = 0
K = 1 = 02K = 1 = 0
＋Ｈ = 005
０Ｌ = 1 ＋Ｈ = 005０Ｌ = 1 ＋Ｈ = 005
０Ｌ = 1 ＋Ｈ = 005０Ｌ = 1 ＋Ｈ = 005
(a)
0010203040506070809
1
0 1 2 3 4 5 6
G(
) K = = 0 ＋Ｈ = 0 ０Ｌ = 071K = 0 = 02
K = = 0 ＋Ｈ = 005 ０Ｌ = 1
K = 1 = 02K = 1 = 0
＋Ｈ = 005 ０Ｌ = 1
＋Ｈ = 005 ０Ｌ = 1 ＋Ｈ = 005 ０Ｌ = 1
Rashidi and Freidooni Mehr [27]Freidoonimehr et al [40]
(b)
0 1 2 3 4 5 6
H(
)
minus07
minus08
minus09
minus1
minus11
minus12
minus13
K = 10 = 02
K = 10 = 0K = 1 = 0
K = 0 = 02
K = = 0 ＋Ｈ = 005 ０Ｌ = 1K = 0 = 0
＋Ｈ = 005 ０Ｌ = 1
K = 1 = 02 ＋Ｈ = 005 ０Ｌ = 1 ＋Ｈ = 005 ０Ｌ = 1
＋Ｈ = 005 ０Ｌ = 1
＋Ｈ = 005 ０Ｌ = 1
＋Ｈ = 0 ０Ｌ = 071
Rashidi and Freidooni Mehr [27]Freidoonimehr et al [40]
(c)
0 2 4 6 8 10 12 140
0102030405060708
(
)
= 02 N = 1
= 0 N = 1
= 0 N = 0 K = 0K = 10 1 0
＋Ｈ = 005 ０Ｌ = 1
Rashidi and Freidooni Mehr [27]
(d)
Figure 2 Effect of variation in the porosity parameter on the (a) radial (b) tangential (c) axial and (d) temperature velocity profiles when119882119904 = minus1 and Re = 100
a maximum value close to the disk surface When we reducethe present problem into the literature available by takingpermeability parameter 119870 = 0 and radiation parameter119873 = 0 then the result obtained is exactly similar to thatof the Rashidi and Freidooni Mehr [27] and Freidoonimehret al [40] This validates the results obtained for presentinvestigation
Figures 3(a)ndash3(d) depict the effect of suction param-eter 119882119904 on the radial tangential axial and temperaturedistribution respectively Figures 3(a)ndash3(c) show that allvelocity components decrease as suction increases Physicalsignificance is that as suction increases adherence of thefluid with the wall increases and as a result boundary layerthickness decreases
Figure 3(d) depicts the effect of suction parameter ontemperature profile It is observed that as suction increasestemperature of fluid decreases because suction cools theboundary layer regime therefore suction is used for coolingthe flow in several engineering processes including MHDpower generators and nuclear energy processes
The effect of Reynolds number Re and Knudsen num-ber Kn on velocity and temperature distribution is plotted
in Figures 4 and 5 respectively Both the parameters areshowing the same effects on slip boundary conditions Itis observed that the increasing values of the Reynolds andKnudsen numbers decrease the fluid velocity componentsand temperature distribution The case when value of Kn liesbetween 0001 and 01 corresponds to slip at the surface of thedisk As slip increases the quantity of the fluid that can stickon the disk decreases Therefore circumferential velocity ofthe fluid reduces and causes reduction in centrifugal forceAs a result inward axial velocity decreases In other words asslip gets stronger flow of fluid drawn or pushed away alongthe velocity directions decreases the heat generation
Figure 6 shows the effect of Prandtl number on tem-perature profile As Prandtl number increases the thermalboundary layer thickness decreases Physical significance isthat Prandtl number precludes dispersal of heat in the fluid
Figure 7 depicts the variation of temperature profilewith radiation It is observed that as radiation increasestemperature of the fluid increases Also radiation parameterincreases temperature gradient near the surface of the disk
Figure 8 demonstrates the comparison of both constantand fluid properties on flow over a rotating disk with slip and
8 Advances in Mathematical Physics
0 1 2 3 4 5 6
0
0005
001
0015
002
0025F
()
Ws = minus1 minus2 minus4
= 0
= 02
(a)
0005
01015
02025
03035
04045
05
0 1 2 3 4 5 6
Ws = minus1 minus2 minus4
= 0
= 02
G(
)
(b)
0 1 2 3 4 5 6
= 0
= 02
Ws = minus1
Ws = minus2
Ws = minus4
H(
)
minus1
minus15
minus2
minus25
minus3
minus35
minus4
minus45
(c)
00102030405060708
Ws = minus1 minus2 minus4
= 0
= 02
10 1282 640 14
(
)
(d)
Figure 3 Effect of variation in the suction parameter on the (a) radial (b) tangential (c) axial and (d) temperature velocity profiles when119870 = 1 Kn = 005 Re = 100119873 = 1 and Pr = 1
Table 2 Comparison between the results of present study with the results reported by Kelson and Desseaux [48] Maleque and Sattar [33]and Alam et al [49] for the radial 1198651015840(0) and tangential minus1198661015840(0) skin friction coefficients for Pr = 071119872 = 0 120576 = 0 and Kn = 0
temperature jump The radial velocity achieves a maximumvalue near to the surface of the disk for all values of 120576 It is alsonoted that an increment in relative temperature differenceparameter causes shifting of maximum point from the diskThe velocity along the tangential direction increases asvalue of relative temperature difference parameter increaseswhereas the velocity along axial direction decreases Temper-ature increases as the value of relative temperature differenceparameter increasesThese results are similar to that obtainedby Rashidi et al [39]
Table 1 illustrates the effect of the parameters 120576 119870Kn Re Pr and 119873 on constant suction parameter on thenumerical values of (1198651015840(0) minus1198661015840(0)) and (minus1205791015840(0)) We observethat numerical value of 1198651015840(0) minus1198661015840(0) and minus1205791015840(0) decreases
with the increasing value of Kn Re and radiation parameter119873Tables 2 and 3 depict the comparison of skin friction coef-
ficients and rate of heat transfer between the results obtainedin the present investigation and the literature available [3348 49] It is observed that results obtained in present studyare very well in agreement with the existing results
5 Conclusions
In this study we have investigated radiation effect on velocityprofile for all components and temperature profile throughrotating disk in porous medium for variable fluid propertiesand in particular case for constant fluid properties also By
Advances in Mathematical Physics 9
0 1 2 3 4 5 6
= 0
= 02
Ws = minus1
２ = 1 10 100 1000 10000
00005
0010015
0020025
0030035
004
F(
)
(a)
= 0
= 02
5 63 41 20
Ws = minus1
２ = 1 10 100 1000 10000
0010203040506070809
1
G(
)
(b)
0 2 3 41 65
Ws = minus2 = 0
２ = 1
２ = 10
２ = 100
２ = 1000
２ = 10000
minus204minus2035minus203
minus2025minus202
minus2015minus201
minus2005minus2
minus1995minus199
H(
)
(c)
0 2 3 41 65
Ws = minus2 = 02 ２ = 1
２ = 10
２ = 100
２ = 1000
２ = 10000
minus2
minus195
minus19
minus185
minus18
minus175
minus17
H(
)
(d)
0 2 4 6 8 10 12 14
= 0
= 02
Ws = minus1
２ = 1 10 100 1000 10000
0010203040506070809
1
(
)
(e)
Figure 4 Effect of variation in the Reynolds number on the (a) radial (b) tangential (c) amp (d) axial and (e) temperature velocity profileswhen 119870 = 1 Kn = 005119873 = 1 and Pr = 1
Table 3 Comparison between the results of present study with the results reported by Kelson and Desseaux [48] Maleque and Sattar [33]and Alam et al [49] for rate of heat transfer minus1205791015840(0) for Pr = 071119872 = 0 120576 = 0 Kn = 0 and119873 = 0119882119904 Present Alam et al [49] Maleque and Sattar [33] Kelson and Desseaux [48]minus1205791015840(0) minus1205791015840(0) minus1205791015840(0) minus1205791015840(0)0 0326798372 032637889 032576 0325856minus2 1438764651 143876482 144212 1437782minus4 2842381877 284369011 284470 2842381minus5 3551223146 355222471 355411 3551223
10 Advances in Mathematical Physics
0 1 2 3 4 5 6
= 0
= 02
00005
0010015
0020025
0030035
004F
()
Ws = minus1
＋Ｈ = 0 002 005 01
(a)
0 1 2 3 4 5 6
= 0
= 02
0010203040506070809
1
G(
)
Ws = minus1
＋Ｈ = 0 002 005 01
(b)
0 1 2 3 4 5 6
= 0
= 02
minus2
minus195
minus19
minus185
minus18
minus175
minus17
H(
)
Ws = minus2
＋Ｈ = 005 002 0
(c)
= 0
= 02
0010203040506070809
1
(
)
104 6 8 12 1420
Ws = minus1
＋Ｈ = 0 002 005 01N = 0 ＋Ｈ = 005
(d)
Figure 5 Effect of variation in the Knudsen number on the (a) radial (b) tangential (c) axial and (d) temperature velocity profiles when119870 = 1 Re = 100119873 = 1 and Pr = 1
= 0
= 02
2 4 6 8 10 12 140
00102030405060708
(
)
N = 0 ０Ｌ = 4０Ｌ = 1 2 3 4
Figure 6 Effect of Prandtl number on temperature distributionwhen 119870 = 1119882119904 = minus1 Re = 100 Kn = 005 and119873 = 1
similarity transformation governing equations transformedinto nonlinear ordinary differential equations which aresolved numerically by usingRunge-Kuttamethodwith shoot-ing technique Based on the resulting solutions the followingconclusions can be drawn
= 0
= 02
2 4 6 8 10 12 140
N = 4 3 2 1 0
0010203040506070809
(
)
Figure 7 Effect of radiation parameter on temperature distributionwhen 119870 = 1119882119904 = minus1 Re = 100 Kn = 005 and Pr = 1
(i) The radial tangential and axial velocity profilesdecrease while the temperature increases with theincreasing values of porosity parameter
(ii) The increasing value of Reynolds and Knudsen num-ber decreases the fluid velocity components and
Advances in Mathematical Physics 11
1 2 3 4 5 60
= 0 02 04 10
0005
001
0015
002
0025F
()
(a)
1 2 54 60 3
0
01
02
03
04
05
06
G(
)
= 1 04 02 0
(b)
1 2 3 4 5 60
= 0
= 02
= 04
= 1
minus11
minus1
minus09
minus08
minus07
minus06
minus05
minus04
H(
)
(c)
00102030405060708
(
)
2 4 6 8 10 12 140
N = 0 = 02
= 1 04 02 0
(d)
Figure 8 Effect of variation in the relative temperature difference parameter on the (a) radial (b) tangential (c) axial and (d) temperaturevelocity profiles when119870 = 1119882119904 = minus1 Kn = 005 Re = 100119873 = 1 and Pr = 1
temperature and suction parameter also shows thesame effect
(iii) For the effect of the radiation parameter on the tem-perature distribution it is seen that the temperaturedistribution decreases with the increasing values ofradiation parameter and also it has been observed thatthe radial and tangential skin friction values decreasewith increase in the radiation parameter
Competing Interests
The authors declare that they have no competing interests
References
[1] Th V Karman ldquoUber laminare und turbulente reibunrdquo Zeits-chrift fur Angewandte Mathematik und Mechanik vol 1 no 4pp 233ndash252 1921
[2] W G Cochran ldquoThe flow due to a rotating diskrdquoMathematicalProceedings of the Cambridge Philosophical Society vol 30 no3 pp 365ndash375 1934
[3] E R Benton ldquoOn the flow due to a rotating diskrdquo Journal ofFluid Mechanics vol 24 no 4 pp 781ndash800 1966
[4] M Turkyilmazoglu ldquoPurely analytic solutions of the compress-ible boundary layer flow due to a porous rotating disk with heattransferrdquoPhysics of Fluids vol 21 no 10 Article ID 106104 2009
[5] D S Chauhan and S Gupta ldquoSteady flow and heat transferbetween two stationary naturally permeable disksrdquo Journal ofUltra Scientist of Physical Sciences vol 12 no 1 pp 45ndash52 2000
[6] C Wagner ldquoHeat transfer from a rotating disk to ambient airrdquoJournal of Applied Physics vol 19 no 9 pp 837ndash839 1948
[7] K Millsaps and K Pohlhausen ldquoHeat transfer by laminar flowfrom a rotating platerdquo Journal of the Aeronautical Sciences vol19 pp 120ndash126 1952
[8] F Kreith and J H Taylor ldquoHeat transfer from a rotating disk inturbulent flow no 1956rdquo ASME Paper 56-A-146 1956
[9] H-T Lin and L-K Lin ldquoHeat transfer from a rotating cone ordisk to fluids of any Prandtl numberrdquo International Communi-cations in Heat and Mass Transfer vol 14 no 3 pp 323ndash3321987
[10] P D Verma and D S Chauhan ldquoFlow between a torsionallyoscillating impermeable disc and a stationary naturally perme-able discrdquo Indian Journal of Pure and Applied Mathematics vol10 pp 1351ndash1361 1979
[11] D S Chauhan and S Jain ldquoSteady flow between highlypermeable rotating disksrdquo Indian Journal of Theoretical Physicsvol 52 no 1 pp 39ndash50 2004
[12] M Turkyilmazoglu ldquoNanofluid flow and heat transfer due to arotating diskrdquo Computers amp Fluids vol 94 pp 139ndash146 2014
[13] M Turkyilmazoglu ldquoMHD fluid flow and heat transfer due to ashrinking rotating diskrdquo Computers amp Fluids vol 90 pp 51ndash562014
[14] M Azimi and R Riazi ldquoHeat transfer analysis of GO-waternanofluid flow between two parallel disksrdquo Propulsion andPower Research vol 4 no 1 pp 23ndash30 2015
12 Advances in Mathematical Physics
[15] B R Sharma and H Konwar ldquoEffect of chemical reaction onmass transfer due to a permeable rotating heated diskrdquo Interna-tional Journal of Computer Applications vol 119 no 21 pp 5ndash92015
[16] H Shahmohamadi and M Mohammadpour ldquoA series solutionfor three-dimensional navier-stokes equations of flow near aninfinite rotating diskrdquo World Journal of Mechanics vol 4 pp117ndash127 2014
[17] S Srinivas A S Reddy T R Ramamohan and A K ShuklaldquoThermal-diffusion and diffusion-thermo effects onMHD flowof viscous fluid between expanding or contracting rotatingporous disks with viscous dissipationrdquo Journal of the EgyptianMathematical Society vol 24 no 1 pp 100ndash107 2016
[18] M A Mansour ldquoRadiative and free-convection effects on theoscillatory flow past a vertical platerdquo Astrophysics and SpaceScience vol 166 no 2 pp 269ndash275 1990
[19] M A Hossain M A Alim and D A S Rees ldquoThe effectof radiation on free convection from a porous vertical platerdquoInternational Journal of Heat and Mass Transfer vol 42 no 1pp 181ndash191 1999
[20] A Raptis and C Perdikis ldquoMHD free convection flow by thepresence of radiationrdquo International Journal of Magnetohydro-dynamics Plasma and Space Research vol 9 pp 237ndash252 2000
[21] M A Hossain and H S Takhar ldquoRadiation effect on mixedconvection along a vertical plate with uniform surface temper-aturerdquoHeat and Mass Transfer vol 31 no 4 pp 243ndash248 1996
[22] A Devi and R U Devi ldquoEffects of thermal radiation onhydromagnetic flow due to a porous rotating disk with halleffectrdquo Journal of Applied Fluid Mechanics vol 5 no 2 pp 1ndash7 2012
[23] C Y Wang ldquoFlow due to a stretching boundary with partialslipmdashan exact solution of the Navier-Stokes equationsrdquo Chemi-cal Engineering Science vol 57 no 17 pp 3745ndash3747 2002
[24] E Osalusi ldquoEffects of thermal radiation on MHD and slip flowover a porous rotating disk with variable propertiesrdquo RomanianJournal of Physics vol 52 no 3-4 pp 217ndash229 2007
[25] A A Khidir ldquoViscous dissipation Ohmic heating and radiationeffects on MHD flow past a rotating disk embedded in aporous medium with variable propertiesrdquo Arabian Journal ofMathematics vol 2 no 3 pp 263ndash277 2013
[26] E M Sparrow G S Beavers and L Y Hung ldquoFlow about aporous-surfaced rotating diskrdquo International Journal of Heatand Mass Transfer vol 14 no 7 pp 993ndash996 1971
[27] M M Rashidi and N Freidooni Mehr ldquoEffects of velocity slipand temperature jump on the entropy generation in magne-tohydrodynamic flow over a porous rotating diskrdquo Journal ofMechanical Engineering vol 1 no 3 2012
[28] A Arikoglu G Komurgoz I Ozkol and A Y Gunes ldquoCom-bined effects of temperature and velocity jump on the heattransfer fluid flow and entropy generation over a single rotatingdiskrdquo Journal of Heat Transfer vol 132 no 11 Article ID 1117032010
[29] THayatM RashidM Imtiaz andAAlsaedi ldquoMagnetohydro-dynamic (MHD) flow of Cu-water nanofluid due to a rotatingdisk with partial sliprdquo AIP Advances vol 5 no 6 Article ID067169 2015
[30] M Zakerullah and J A D Ackroyd ldquoLaminar natural convec-tion boundary-layers on Horizontal Circular disksrdquo Journal ofApplied Mathematics and Physics vol 30 pp 427ndash435 1979
[31] HHerwig ldquoThe effect of variable properties onmomentumandheat transfer in a tube with constant heat flux across the wallrdquoInternational Journal of Heat and Mass Transfer vol 28 no 2pp 423ndash431 1985
[32] H Herwig and K Klemp ldquoVariable property effects of fullydeveloped laminar flow in concentric annulirdquo Journal of HeatTransfer vol 110 no 2 pp 314ndash320 1988
[33] K AMaleque andMA Sattar ldquoSteady laminar convective flowwith variable properties due to a porous rotating diskrdquo Journalof Heat Transfer vol 127 no 12 pp 1406ndash1409 2005
[34] K A Maleque and M A Sattar ldquoThe effects of variable proper-ties and hall current on steady MHD laminar convective fluidflow due to a porous rotating diskrdquo International Journal of Heatand Mass Transfer vol 48 no 23-24 pp 4963ndash4972 2005
[35] E Osalusi and P Sibanda ldquoOn variable laminar convective flowproperties due to a porous rotating disk in a magnetic fieldrdquoRomanian Journal of Physics vol 9 no 10 pp 933ndash944 2006
[36] M M Rahman ldquoConvective Hydromagnetic slip flow withvariable properties due to a porous rotating diskrdquo The SultanQaboos University Journal for Science vol 15 pp 55ndash79 2010
[37] M M Rashidi S A Mohimanian Pour T Hayat and SObaidat ldquoAnalytic approximate solutions for steady flow over arotating disk in porousmediumwith heat transfer by homotopyanalysis methodrdquo Computers amp Fluids vol 54 no 1 pp 1ndash92012
[38] S Hussain F Ahmad M Shafique and S Hussain ldquoNumericalsolution for accelerated rotating disk in a viscous fluidrdquoAppliedMathematics vol 4 no 6 pp 899ndash902 2013
[39] M M Rashidi N Kavyani and S Abelman ldquoInvestigation ofentropy generation inMHDand slip flow over a rotating porousdisk with variable propertiesrdquo International Journal of Heat andMass Transfer vol 70 pp 892ndash917 2014
[40] N Freidoonimehr M M Rashidi S Abelman and G Loren-zini ldquoAnalytical modeling of MHD flow over a permeablerotating disk in the presence of Soret and Dufour effectsentropy analysisrdquo Entropy vol 18 no 5 article 131 2016
[41] M S Alam S M Chapal Hossain and M M Rahman ldquoTran-sient thermophoretic particle deposition on forced convectiveheat and mass transfer flow due to a rotating diskrdquo Ain ShamsEngineering Journal vol 7 no 1 pp 441ndash452 2016
[42] S Jayaraj ldquoThermophoresis in laminar flow over cold inclinedplates with variable propertiesrdquoHeat andMass Transfer vol 30no 3 pp 167ndash173 1995
[43] M M Rahman ldquoThermophoretic deposition of nanoparticlesdue to a permeable rotating disk effects of partial slip mag-netic field thermal radiation thermal-diffusion and diffusion-thermordquo International Journal of Mathematical ComputationalPhysical Electrical and Computer Engineering vol 7 no 5 2013
[44] I V Shevchuk Convective Heat and Mass Transfer in RotatingDisk Systems Springer Berlin Germany 2009
[45] C L Tien and D T Campbell ldquoHeat and mass transfer fromrotating conesrdquo Journal of FluidMechanics vol 17 no 1 pp 105ndash112 1963
[46] C J Elkins and J K Eaton ldquoHeat transfer in the rotatingdisk boundary layerrdquo Tech Rep TSD-103 Stanford UniversityDepartment of Mechanical EngineeringThermosciences Divi-sion Stanford Calif USA 1997
[47] G Karniadakis A Beskok and N Aluru Microflows Funda-mentals and Simulation Springer New York NY USA 2001
[48] N Kelson andA Desseaux ldquoNote on porous rotating disk flowrdquoAustralian amp New Zealand Industrial and Applied MathematicsJournal vol 42 pp 837ndash855 2000
[49] M Alam N Poddar M Rahman and K Vajravelu ldquoTransienthydromagnetic forced convective heat transfer slip flow due toa porous rotating disk with variable fluid propertiesrdquo AmericanJournal of Heat andMass Transfer vol 2 no 3 pp 165ndash189 2015
Freidoonimehr et al [40]Rashidi and Freidooni Mehr [27]
F(
)
K = = 0 ＋Ｈ = 0 ０Ｌ = 071
K = 0 = 02 ０Ｌ = 1K = = 0 ＋Ｈ = 005 ０Ｌ = 1
K = 05 = 02K = 05 = 0
K = 1 = 02K = 1 = 0
＋Ｈ = 005
０Ｌ = 1 ＋Ｈ = 005０Ｌ = 1 ＋Ｈ = 005
０Ｌ = 1 ＋Ｈ = 005０Ｌ = 1 ＋Ｈ = 005
(a)
0010203040506070809
1
0 1 2 3 4 5 6
G(
) K = = 0 ＋Ｈ = 0 ０Ｌ = 071K = 0 = 02
K = = 0 ＋Ｈ = 005 ０Ｌ = 1
K = 1 = 02K = 1 = 0
＋Ｈ = 005 ０Ｌ = 1
＋Ｈ = 005 ０Ｌ = 1 ＋Ｈ = 005 ０Ｌ = 1
Rashidi and Freidooni Mehr [27]Freidoonimehr et al [40]
(b)
0 1 2 3 4 5 6
H(
)
minus07
minus08
minus09
minus1
minus11
minus12
minus13
K = 10 = 02
K = 10 = 0K = 1 = 0
K = 0 = 02
K = = 0 ＋Ｈ = 005 ０Ｌ = 1K = 0 = 0
＋Ｈ = 005 ０Ｌ = 1
K = 1 = 02 ＋Ｈ = 005 ０Ｌ = 1 ＋Ｈ = 005 ０Ｌ = 1
＋Ｈ = 005 ０Ｌ = 1
＋Ｈ = 005 ０Ｌ = 1
＋Ｈ = 0 ０Ｌ = 071
Rashidi and Freidooni Mehr [27]Freidoonimehr et al [40]
(c)
0 2 4 6 8 10 12 140
0102030405060708
(
)
= 02 N = 1
= 0 N = 1
= 0 N = 0 K = 0K = 10 1 0
＋Ｈ = 005 ０Ｌ = 1
Rashidi and Freidooni Mehr [27]
(d)
Figure 2 Effect of variation in the porosity parameter on the (a) radial (b) tangential (c) axial and (d) temperature velocity profiles when119882119904 = minus1 and Re = 100
a maximum value close to the disk surface When we reducethe present problem into the literature available by takingpermeability parameter 119870 = 0 and radiation parameter119873 = 0 then the result obtained is exactly similar to thatof the Rashidi and Freidooni Mehr [27] and Freidoonimehret al [40] This validates the results obtained for presentinvestigation
Figures 3(a)ndash3(d) depict the effect of suction param-eter 119882119904 on the radial tangential axial and temperaturedistribution respectively Figures 3(a)ndash3(c) show that allvelocity components decrease as suction increases Physicalsignificance is that as suction increases adherence of thefluid with the wall increases and as a result boundary layerthickness decreases
Figure 3(d) depicts the effect of suction parameter ontemperature profile It is observed that as suction increasestemperature of fluid decreases because suction cools theboundary layer regime therefore suction is used for coolingthe flow in several engineering processes including MHDpower generators and nuclear energy processes
The effect of Reynolds number Re and Knudsen num-ber Kn on velocity and temperature distribution is plotted
in Figures 4 and 5 respectively Both the parameters areshowing the same effects on slip boundary conditions Itis observed that the increasing values of the Reynolds andKnudsen numbers decrease the fluid velocity componentsand temperature distribution The case when value of Kn liesbetween 0001 and 01 corresponds to slip at the surface of thedisk As slip increases the quantity of the fluid that can stickon the disk decreases Therefore circumferential velocity ofthe fluid reduces and causes reduction in centrifugal forceAs a result inward axial velocity decreases In other words asslip gets stronger flow of fluid drawn or pushed away alongthe velocity directions decreases the heat generation
Figure 6 shows the effect of Prandtl number on tem-perature profile As Prandtl number increases the thermalboundary layer thickness decreases Physical significance isthat Prandtl number precludes dispersal of heat in the fluid
Figure 7 depicts the variation of temperature profilewith radiation It is observed that as radiation increasestemperature of the fluid increases Also radiation parameterincreases temperature gradient near the surface of the disk
Figure 8 demonstrates the comparison of both constantand fluid properties on flow over a rotating disk with slip and
8 Advances in Mathematical Physics
0 1 2 3 4 5 6
0
0005
001
0015
002
0025F
()
Ws = minus1 minus2 minus4
= 0
= 02
(a)
0005
01015
02025
03035
04045
05
0 1 2 3 4 5 6
Ws = minus1 minus2 minus4
= 0
= 02
G(
)
(b)
0 1 2 3 4 5 6
= 0
= 02
Ws = minus1
Ws = minus2
Ws = minus4
H(
)
minus1
minus15
minus2
minus25
minus3
minus35
minus4
minus45
(c)
00102030405060708
Ws = minus1 minus2 minus4
= 0
= 02
10 1282 640 14
(
)
(d)
Figure 3 Effect of variation in the suction parameter on the (a) radial (b) tangential (c) axial and (d) temperature velocity profiles when119870 = 1 Kn = 005 Re = 100119873 = 1 and Pr = 1
Table 2 Comparison between the results of present study with the results reported by Kelson and Desseaux [48] Maleque and Sattar [33]and Alam et al [49] for the radial 1198651015840(0) and tangential minus1198661015840(0) skin friction coefficients for Pr = 071119872 = 0 120576 = 0 and Kn = 0
temperature jump The radial velocity achieves a maximumvalue near to the surface of the disk for all values of 120576 It is alsonoted that an increment in relative temperature differenceparameter causes shifting of maximum point from the diskThe velocity along the tangential direction increases asvalue of relative temperature difference parameter increaseswhereas the velocity along axial direction decreases Temper-ature increases as the value of relative temperature differenceparameter increasesThese results are similar to that obtainedby Rashidi et al [39]
Table 1 illustrates the effect of the parameters 120576 119870Kn Re Pr and 119873 on constant suction parameter on thenumerical values of (1198651015840(0) minus1198661015840(0)) and (minus1205791015840(0)) We observethat numerical value of 1198651015840(0) minus1198661015840(0) and minus1205791015840(0) decreases
with the increasing value of Kn Re and radiation parameter119873Tables 2 and 3 depict the comparison of skin friction coef-
ficients and rate of heat transfer between the results obtainedin the present investigation and the literature available [3348 49] It is observed that results obtained in present studyare very well in agreement with the existing results
5 Conclusions
In this study we have investigated radiation effect on velocityprofile for all components and temperature profile throughrotating disk in porous medium for variable fluid propertiesand in particular case for constant fluid properties also By
Advances in Mathematical Physics 9
0 1 2 3 4 5 6
= 0
= 02
Ws = minus1
２ = 1 10 100 1000 10000
00005
0010015
0020025
0030035
004
F(
)
(a)
= 0
= 02
5 63 41 20
Ws = minus1
２ = 1 10 100 1000 10000
0010203040506070809
1
G(
)
(b)
0 2 3 41 65
Ws = minus2 = 0
２ = 1
２ = 10
２ = 100
２ = 1000
２ = 10000
minus204minus2035minus203
minus2025minus202
minus2015minus201
minus2005minus2
minus1995minus199
H(
)
(c)
0 2 3 41 65
Ws = minus2 = 02 ２ = 1
２ = 10
２ = 100
２ = 1000
２ = 10000
minus2
minus195
minus19
minus185
minus18
minus175
minus17
H(
)
(d)
0 2 4 6 8 10 12 14
= 0
= 02
Ws = minus1
２ = 1 10 100 1000 10000
0010203040506070809
1
(
)
(e)
Figure 4 Effect of variation in the Reynolds number on the (a) radial (b) tangential (c) amp (d) axial and (e) temperature velocity profileswhen 119870 = 1 Kn = 005119873 = 1 and Pr = 1
Table 3 Comparison between the results of present study with the results reported by Kelson and Desseaux [48] Maleque and Sattar [33]and Alam et al [49] for rate of heat transfer minus1205791015840(0) for Pr = 071119872 = 0 120576 = 0 Kn = 0 and119873 = 0119882119904 Present Alam et al [49] Maleque and Sattar [33] Kelson and Desseaux [48]minus1205791015840(0) minus1205791015840(0) minus1205791015840(0) minus1205791015840(0)0 0326798372 032637889 032576 0325856minus2 1438764651 143876482 144212 1437782minus4 2842381877 284369011 284470 2842381minus5 3551223146 355222471 355411 3551223
10 Advances in Mathematical Physics
0 1 2 3 4 5 6
= 0
= 02
00005
0010015
0020025
0030035
004F
()
Ws = minus1
＋Ｈ = 0 002 005 01
(a)
0 1 2 3 4 5 6
= 0
= 02
0010203040506070809
1
G(
)
Ws = minus1
＋Ｈ = 0 002 005 01
(b)
0 1 2 3 4 5 6
= 0
= 02
minus2
minus195
minus19
minus185
minus18
minus175
minus17
H(
)
Ws = minus2
＋Ｈ = 005 002 0
(c)
= 0
= 02
0010203040506070809
1
(
)
104 6 8 12 1420
Ws = minus1
＋Ｈ = 0 002 005 01N = 0 ＋Ｈ = 005
(d)
Figure 5 Effect of variation in the Knudsen number on the (a) radial (b) tangential (c) axial and (d) temperature velocity profiles when119870 = 1 Re = 100119873 = 1 and Pr = 1
= 0
= 02
2 4 6 8 10 12 140
00102030405060708
(
)
N = 0 ０Ｌ = 4０Ｌ = 1 2 3 4
Figure 6 Effect of Prandtl number on temperature distributionwhen 119870 = 1119882119904 = minus1 Re = 100 Kn = 005 and119873 = 1
similarity transformation governing equations transformedinto nonlinear ordinary differential equations which aresolved numerically by usingRunge-Kuttamethodwith shoot-ing technique Based on the resulting solutions the followingconclusions can be drawn
= 0
= 02
2 4 6 8 10 12 140
N = 4 3 2 1 0
0010203040506070809
(
)
Figure 7 Effect of radiation parameter on temperature distributionwhen 119870 = 1119882119904 = minus1 Re = 100 Kn = 005 and Pr = 1
(i) The radial tangential and axial velocity profilesdecrease while the temperature increases with theincreasing values of porosity parameter
(ii) The increasing value of Reynolds and Knudsen num-ber decreases the fluid velocity components and
Advances in Mathematical Physics 11
1 2 3 4 5 60
= 0 02 04 10
0005
001
0015
002
0025F
()
(a)
1 2 54 60 3
0
01
02
03
04
05
06
G(
)
= 1 04 02 0
(b)
1 2 3 4 5 60
= 0
= 02
= 04
= 1
minus11
minus1
minus09
minus08
minus07
minus06
minus05
minus04
H(
)
(c)
00102030405060708
(
)
2 4 6 8 10 12 140
N = 0 = 02
= 1 04 02 0
(d)
Figure 8 Effect of variation in the relative temperature difference parameter on the (a) radial (b) tangential (c) axial and (d) temperaturevelocity profiles when119870 = 1119882119904 = minus1 Kn = 005 Re = 100119873 = 1 and Pr = 1
temperature and suction parameter also shows thesame effect
(iii) For the effect of the radiation parameter on the tem-perature distribution it is seen that the temperaturedistribution decreases with the increasing values ofradiation parameter and also it has been observed thatthe radial and tangential skin friction values decreasewith increase in the radiation parameter
Competing Interests
The authors declare that they have no competing interests
References
[1] Th V Karman ldquoUber laminare und turbulente reibunrdquo Zeits-chrift fur Angewandte Mathematik und Mechanik vol 1 no 4pp 233ndash252 1921
[2] W G Cochran ldquoThe flow due to a rotating diskrdquoMathematicalProceedings of the Cambridge Philosophical Society vol 30 no3 pp 365ndash375 1934
[3] E R Benton ldquoOn the flow due to a rotating diskrdquo Journal ofFluid Mechanics vol 24 no 4 pp 781ndash800 1966
[4] M Turkyilmazoglu ldquoPurely analytic solutions of the compress-ible boundary layer flow due to a porous rotating disk with heattransferrdquoPhysics of Fluids vol 21 no 10 Article ID 106104 2009
[5] D S Chauhan and S Gupta ldquoSteady flow and heat transferbetween two stationary naturally permeable disksrdquo Journal ofUltra Scientist of Physical Sciences vol 12 no 1 pp 45ndash52 2000
[6] C Wagner ldquoHeat transfer from a rotating disk to ambient airrdquoJournal of Applied Physics vol 19 no 9 pp 837ndash839 1948
[7] K Millsaps and K Pohlhausen ldquoHeat transfer by laminar flowfrom a rotating platerdquo Journal of the Aeronautical Sciences vol19 pp 120ndash126 1952
[8] F Kreith and J H Taylor ldquoHeat transfer from a rotating disk inturbulent flow no 1956rdquo ASME Paper 56-A-146 1956
[9] H-T Lin and L-K Lin ldquoHeat transfer from a rotating cone ordisk to fluids of any Prandtl numberrdquo International Communi-cations in Heat and Mass Transfer vol 14 no 3 pp 323ndash3321987
[10] P D Verma and D S Chauhan ldquoFlow between a torsionallyoscillating impermeable disc and a stationary naturally perme-able discrdquo Indian Journal of Pure and Applied Mathematics vol10 pp 1351ndash1361 1979
[11] D S Chauhan and S Jain ldquoSteady flow between highlypermeable rotating disksrdquo Indian Journal of Theoretical Physicsvol 52 no 1 pp 39ndash50 2004
[12] M Turkyilmazoglu ldquoNanofluid flow and heat transfer due to arotating diskrdquo Computers amp Fluids vol 94 pp 139ndash146 2014
[13] M Turkyilmazoglu ldquoMHD fluid flow and heat transfer due to ashrinking rotating diskrdquo Computers amp Fluids vol 90 pp 51ndash562014
[14] M Azimi and R Riazi ldquoHeat transfer analysis of GO-waternanofluid flow between two parallel disksrdquo Propulsion andPower Research vol 4 no 1 pp 23ndash30 2015
12 Advances in Mathematical Physics
[15] B R Sharma and H Konwar ldquoEffect of chemical reaction onmass transfer due to a permeable rotating heated diskrdquo Interna-tional Journal of Computer Applications vol 119 no 21 pp 5ndash92015
[16] H Shahmohamadi and M Mohammadpour ldquoA series solutionfor three-dimensional navier-stokes equations of flow near aninfinite rotating diskrdquo World Journal of Mechanics vol 4 pp117ndash127 2014
[17] S Srinivas A S Reddy T R Ramamohan and A K ShuklaldquoThermal-diffusion and diffusion-thermo effects onMHD flowof viscous fluid between expanding or contracting rotatingporous disks with viscous dissipationrdquo Journal of the EgyptianMathematical Society vol 24 no 1 pp 100ndash107 2016
[18] M A Mansour ldquoRadiative and free-convection effects on theoscillatory flow past a vertical platerdquo Astrophysics and SpaceScience vol 166 no 2 pp 269ndash275 1990
[19] M A Hossain M A Alim and D A S Rees ldquoThe effectof radiation on free convection from a porous vertical platerdquoInternational Journal of Heat and Mass Transfer vol 42 no 1pp 181ndash191 1999
[20] A Raptis and C Perdikis ldquoMHD free convection flow by thepresence of radiationrdquo International Journal of Magnetohydro-dynamics Plasma and Space Research vol 9 pp 237ndash252 2000
[21] M A Hossain and H S Takhar ldquoRadiation effect on mixedconvection along a vertical plate with uniform surface temper-aturerdquoHeat and Mass Transfer vol 31 no 4 pp 243ndash248 1996
[22] A Devi and R U Devi ldquoEffects of thermal radiation onhydromagnetic flow due to a porous rotating disk with halleffectrdquo Journal of Applied Fluid Mechanics vol 5 no 2 pp 1ndash7 2012
[23] C Y Wang ldquoFlow due to a stretching boundary with partialslipmdashan exact solution of the Navier-Stokes equationsrdquo Chemi-cal Engineering Science vol 57 no 17 pp 3745ndash3747 2002
[24] E Osalusi ldquoEffects of thermal radiation on MHD and slip flowover a porous rotating disk with variable propertiesrdquo RomanianJournal of Physics vol 52 no 3-4 pp 217ndash229 2007
[25] A A Khidir ldquoViscous dissipation Ohmic heating and radiationeffects on MHD flow past a rotating disk embedded in aporous medium with variable propertiesrdquo Arabian Journal ofMathematics vol 2 no 3 pp 263ndash277 2013
[26] E M Sparrow G S Beavers and L Y Hung ldquoFlow about aporous-surfaced rotating diskrdquo International Journal of Heatand Mass Transfer vol 14 no 7 pp 993ndash996 1971
[27] M M Rashidi and N Freidooni Mehr ldquoEffects of velocity slipand temperature jump on the entropy generation in magne-tohydrodynamic flow over a porous rotating diskrdquo Journal ofMechanical Engineering vol 1 no 3 2012
[28] A Arikoglu G Komurgoz I Ozkol and A Y Gunes ldquoCom-bined effects of temperature and velocity jump on the heattransfer fluid flow and entropy generation over a single rotatingdiskrdquo Journal of Heat Transfer vol 132 no 11 Article ID 1117032010
[29] THayatM RashidM Imtiaz andAAlsaedi ldquoMagnetohydro-dynamic (MHD) flow of Cu-water nanofluid due to a rotatingdisk with partial sliprdquo AIP Advances vol 5 no 6 Article ID067169 2015
[30] M Zakerullah and J A D Ackroyd ldquoLaminar natural convec-tion boundary-layers on Horizontal Circular disksrdquo Journal ofApplied Mathematics and Physics vol 30 pp 427ndash435 1979
[31] HHerwig ldquoThe effect of variable properties onmomentumandheat transfer in a tube with constant heat flux across the wallrdquoInternational Journal of Heat and Mass Transfer vol 28 no 2pp 423ndash431 1985
[32] H Herwig and K Klemp ldquoVariable property effects of fullydeveloped laminar flow in concentric annulirdquo Journal of HeatTransfer vol 110 no 2 pp 314ndash320 1988
[33] K AMaleque andMA Sattar ldquoSteady laminar convective flowwith variable properties due to a porous rotating diskrdquo Journalof Heat Transfer vol 127 no 12 pp 1406ndash1409 2005
[34] K A Maleque and M A Sattar ldquoThe effects of variable proper-ties and hall current on steady MHD laminar convective fluidflow due to a porous rotating diskrdquo International Journal of Heatand Mass Transfer vol 48 no 23-24 pp 4963ndash4972 2005
[35] E Osalusi and P Sibanda ldquoOn variable laminar convective flowproperties due to a porous rotating disk in a magnetic fieldrdquoRomanian Journal of Physics vol 9 no 10 pp 933ndash944 2006
[36] M M Rahman ldquoConvective Hydromagnetic slip flow withvariable properties due to a porous rotating diskrdquo The SultanQaboos University Journal for Science vol 15 pp 55ndash79 2010
[37] M M Rashidi S A Mohimanian Pour T Hayat and SObaidat ldquoAnalytic approximate solutions for steady flow over arotating disk in porousmediumwith heat transfer by homotopyanalysis methodrdquo Computers amp Fluids vol 54 no 1 pp 1ndash92012
[38] S Hussain F Ahmad M Shafique and S Hussain ldquoNumericalsolution for accelerated rotating disk in a viscous fluidrdquoAppliedMathematics vol 4 no 6 pp 899ndash902 2013
[39] M M Rashidi N Kavyani and S Abelman ldquoInvestigation ofentropy generation inMHDand slip flow over a rotating porousdisk with variable propertiesrdquo International Journal of Heat andMass Transfer vol 70 pp 892ndash917 2014
[40] N Freidoonimehr M M Rashidi S Abelman and G Loren-zini ldquoAnalytical modeling of MHD flow over a permeablerotating disk in the presence of Soret and Dufour effectsentropy analysisrdquo Entropy vol 18 no 5 article 131 2016
[41] M S Alam S M Chapal Hossain and M M Rahman ldquoTran-sient thermophoretic particle deposition on forced convectiveheat and mass transfer flow due to a rotating diskrdquo Ain ShamsEngineering Journal vol 7 no 1 pp 441ndash452 2016
[42] S Jayaraj ldquoThermophoresis in laminar flow over cold inclinedplates with variable propertiesrdquoHeat andMass Transfer vol 30no 3 pp 167ndash173 1995
[43] M M Rahman ldquoThermophoretic deposition of nanoparticlesdue to a permeable rotating disk effects of partial slip mag-netic field thermal radiation thermal-diffusion and diffusion-thermordquo International Journal of Mathematical ComputationalPhysical Electrical and Computer Engineering vol 7 no 5 2013
[44] I V Shevchuk Convective Heat and Mass Transfer in RotatingDisk Systems Springer Berlin Germany 2009
[45] C L Tien and D T Campbell ldquoHeat and mass transfer fromrotating conesrdquo Journal of FluidMechanics vol 17 no 1 pp 105ndash112 1963
[46] C J Elkins and J K Eaton ldquoHeat transfer in the rotatingdisk boundary layerrdquo Tech Rep TSD-103 Stanford UniversityDepartment of Mechanical EngineeringThermosciences Divi-sion Stanford Calif USA 1997
[47] G Karniadakis A Beskok and N Aluru Microflows Funda-mentals and Simulation Springer New York NY USA 2001
[48] N Kelson andA Desseaux ldquoNote on porous rotating disk flowrdquoAustralian amp New Zealand Industrial and Applied MathematicsJournal vol 42 pp 837ndash855 2000
[49] M Alam N Poddar M Rahman and K Vajravelu ldquoTransienthydromagnetic forced convective heat transfer slip flow due toa porous rotating disk with variable fluid propertiesrdquo AmericanJournal of Heat andMass Transfer vol 2 no 3 pp 165ndash189 2015
Figure 3 Effect of variation in the suction parameter on the (a) radial (b) tangential (c) axial and (d) temperature velocity profiles when119870 = 1 Kn = 005 Re = 100119873 = 1 and Pr = 1
Table 2 Comparison between the results of present study with the results reported by Kelson and Desseaux [48] Maleque and Sattar [33]and Alam et al [49] for the radial 1198651015840(0) and tangential minus1198661015840(0) skin friction coefficients for Pr = 071119872 = 0 120576 = 0 and Kn = 0
temperature jump The radial velocity achieves a maximumvalue near to the surface of the disk for all values of 120576 It is alsonoted that an increment in relative temperature differenceparameter causes shifting of maximum point from the diskThe velocity along the tangential direction increases asvalue of relative temperature difference parameter increaseswhereas the velocity along axial direction decreases Temper-ature increases as the value of relative temperature differenceparameter increasesThese results are similar to that obtainedby Rashidi et al [39]
Table 1 illustrates the effect of the parameters 120576 119870Kn Re Pr and 119873 on constant suction parameter on thenumerical values of (1198651015840(0) minus1198661015840(0)) and (minus1205791015840(0)) We observethat numerical value of 1198651015840(0) minus1198661015840(0) and minus1205791015840(0) decreases
with the increasing value of Kn Re and radiation parameter119873Tables 2 and 3 depict the comparison of skin friction coef-
ficients and rate of heat transfer between the results obtainedin the present investigation and the literature available [3348 49] It is observed that results obtained in present studyare very well in agreement with the existing results
5 Conclusions
In this study we have investigated radiation effect on velocityprofile for all components and temperature profile throughrotating disk in porous medium for variable fluid propertiesand in particular case for constant fluid properties also By
Advances in Mathematical Physics 9
0 1 2 3 4 5 6
= 0
= 02
Ws = minus1
２ = 1 10 100 1000 10000
00005
0010015
0020025
0030035
004
F(
)
(a)
= 0
= 02
5 63 41 20
Ws = minus1
２ = 1 10 100 1000 10000
0010203040506070809
1
G(
)
(b)
0 2 3 41 65
Ws = minus2 = 0
２ = 1
２ = 10
２ = 100
２ = 1000
２ = 10000
minus204minus2035minus203
minus2025minus202
minus2015minus201
minus2005minus2
minus1995minus199
H(
)
(c)
0 2 3 41 65
Ws = minus2 = 02 ２ = 1
２ = 10
２ = 100
２ = 1000
２ = 10000
minus2
minus195
minus19
minus185
minus18
minus175
minus17
H(
)
(d)
0 2 4 6 8 10 12 14
= 0
= 02
Ws = minus1
２ = 1 10 100 1000 10000
0010203040506070809
1
(
)
(e)
Figure 4 Effect of variation in the Reynolds number on the (a) radial (b) tangential (c) amp (d) axial and (e) temperature velocity profileswhen 119870 = 1 Kn = 005119873 = 1 and Pr = 1
Table 3 Comparison between the results of present study with the results reported by Kelson and Desseaux [48] Maleque and Sattar [33]and Alam et al [49] for rate of heat transfer minus1205791015840(0) for Pr = 071119872 = 0 120576 = 0 Kn = 0 and119873 = 0119882119904 Present Alam et al [49] Maleque and Sattar [33] Kelson and Desseaux [48]minus1205791015840(0) minus1205791015840(0) minus1205791015840(0) minus1205791015840(0)0 0326798372 032637889 032576 0325856minus2 1438764651 143876482 144212 1437782minus4 2842381877 284369011 284470 2842381minus5 3551223146 355222471 355411 3551223
10 Advances in Mathematical Physics
0 1 2 3 4 5 6
= 0
= 02
00005
0010015
0020025
0030035
004F
()
Ws = minus1
＋Ｈ = 0 002 005 01
(a)
0 1 2 3 4 5 6
= 0
= 02
0010203040506070809
1
G(
)
Ws = minus1
＋Ｈ = 0 002 005 01
(b)
0 1 2 3 4 5 6
= 0
= 02
minus2
minus195
minus19
minus185
minus18
minus175
minus17
H(
)
Ws = minus2
＋Ｈ = 005 002 0
(c)
= 0
= 02
0010203040506070809
1
(
)
104 6 8 12 1420
Ws = minus1
＋Ｈ = 0 002 005 01N = 0 ＋Ｈ = 005
(d)
Figure 5 Effect of variation in the Knudsen number on the (a) radial (b) tangential (c) axial and (d) temperature velocity profiles when119870 = 1 Re = 100119873 = 1 and Pr = 1
= 0
= 02
2 4 6 8 10 12 140
00102030405060708
(
)
N = 0 ０Ｌ = 4０Ｌ = 1 2 3 4
Figure 6 Effect of Prandtl number on temperature distributionwhen 119870 = 1119882119904 = minus1 Re = 100 Kn = 005 and119873 = 1
similarity transformation governing equations transformedinto nonlinear ordinary differential equations which aresolved numerically by usingRunge-Kuttamethodwith shoot-ing technique Based on the resulting solutions the followingconclusions can be drawn
= 0
= 02
2 4 6 8 10 12 140
N = 4 3 2 1 0
0010203040506070809
(
)
Figure 7 Effect of radiation parameter on temperature distributionwhen 119870 = 1119882119904 = minus1 Re = 100 Kn = 005 and Pr = 1
(i) The radial tangential and axial velocity profilesdecrease while the temperature increases with theincreasing values of porosity parameter
(ii) The increasing value of Reynolds and Knudsen num-ber decreases the fluid velocity components and
Advances in Mathematical Physics 11
1 2 3 4 5 60
= 0 02 04 10
0005
001
0015
002
0025F
()
(a)
1 2 54 60 3
0
01
02
03
04
05
06
G(
)
= 1 04 02 0
(b)
1 2 3 4 5 60
= 0
= 02
= 04
= 1
minus11
minus1
minus09
minus08
minus07
minus06
minus05
minus04
H(
)
(c)
00102030405060708
(
)
2 4 6 8 10 12 140
N = 0 = 02
= 1 04 02 0
(d)
Figure 8 Effect of variation in the relative temperature difference parameter on the (a) radial (b) tangential (c) axial and (d) temperaturevelocity profiles when119870 = 1119882119904 = minus1 Kn = 005 Re = 100119873 = 1 and Pr = 1
temperature and suction parameter also shows thesame effect
(iii) For the effect of the radiation parameter on the tem-perature distribution it is seen that the temperaturedistribution decreases with the increasing values ofradiation parameter and also it has been observed thatthe radial and tangential skin friction values decreasewith increase in the radiation parameter
Competing Interests
The authors declare that they have no competing interests
References
[1] Th V Karman ldquoUber laminare und turbulente reibunrdquo Zeits-chrift fur Angewandte Mathematik und Mechanik vol 1 no 4pp 233ndash252 1921
[2] W G Cochran ldquoThe flow due to a rotating diskrdquoMathematicalProceedings of the Cambridge Philosophical Society vol 30 no3 pp 365ndash375 1934
[3] E R Benton ldquoOn the flow due to a rotating diskrdquo Journal ofFluid Mechanics vol 24 no 4 pp 781ndash800 1966
[4] M Turkyilmazoglu ldquoPurely analytic solutions of the compress-ible boundary layer flow due to a porous rotating disk with heattransferrdquoPhysics of Fluids vol 21 no 10 Article ID 106104 2009
[5] D S Chauhan and S Gupta ldquoSteady flow and heat transferbetween two stationary naturally permeable disksrdquo Journal ofUltra Scientist of Physical Sciences vol 12 no 1 pp 45ndash52 2000
[6] C Wagner ldquoHeat transfer from a rotating disk to ambient airrdquoJournal of Applied Physics vol 19 no 9 pp 837ndash839 1948
[7] K Millsaps and K Pohlhausen ldquoHeat transfer by laminar flowfrom a rotating platerdquo Journal of the Aeronautical Sciences vol19 pp 120ndash126 1952
[8] F Kreith and J H Taylor ldquoHeat transfer from a rotating disk inturbulent flow no 1956rdquo ASME Paper 56-A-146 1956
[9] H-T Lin and L-K Lin ldquoHeat transfer from a rotating cone ordisk to fluids of any Prandtl numberrdquo International Communi-cations in Heat and Mass Transfer vol 14 no 3 pp 323ndash3321987
[10] P D Verma and D S Chauhan ldquoFlow between a torsionallyoscillating impermeable disc and a stationary naturally perme-able discrdquo Indian Journal of Pure and Applied Mathematics vol10 pp 1351ndash1361 1979
[11] D S Chauhan and S Jain ldquoSteady flow between highlypermeable rotating disksrdquo Indian Journal of Theoretical Physicsvol 52 no 1 pp 39ndash50 2004
[12] M Turkyilmazoglu ldquoNanofluid flow and heat transfer due to arotating diskrdquo Computers amp Fluids vol 94 pp 139ndash146 2014
[13] M Turkyilmazoglu ldquoMHD fluid flow and heat transfer due to ashrinking rotating diskrdquo Computers amp Fluids vol 90 pp 51ndash562014
[14] M Azimi and R Riazi ldquoHeat transfer analysis of GO-waternanofluid flow between two parallel disksrdquo Propulsion andPower Research vol 4 no 1 pp 23ndash30 2015
12 Advances in Mathematical Physics
[15] B R Sharma and H Konwar ldquoEffect of chemical reaction onmass transfer due to a permeable rotating heated diskrdquo Interna-tional Journal of Computer Applications vol 119 no 21 pp 5ndash92015
[16] H Shahmohamadi and M Mohammadpour ldquoA series solutionfor three-dimensional navier-stokes equations of flow near aninfinite rotating diskrdquo World Journal of Mechanics vol 4 pp117ndash127 2014
[17] S Srinivas A S Reddy T R Ramamohan and A K ShuklaldquoThermal-diffusion and diffusion-thermo effects onMHD flowof viscous fluid between expanding or contracting rotatingporous disks with viscous dissipationrdquo Journal of the EgyptianMathematical Society vol 24 no 1 pp 100ndash107 2016
[18] M A Mansour ldquoRadiative and free-convection effects on theoscillatory flow past a vertical platerdquo Astrophysics and SpaceScience vol 166 no 2 pp 269ndash275 1990
[19] M A Hossain M A Alim and D A S Rees ldquoThe effectof radiation on free convection from a porous vertical platerdquoInternational Journal of Heat and Mass Transfer vol 42 no 1pp 181ndash191 1999
[20] A Raptis and C Perdikis ldquoMHD free convection flow by thepresence of radiationrdquo International Journal of Magnetohydro-dynamics Plasma and Space Research vol 9 pp 237ndash252 2000
[21] M A Hossain and H S Takhar ldquoRadiation effect on mixedconvection along a vertical plate with uniform surface temper-aturerdquoHeat and Mass Transfer vol 31 no 4 pp 243ndash248 1996
[22] A Devi and R U Devi ldquoEffects of thermal radiation onhydromagnetic flow due to a porous rotating disk with halleffectrdquo Journal of Applied Fluid Mechanics vol 5 no 2 pp 1ndash7 2012
[23] C Y Wang ldquoFlow due to a stretching boundary with partialslipmdashan exact solution of the Navier-Stokes equationsrdquo Chemi-cal Engineering Science vol 57 no 17 pp 3745ndash3747 2002
[24] E Osalusi ldquoEffects of thermal radiation on MHD and slip flowover a porous rotating disk with variable propertiesrdquo RomanianJournal of Physics vol 52 no 3-4 pp 217ndash229 2007
[25] A A Khidir ldquoViscous dissipation Ohmic heating and radiationeffects on MHD flow past a rotating disk embedded in aporous medium with variable propertiesrdquo Arabian Journal ofMathematics vol 2 no 3 pp 263ndash277 2013
[26] E M Sparrow G S Beavers and L Y Hung ldquoFlow about aporous-surfaced rotating diskrdquo International Journal of Heatand Mass Transfer vol 14 no 7 pp 993ndash996 1971
[27] M M Rashidi and N Freidooni Mehr ldquoEffects of velocity slipand temperature jump on the entropy generation in magne-tohydrodynamic flow over a porous rotating diskrdquo Journal ofMechanical Engineering vol 1 no 3 2012
[28] A Arikoglu G Komurgoz I Ozkol and A Y Gunes ldquoCom-bined effects of temperature and velocity jump on the heattransfer fluid flow and entropy generation over a single rotatingdiskrdquo Journal of Heat Transfer vol 132 no 11 Article ID 1117032010
[29] THayatM RashidM Imtiaz andAAlsaedi ldquoMagnetohydro-dynamic (MHD) flow of Cu-water nanofluid due to a rotatingdisk with partial sliprdquo AIP Advances vol 5 no 6 Article ID067169 2015
[30] M Zakerullah and J A D Ackroyd ldquoLaminar natural convec-tion boundary-layers on Horizontal Circular disksrdquo Journal ofApplied Mathematics and Physics vol 30 pp 427ndash435 1979
[31] HHerwig ldquoThe effect of variable properties onmomentumandheat transfer in a tube with constant heat flux across the wallrdquoInternational Journal of Heat and Mass Transfer vol 28 no 2pp 423ndash431 1985
[32] H Herwig and K Klemp ldquoVariable property effects of fullydeveloped laminar flow in concentric annulirdquo Journal of HeatTransfer vol 110 no 2 pp 314ndash320 1988
[33] K AMaleque andMA Sattar ldquoSteady laminar convective flowwith variable properties due to a porous rotating diskrdquo Journalof Heat Transfer vol 127 no 12 pp 1406ndash1409 2005
[34] K A Maleque and M A Sattar ldquoThe effects of variable proper-ties and hall current on steady MHD laminar convective fluidflow due to a porous rotating diskrdquo International Journal of Heatand Mass Transfer vol 48 no 23-24 pp 4963ndash4972 2005
[35] E Osalusi and P Sibanda ldquoOn variable laminar convective flowproperties due to a porous rotating disk in a magnetic fieldrdquoRomanian Journal of Physics vol 9 no 10 pp 933ndash944 2006
[36] M M Rahman ldquoConvective Hydromagnetic slip flow withvariable properties due to a porous rotating diskrdquo The SultanQaboos University Journal for Science vol 15 pp 55ndash79 2010
[37] M M Rashidi S A Mohimanian Pour T Hayat and SObaidat ldquoAnalytic approximate solutions for steady flow over arotating disk in porousmediumwith heat transfer by homotopyanalysis methodrdquo Computers amp Fluids vol 54 no 1 pp 1ndash92012
[38] S Hussain F Ahmad M Shafique and S Hussain ldquoNumericalsolution for accelerated rotating disk in a viscous fluidrdquoAppliedMathematics vol 4 no 6 pp 899ndash902 2013
[39] M M Rashidi N Kavyani and S Abelman ldquoInvestigation ofentropy generation inMHDand slip flow over a rotating porousdisk with variable propertiesrdquo International Journal of Heat andMass Transfer vol 70 pp 892ndash917 2014
[40] N Freidoonimehr M M Rashidi S Abelman and G Loren-zini ldquoAnalytical modeling of MHD flow over a permeablerotating disk in the presence of Soret and Dufour effectsentropy analysisrdquo Entropy vol 18 no 5 article 131 2016
[41] M S Alam S M Chapal Hossain and M M Rahman ldquoTran-sient thermophoretic particle deposition on forced convectiveheat and mass transfer flow due to a rotating diskrdquo Ain ShamsEngineering Journal vol 7 no 1 pp 441ndash452 2016
[42] S Jayaraj ldquoThermophoresis in laminar flow over cold inclinedplates with variable propertiesrdquoHeat andMass Transfer vol 30no 3 pp 167ndash173 1995
[43] M M Rahman ldquoThermophoretic deposition of nanoparticlesdue to a permeable rotating disk effects of partial slip mag-netic field thermal radiation thermal-diffusion and diffusion-thermordquo International Journal of Mathematical ComputationalPhysical Electrical and Computer Engineering vol 7 no 5 2013
[44] I V Shevchuk Convective Heat and Mass Transfer in RotatingDisk Systems Springer Berlin Germany 2009
[45] C L Tien and D T Campbell ldquoHeat and mass transfer fromrotating conesrdquo Journal of FluidMechanics vol 17 no 1 pp 105ndash112 1963
[46] C J Elkins and J K Eaton ldquoHeat transfer in the rotatingdisk boundary layerrdquo Tech Rep TSD-103 Stanford UniversityDepartment of Mechanical EngineeringThermosciences Divi-sion Stanford Calif USA 1997
[47] G Karniadakis A Beskok and N Aluru Microflows Funda-mentals and Simulation Springer New York NY USA 2001
[48] N Kelson andA Desseaux ldquoNote on porous rotating disk flowrdquoAustralian amp New Zealand Industrial and Applied MathematicsJournal vol 42 pp 837ndash855 2000
[49] M Alam N Poddar M Rahman and K Vajravelu ldquoTransienthydromagnetic forced convective heat transfer slip flow due toa porous rotating disk with variable fluid propertiesrdquo AmericanJournal of Heat andMass Transfer vol 2 no 3 pp 165ndash189 2015
Figure 4 Effect of variation in the Reynolds number on the (a) radial (b) tangential (c) amp (d) axial and (e) temperature velocity profileswhen 119870 = 1 Kn = 005119873 = 1 and Pr = 1
Table 3 Comparison between the results of present study with the results reported by Kelson and Desseaux [48] Maleque and Sattar [33]and Alam et al [49] for rate of heat transfer minus1205791015840(0) for Pr = 071119872 = 0 120576 = 0 Kn = 0 and119873 = 0119882119904 Present Alam et al [49] Maleque and Sattar [33] Kelson and Desseaux [48]minus1205791015840(0) minus1205791015840(0) minus1205791015840(0) minus1205791015840(0)0 0326798372 032637889 032576 0325856minus2 1438764651 143876482 144212 1437782minus4 2842381877 284369011 284470 2842381minus5 3551223146 355222471 355411 3551223
10 Advances in Mathematical Physics
0 1 2 3 4 5 6
= 0
= 02
00005
0010015
0020025
0030035
004F
()
Ws = minus1
＋Ｈ = 0 002 005 01
(a)
0 1 2 3 4 5 6
= 0
= 02
0010203040506070809
1
G(
)
Ws = minus1
＋Ｈ = 0 002 005 01
(b)
0 1 2 3 4 5 6
= 0
= 02
minus2
minus195
minus19
minus185
minus18
minus175
minus17
H(
)
Ws = minus2
＋Ｈ = 005 002 0
(c)
= 0
= 02
0010203040506070809
1
(
)
104 6 8 12 1420
Ws = minus1
＋Ｈ = 0 002 005 01N = 0 ＋Ｈ = 005
(d)
Figure 5 Effect of variation in the Knudsen number on the (a) radial (b) tangential (c) axial and (d) temperature velocity profiles when119870 = 1 Re = 100119873 = 1 and Pr = 1
= 0
= 02
2 4 6 8 10 12 140
00102030405060708
(
)
N = 0 ０Ｌ = 4０Ｌ = 1 2 3 4
Figure 6 Effect of Prandtl number on temperature distributionwhen 119870 = 1119882119904 = minus1 Re = 100 Kn = 005 and119873 = 1
similarity transformation governing equations transformedinto nonlinear ordinary differential equations which aresolved numerically by usingRunge-Kuttamethodwith shoot-ing technique Based on the resulting solutions the followingconclusions can be drawn
= 0
= 02
2 4 6 8 10 12 140
N = 4 3 2 1 0
0010203040506070809
(
)
Figure 7 Effect of radiation parameter on temperature distributionwhen 119870 = 1119882119904 = minus1 Re = 100 Kn = 005 and Pr = 1
(i) The radial tangential and axial velocity profilesdecrease while the temperature increases with theincreasing values of porosity parameter
(ii) The increasing value of Reynolds and Knudsen num-ber decreases the fluid velocity components and
Advances in Mathematical Physics 11
1 2 3 4 5 60
= 0 02 04 10
0005
001
0015
002
0025F
()
(a)
1 2 54 60 3
0
01
02
03
04
05
06
G(
)
= 1 04 02 0
(b)
1 2 3 4 5 60
= 0
= 02
= 04
= 1
minus11
minus1
minus09
minus08
minus07
minus06
minus05
minus04
H(
)
(c)
00102030405060708
(
)
2 4 6 8 10 12 140
N = 0 = 02
= 1 04 02 0
(d)
Figure 8 Effect of variation in the relative temperature difference parameter on the (a) radial (b) tangential (c) axial and (d) temperaturevelocity profiles when119870 = 1119882119904 = minus1 Kn = 005 Re = 100119873 = 1 and Pr = 1
temperature and suction parameter also shows thesame effect
(iii) For the effect of the radiation parameter on the tem-perature distribution it is seen that the temperaturedistribution decreases with the increasing values ofradiation parameter and also it has been observed thatthe radial and tangential skin friction values decreasewith increase in the radiation parameter
Competing Interests
The authors declare that they have no competing interests
References
[1] Th V Karman ldquoUber laminare und turbulente reibunrdquo Zeits-chrift fur Angewandte Mathematik und Mechanik vol 1 no 4pp 233ndash252 1921
[2] W G Cochran ldquoThe flow due to a rotating diskrdquoMathematicalProceedings of the Cambridge Philosophical Society vol 30 no3 pp 365ndash375 1934
[3] E R Benton ldquoOn the flow due to a rotating diskrdquo Journal ofFluid Mechanics vol 24 no 4 pp 781ndash800 1966
[4] M Turkyilmazoglu ldquoPurely analytic solutions of the compress-ible boundary layer flow due to a porous rotating disk with heattransferrdquoPhysics of Fluids vol 21 no 10 Article ID 106104 2009
[5] D S Chauhan and S Gupta ldquoSteady flow and heat transferbetween two stationary naturally permeable disksrdquo Journal ofUltra Scientist of Physical Sciences vol 12 no 1 pp 45ndash52 2000
[6] C Wagner ldquoHeat transfer from a rotating disk to ambient airrdquoJournal of Applied Physics vol 19 no 9 pp 837ndash839 1948
[7] K Millsaps and K Pohlhausen ldquoHeat transfer by laminar flowfrom a rotating platerdquo Journal of the Aeronautical Sciences vol19 pp 120ndash126 1952
[8] F Kreith and J H Taylor ldquoHeat transfer from a rotating disk inturbulent flow no 1956rdquo ASME Paper 56-A-146 1956
[9] H-T Lin and L-K Lin ldquoHeat transfer from a rotating cone ordisk to fluids of any Prandtl numberrdquo International Communi-cations in Heat and Mass Transfer vol 14 no 3 pp 323ndash3321987
[10] P D Verma and D S Chauhan ldquoFlow between a torsionallyoscillating impermeable disc and a stationary naturally perme-able discrdquo Indian Journal of Pure and Applied Mathematics vol10 pp 1351ndash1361 1979
[11] D S Chauhan and S Jain ldquoSteady flow between highlypermeable rotating disksrdquo Indian Journal of Theoretical Physicsvol 52 no 1 pp 39ndash50 2004
[12] M Turkyilmazoglu ldquoNanofluid flow and heat transfer due to arotating diskrdquo Computers amp Fluids vol 94 pp 139ndash146 2014
[13] M Turkyilmazoglu ldquoMHD fluid flow and heat transfer due to ashrinking rotating diskrdquo Computers amp Fluids vol 90 pp 51ndash562014
[14] M Azimi and R Riazi ldquoHeat transfer analysis of GO-waternanofluid flow between two parallel disksrdquo Propulsion andPower Research vol 4 no 1 pp 23ndash30 2015
12 Advances in Mathematical Physics
[15] B R Sharma and H Konwar ldquoEffect of chemical reaction onmass transfer due to a permeable rotating heated diskrdquo Interna-tional Journal of Computer Applications vol 119 no 21 pp 5ndash92015
[16] H Shahmohamadi and M Mohammadpour ldquoA series solutionfor three-dimensional navier-stokes equations of flow near aninfinite rotating diskrdquo World Journal of Mechanics vol 4 pp117ndash127 2014
[17] S Srinivas A S Reddy T R Ramamohan and A K ShuklaldquoThermal-diffusion and diffusion-thermo effects onMHD flowof viscous fluid between expanding or contracting rotatingporous disks with viscous dissipationrdquo Journal of the EgyptianMathematical Society vol 24 no 1 pp 100ndash107 2016
[18] M A Mansour ldquoRadiative and free-convection effects on theoscillatory flow past a vertical platerdquo Astrophysics and SpaceScience vol 166 no 2 pp 269ndash275 1990
[19] M A Hossain M A Alim and D A S Rees ldquoThe effectof radiation on free convection from a porous vertical platerdquoInternational Journal of Heat and Mass Transfer vol 42 no 1pp 181ndash191 1999
[20] A Raptis and C Perdikis ldquoMHD free convection flow by thepresence of radiationrdquo International Journal of Magnetohydro-dynamics Plasma and Space Research vol 9 pp 237ndash252 2000
[21] M A Hossain and H S Takhar ldquoRadiation effect on mixedconvection along a vertical plate with uniform surface temper-aturerdquoHeat and Mass Transfer vol 31 no 4 pp 243ndash248 1996
[22] A Devi and R U Devi ldquoEffects of thermal radiation onhydromagnetic flow due to a porous rotating disk with halleffectrdquo Journal of Applied Fluid Mechanics vol 5 no 2 pp 1ndash7 2012
[23] C Y Wang ldquoFlow due to a stretching boundary with partialslipmdashan exact solution of the Navier-Stokes equationsrdquo Chemi-cal Engineering Science vol 57 no 17 pp 3745ndash3747 2002
[24] E Osalusi ldquoEffects of thermal radiation on MHD and slip flowover a porous rotating disk with variable propertiesrdquo RomanianJournal of Physics vol 52 no 3-4 pp 217ndash229 2007
[25] A A Khidir ldquoViscous dissipation Ohmic heating and radiationeffects on MHD flow past a rotating disk embedded in aporous medium with variable propertiesrdquo Arabian Journal ofMathematics vol 2 no 3 pp 263ndash277 2013
[26] E M Sparrow G S Beavers and L Y Hung ldquoFlow about aporous-surfaced rotating diskrdquo International Journal of Heatand Mass Transfer vol 14 no 7 pp 993ndash996 1971
[27] M M Rashidi and N Freidooni Mehr ldquoEffects of velocity slipand temperature jump on the entropy generation in magne-tohydrodynamic flow over a porous rotating diskrdquo Journal ofMechanical Engineering vol 1 no 3 2012
[28] A Arikoglu G Komurgoz I Ozkol and A Y Gunes ldquoCom-bined effects of temperature and velocity jump on the heattransfer fluid flow and entropy generation over a single rotatingdiskrdquo Journal of Heat Transfer vol 132 no 11 Article ID 1117032010
[29] THayatM RashidM Imtiaz andAAlsaedi ldquoMagnetohydro-dynamic (MHD) flow of Cu-water nanofluid due to a rotatingdisk with partial sliprdquo AIP Advances vol 5 no 6 Article ID067169 2015
[30] M Zakerullah and J A D Ackroyd ldquoLaminar natural convec-tion boundary-layers on Horizontal Circular disksrdquo Journal ofApplied Mathematics and Physics vol 30 pp 427ndash435 1979
[31] HHerwig ldquoThe effect of variable properties onmomentumandheat transfer in a tube with constant heat flux across the wallrdquoInternational Journal of Heat and Mass Transfer vol 28 no 2pp 423ndash431 1985
[32] H Herwig and K Klemp ldquoVariable property effects of fullydeveloped laminar flow in concentric annulirdquo Journal of HeatTransfer vol 110 no 2 pp 314ndash320 1988
[33] K AMaleque andMA Sattar ldquoSteady laminar convective flowwith variable properties due to a porous rotating diskrdquo Journalof Heat Transfer vol 127 no 12 pp 1406ndash1409 2005
[34] K A Maleque and M A Sattar ldquoThe effects of variable proper-ties and hall current on steady MHD laminar convective fluidflow due to a porous rotating diskrdquo International Journal of Heatand Mass Transfer vol 48 no 23-24 pp 4963ndash4972 2005
[35] E Osalusi and P Sibanda ldquoOn variable laminar convective flowproperties due to a porous rotating disk in a magnetic fieldrdquoRomanian Journal of Physics vol 9 no 10 pp 933ndash944 2006
[36] M M Rahman ldquoConvective Hydromagnetic slip flow withvariable properties due to a porous rotating diskrdquo The SultanQaboos University Journal for Science vol 15 pp 55ndash79 2010
[37] M M Rashidi S A Mohimanian Pour T Hayat and SObaidat ldquoAnalytic approximate solutions for steady flow over arotating disk in porousmediumwith heat transfer by homotopyanalysis methodrdquo Computers amp Fluids vol 54 no 1 pp 1ndash92012
[38] S Hussain F Ahmad M Shafique and S Hussain ldquoNumericalsolution for accelerated rotating disk in a viscous fluidrdquoAppliedMathematics vol 4 no 6 pp 899ndash902 2013
[39] M M Rashidi N Kavyani and S Abelman ldquoInvestigation ofentropy generation inMHDand slip flow over a rotating porousdisk with variable propertiesrdquo International Journal of Heat andMass Transfer vol 70 pp 892ndash917 2014
[40] N Freidoonimehr M M Rashidi S Abelman and G Loren-zini ldquoAnalytical modeling of MHD flow over a permeablerotating disk in the presence of Soret and Dufour effectsentropy analysisrdquo Entropy vol 18 no 5 article 131 2016
[41] M S Alam S M Chapal Hossain and M M Rahman ldquoTran-sient thermophoretic particle deposition on forced convectiveheat and mass transfer flow due to a rotating diskrdquo Ain ShamsEngineering Journal vol 7 no 1 pp 441ndash452 2016
[42] S Jayaraj ldquoThermophoresis in laminar flow over cold inclinedplates with variable propertiesrdquoHeat andMass Transfer vol 30no 3 pp 167ndash173 1995
[43] M M Rahman ldquoThermophoretic deposition of nanoparticlesdue to a permeable rotating disk effects of partial slip mag-netic field thermal radiation thermal-diffusion and diffusion-thermordquo International Journal of Mathematical ComputationalPhysical Electrical and Computer Engineering vol 7 no 5 2013
[44] I V Shevchuk Convective Heat and Mass Transfer in RotatingDisk Systems Springer Berlin Germany 2009
[45] C L Tien and D T Campbell ldquoHeat and mass transfer fromrotating conesrdquo Journal of FluidMechanics vol 17 no 1 pp 105ndash112 1963
[46] C J Elkins and J K Eaton ldquoHeat transfer in the rotatingdisk boundary layerrdquo Tech Rep TSD-103 Stanford UniversityDepartment of Mechanical EngineeringThermosciences Divi-sion Stanford Calif USA 1997
[47] G Karniadakis A Beskok and N Aluru Microflows Funda-mentals and Simulation Springer New York NY USA 2001
[48] N Kelson andA Desseaux ldquoNote on porous rotating disk flowrdquoAustralian amp New Zealand Industrial and Applied MathematicsJournal vol 42 pp 837ndash855 2000
[49] M Alam N Poddar M Rahman and K Vajravelu ldquoTransienthydromagnetic forced convective heat transfer slip flow due toa porous rotating disk with variable fluid propertiesrdquo AmericanJournal of Heat andMass Transfer vol 2 no 3 pp 165ndash189 2015
Figure 5 Effect of variation in the Knudsen number on the (a) radial (b) tangential (c) axial and (d) temperature velocity profiles when119870 = 1 Re = 100119873 = 1 and Pr = 1
= 0
= 02
2 4 6 8 10 12 140
00102030405060708
(
)
N = 0 ０Ｌ = 4０Ｌ = 1 2 3 4
Figure 6 Effect of Prandtl number on temperature distributionwhen 119870 = 1119882119904 = minus1 Re = 100 Kn = 005 and119873 = 1
similarity transformation governing equations transformedinto nonlinear ordinary differential equations which aresolved numerically by usingRunge-Kuttamethodwith shoot-ing technique Based on the resulting solutions the followingconclusions can be drawn
= 0
= 02
2 4 6 8 10 12 140
N = 4 3 2 1 0
0010203040506070809
(
)
Figure 7 Effect of radiation parameter on temperature distributionwhen 119870 = 1119882119904 = minus1 Re = 100 Kn = 005 and Pr = 1
(i) The radial tangential and axial velocity profilesdecrease while the temperature increases with theincreasing values of porosity parameter
(ii) The increasing value of Reynolds and Knudsen num-ber decreases the fluid velocity components and
Advances in Mathematical Physics 11
1 2 3 4 5 60
= 0 02 04 10
0005
001
0015
002
0025F
()
(a)
1 2 54 60 3
0
01
02
03
04
05
06
G(
)
= 1 04 02 0
(b)
1 2 3 4 5 60
= 0
= 02
= 04
= 1
minus11
minus1
minus09
minus08
minus07
minus06
minus05
minus04
H(
)
(c)
00102030405060708
(
)
2 4 6 8 10 12 140
N = 0 = 02
= 1 04 02 0
(d)
Figure 8 Effect of variation in the relative temperature difference parameter on the (a) radial (b) tangential (c) axial and (d) temperaturevelocity profiles when119870 = 1119882119904 = minus1 Kn = 005 Re = 100119873 = 1 and Pr = 1
temperature and suction parameter also shows thesame effect
(iii) For the effect of the radiation parameter on the tem-perature distribution it is seen that the temperaturedistribution decreases with the increasing values ofradiation parameter and also it has been observed thatthe radial and tangential skin friction values decreasewith increase in the radiation parameter
Competing Interests
The authors declare that they have no competing interests
References
[1] Th V Karman ldquoUber laminare und turbulente reibunrdquo Zeits-chrift fur Angewandte Mathematik und Mechanik vol 1 no 4pp 233ndash252 1921
[2] W G Cochran ldquoThe flow due to a rotating diskrdquoMathematicalProceedings of the Cambridge Philosophical Society vol 30 no3 pp 365ndash375 1934
[3] E R Benton ldquoOn the flow due to a rotating diskrdquo Journal ofFluid Mechanics vol 24 no 4 pp 781ndash800 1966
[4] M Turkyilmazoglu ldquoPurely analytic solutions of the compress-ible boundary layer flow due to a porous rotating disk with heattransferrdquoPhysics of Fluids vol 21 no 10 Article ID 106104 2009
[5] D S Chauhan and S Gupta ldquoSteady flow and heat transferbetween two stationary naturally permeable disksrdquo Journal ofUltra Scientist of Physical Sciences vol 12 no 1 pp 45ndash52 2000
[6] C Wagner ldquoHeat transfer from a rotating disk to ambient airrdquoJournal of Applied Physics vol 19 no 9 pp 837ndash839 1948
[7] K Millsaps and K Pohlhausen ldquoHeat transfer by laminar flowfrom a rotating platerdquo Journal of the Aeronautical Sciences vol19 pp 120ndash126 1952
[8] F Kreith and J H Taylor ldquoHeat transfer from a rotating disk inturbulent flow no 1956rdquo ASME Paper 56-A-146 1956
[9] H-T Lin and L-K Lin ldquoHeat transfer from a rotating cone ordisk to fluids of any Prandtl numberrdquo International Communi-cations in Heat and Mass Transfer vol 14 no 3 pp 323ndash3321987
[10] P D Verma and D S Chauhan ldquoFlow between a torsionallyoscillating impermeable disc and a stationary naturally perme-able discrdquo Indian Journal of Pure and Applied Mathematics vol10 pp 1351ndash1361 1979
[11] D S Chauhan and S Jain ldquoSteady flow between highlypermeable rotating disksrdquo Indian Journal of Theoretical Physicsvol 52 no 1 pp 39ndash50 2004
[12] M Turkyilmazoglu ldquoNanofluid flow and heat transfer due to arotating diskrdquo Computers amp Fluids vol 94 pp 139ndash146 2014
[13] M Turkyilmazoglu ldquoMHD fluid flow and heat transfer due to ashrinking rotating diskrdquo Computers amp Fluids vol 90 pp 51ndash562014
[14] M Azimi and R Riazi ldquoHeat transfer analysis of GO-waternanofluid flow between two parallel disksrdquo Propulsion andPower Research vol 4 no 1 pp 23ndash30 2015
12 Advances in Mathematical Physics
[15] B R Sharma and H Konwar ldquoEffect of chemical reaction onmass transfer due to a permeable rotating heated diskrdquo Interna-tional Journal of Computer Applications vol 119 no 21 pp 5ndash92015
[16] H Shahmohamadi and M Mohammadpour ldquoA series solutionfor three-dimensional navier-stokes equations of flow near aninfinite rotating diskrdquo World Journal of Mechanics vol 4 pp117ndash127 2014
[17] S Srinivas A S Reddy T R Ramamohan and A K ShuklaldquoThermal-diffusion and diffusion-thermo effects onMHD flowof viscous fluid between expanding or contracting rotatingporous disks with viscous dissipationrdquo Journal of the EgyptianMathematical Society vol 24 no 1 pp 100ndash107 2016
[18] M A Mansour ldquoRadiative and free-convection effects on theoscillatory flow past a vertical platerdquo Astrophysics and SpaceScience vol 166 no 2 pp 269ndash275 1990
[19] M A Hossain M A Alim and D A S Rees ldquoThe effectof radiation on free convection from a porous vertical platerdquoInternational Journal of Heat and Mass Transfer vol 42 no 1pp 181ndash191 1999
[20] A Raptis and C Perdikis ldquoMHD free convection flow by thepresence of radiationrdquo International Journal of Magnetohydro-dynamics Plasma and Space Research vol 9 pp 237ndash252 2000
[21] M A Hossain and H S Takhar ldquoRadiation effect on mixedconvection along a vertical plate with uniform surface temper-aturerdquoHeat and Mass Transfer vol 31 no 4 pp 243ndash248 1996
[22] A Devi and R U Devi ldquoEffects of thermal radiation onhydromagnetic flow due to a porous rotating disk with halleffectrdquo Journal of Applied Fluid Mechanics vol 5 no 2 pp 1ndash7 2012
[23] C Y Wang ldquoFlow due to a stretching boundary with partialslipmdashan exact solution of the Navier-Stokes equationsrdquo Chemi-cal Engineering Science vol 57 no 17 pp 3745ndash3747 2002
[24] E Osalusi ldquoEffects of thermal radiation on MHD and slip flowover a porous rotating disk with variable propertiesrdquo RomanianJournal of Physics vol 52 no 3-4 pp 217ndash229 2007
[25] A A Khidir ldquoViscous dissipation Ohmic heating and radiationeffects on MHD flow past a rotating disk embedded in aporous medium with variable propertiesrdquo Arabian Journal ofMathematics vol 2 no 3 pp 263ndash277 2013
[26] E M Sparrow G S Beavers and L Y Hung ldquoFlow about aporous-surfaced rotating diskrdquo International Journal of Heatand Mass Transfer vol 14 no 7 pp 993ndash996 1971
[27] M M Rashidi and N Freidooni Mehr ldquoEffects of velocity slipand temperature jump on the entropy generation in magne-tohydrodynamic flow over a porous rotating diskrdquo Journal ofMechanical Engineering vol 1 no 3 2012
[28] A Arikoglu G Komurgoz I Ozkol and A Y Gunes ldquoCom-bined effects of temperature and velocity jump on the heattransfer fluid flow and entropy generation over a single rotatingdiskrdquo Journal of Heat Transfer vol 132 no 11 Article ID 1117032010
[29] THayatM RashidM Imtiaz andAAlsaedi ldquoMagnetohydro-dynamic (MHD) flow of Cu-water nanofluid due to a rotatingdisk with partial sliprdquo AIP Advances vol 5 no 6 Article ID067169 2015
[30] M Zakerullah and J A D Ackroyd ldquoLaminar natural convec-tion boundary-layers on Horizontal Circular disksrdquo Journal ofApplied Mathematics and Physics vol 30 pp 427ndash435 1979
[31] HHerwig ldquoThe effect of variable properties onmomentumandheat transfer in a tube with constant heat flux across the wallrdquoInternational Journal of Heat and Mass Transfer vol 28 no 2pp 423ndash431 1985
[32] H Herwig and K Klemp ldquoVariable property effects of fullydeveloped laminar flow in concentric annulirdquo Journal of HeatTransfer vol 110 no 2 pp 314ndash320 1988
[33] K AMaleque andMA Sattar ldquoSteady laminar convective flowwith variable properties due to a porous rotating diskrdquo Journalof Heat Transfer vol 127 no 12 pp 1406ndash1409 2005
[34] K A Maleque and M A Sattar ldquoThe effects of variable proper-ties and hall current on steady MHD laminar convective fluidflow due to a porous rotating diskrdquo International Journal of Heatand Mass Transfer vol 48 no 23-24 pp 4963ndash4972 2005
[35] E Osalusi and P Sibanda ldquoOn variable laminar convective flowproperties due to a porous rotating disk in a magnetic fieldrdquoRomanian Journal of Physics vol 9 no 10 pp 933ndash944 2006
[36] M M Rahman ldquoConvective Hydromagnetic slip flow withvariable properties due to a porous rotating diskrdquo The SultanQaboos University Journal for Science vol 15 pp 55ndash79 2010
[37] M M Rashidi S A Mohimanian Pour T Hayat and SObaidat ldquoAnalytic approximate solutions for steady flow over arotating disk in porousmediumwith heat transfer by homotopyanalysis methodrdquo Computers amp Fluids vol 54 no 1 pp 1ndash92012
[38] S Hussain F Ahmad M Shafique and S Hussain ldquoNumericalsolution for accelerated rotating disk in a viscous fluidrdquoAppliedMathematics vol 4 no 6 pp 899ndash902 2013
[39] M M Rashidi N Kavyani and S Abelman ldquoInvestigation ofentropy generation inMHDand slip flow over a rotating porousdisk with variable propertiesrdquo International Journal of Heat andMass Transfer vol 70 pp 892ndash917 2014
[40] N Freidoonimehr M M Rashidi S Abelman and G Loren-zini ldquoAnalytical modeling of MHD flow over a permeablerotating disk in the presence of Soret and Dufour effectsentropy analysisrdquo Entropy vol 18 no 5 article 131 2016
[41] M S Alam S M Chapal Hossain and M M Rahman ldquoTran-sient thermophoretic particle deposition on forced convectiveheat and mass transfer flow due to a rotating diskrdquo Ain ShamsEngineering Journal vol 7 no 1 pp 441ndash452 2016
[42] S Jayaraj ldquoThermophoresis in laminar flow over cold inclinedplates with variable propertiesrdquoHeat andMass Transfer vol 30no 3 pp 167ndash173 1995
[43] M M Rahman ldquoThermophoretic deposition of nanoparticlesdue to a permeable rotating disk effects of partial slip mag-netic field thermal radiation thermal-diffusion and diffusion-thermordquo International Journal of Mathematical ComputationalPhysical Electrical and Computer Engineering vol 7 no 5 2013
[44] I V Shevchuk Convective Heat and Mass Transfer in RotatingDisk Systems Springer Berlin Germany 2009
[45] C L Tien and D T Campbell ldquoHeat and mass transfer fromrotating conesrdquo Journal of FluidMechanics vol 17 no 1 pp 105ndash112 1963
[46] C J Elkins and J K Eaton ldquoHeat transfer in the rotatingdisk boundary layerrdquo Tech Rep TSD-103 Stanford UniversityDepartment of Mechanical EngineeringThermosciences Divi-sion Stanford Calif USA 1997
[47] G Karniadakis A Beskok and N Aluru Microflows Funda-mentals and Simulation Springer New York NY USA 2001
[48] N Kelson andA Desseaux ldquoNote on porous rotating disk flowrdquoAustralian amp New Zealand Industrial and Applied MathematicsJournal vol 42 pp 837ndash855 2000
[49] M Alam N Poddar M Rahman and K Vajravelu ldquoTransienthydromagnetic forced convective heat transfer slip flow due toa porous rotating disk with variable fluid propertiesrdquo AmericanJournal of Heat andMass Transfer vol 2 no 3 pp 165ndash189 2015
Figure 8 Effect of variation in the relative temperature difference parameter on the (a) radial (b) tangential (c) axial and (d) temperaturevelocity profiles when119870 = 1119882119904 = minus1 Kn = 005 Re = 100119873 = 1 and Pr = 1
temperature and suction parameter also shows thesame effect
(iii) For the effect of the radiation parameter on the tem-perature distribution it is seen that the temperaturedistribution decreases with the increasing values ofradiation parameter and also it has been observed thatthe radial and tangential skin friction values decreasewith increase in the radiation parameter
Competing Interests
The authors declare that they have no competing interests
References
[1] Th V Karman ldquoUber laminare und turbulente reibunrdquo Zeits-chrift fur Angewandte Mathematik und Mechanik vol 1 no 4pp 233ndash252 1921
[2] W G Cochran ldquoThe flow due to a rotating diskrdquoMathematicalProceedings of the Cambridge Philosophical Society vol 30 no3 pp 365ndash375 1934
[3] E R Benton ldquoOn the flow due to a rotating diskrdquo Journal ofFluid Mechanics vol 24 no 4 pp 781ndash800 1966
[4] M Turkyilmazoglu ldquoPurely analytic solutions of the compress-ible boundary layer flow due to a porous rotating disk with heattransferrdquoPhysics of Fluids vol 21 no 10 Article ID 106104 2009
[5] D S Chauhan and S Gupta ldquoSteady flow and heat transferbetween two stationary naturally permeable disksrdquo Journal ofUltra Scientist of Physical Sciences vol 12 no 1 pp 45ndash52 2000
[6] C Wagner ldquoHeat transfer from a rotating disk to ambient airrdquoJournal of Applied Physics vol 19 no 9 pp 837ndash839 1948
[7] K Millsaps and K Pohlhausen ldquoHeat transfer by laminar flowfrom a rotating platerdquo Journal of the Aeronautical Sciences vol19 pp 120ndash126 1952
[8] F Kreith and J H Taylor ldquoHeat transfer from a rotating disk inturbulent flow no 1956rdquo ASME Paper 56-A-146 1956
[9] H-T Lin and L-K Lin ldquoHeat transfer from a rotating cone ordisk to fluids of any Prandtl numberrdquo International Communi-cations in Heat and Mass Transfer vol 14 no 3 pp 323ndash3321987
[10] P D Verma and D S Chauhan ldquoFlow between a torsionallyoscillating impermeable disc and a stationary naturally perme-able discrdquo Indian Journal of Pure and Applied Mathematics vol10 pp 1351ndash1361 1979
[11] D S Chauhan and S Jain ldquoSteady flow between highlypermeable rotating disksrdquo Indian Journal of Theoretical Physicsvol 52 no 1 pp 39ndash50 2004
[12] M Turkyilmazoglu ldquoNanofluid flow and heat transfer due to arotating diskrdquo Computers amp Fluids vol 94 pp 139ndash146 2014
[13] M Turkyilmazoglu ldquoMHD fluid flow and heat transfer due to ashrinking rotating diskrdquo Computers amp Fluids vol 90 pp 51ndash562014
[14] M Azimi and R Riazi ldquoHeat transfer analysis of GO-waternanofluid flow between two parallel disksrdquo Propulsion andPower Research vol 4 no 1 pp 23ndash30 2015
12 Advances in Mathematical Physics
[15] B R Sharma and H Konwar ldquoEffect of chemical reaction onmass transfer due to a permeable rotating heated diskrdquo Interna-tional Journal of Computer Applications vol 119 no 21 pp 5ndash92015
[16] H Shahmohamadi and M Mohammadpour ldquoA series solutionfor three-dimensional navier-stokes equations of flow near aninfinite rotating diskrdquo World Journal of Mechanics vol 4 pp117ndash127 2014
[17] S Srinivas A S Reddy T R Ramamohan and A K ShuklaldquoThermal-diffusion and diffusion-thermo effects onMHD flowof viscous fluid between expanding or contracting rotatingporous disks with viscous dissipationrdquo Journal of the EgyptianMathematical Society vol 24 no 1 pp 100ndash107 2016
[18] M A Mansour ldquoRadiative and free-convection effects on theoscillatory flow past a vertical platerdquo Astrophysics and SpaceScience vol 166 no 2 pp 269ndash275 1990
[19] M A Hossain M A Alim and D A S Rees ldquoThe effectof radiation on free convection from a porous vertical platerdquoInternational Journal of Heat and Mass Transfer vol 42 no 1pp 181ndash191 1999
[20] A Raptis and C Perdikis ldquoMHD free convection flow by thepresence of radiationrdquo International Journal of Magnetohydro-dynamics Plasma and Space Research vol 9 pp 237ndash252 2000
[21] M A Hossain and H S Takhar ldquoRadiation effect on mixedconvection along a vertical plate with uniform surface temper-aturerdquoHeat and Mass Transfer vol 31 no 4 pp 243ndash248 1996
[22] A Devi and R U Devi ldquoEffects of thermal radiation onhydromagnetic flow due to a porous rotating disk with halleffectrdquo Journal of Applied Fluid Mechanics vol 5 no 2 pp 1ndash7 2012
[23] C Y Wang ldquoFlow due to a stretching boundary with partialslipmdashan exact solution of the Navier-Stokes equationsrdquo Chemi-cal Engineering Science vol 57 no 17 pp 3745ndash3747 2002
[24] E Osalusi ldquoEffects of thermal radiation on MHD and slip flowover a porous rotating disk with variable propertiesrdquo RomanianJournal of Physics vol 52 no 3-4 pp 217ndash229 2007
[25] A A Khidir ldquoViscous dissipation Ohmic heating and radiationeffects on MHD flow past a rotating disk embedded in aporous medium with variable propertiesrdquo Arabian Journal ofMathematics vol 2 no 3 pp 263ndash277 2013
[26] E M Sparrow G S Beavers and L Y Hung ldquoFlow about aporous-surfaced rotating diskrdquo International Journal of Heatand Mass Transfer vol 14 no 7 pp 993ndash996 1971
[27] M M Rashidi and N Freidooni Mehr ldquoEffects of velocity slipand temperature jump on the entropy generation in magne-tohydrodynamic flow over a porous rotating diskrdquo Journal ofMechanical Engineering vol 1 no 3 2012
[28] A Arikoglu G Komurgoz I Ozkol and A Y Gunes ldquoCom-bined effects of temperature and velocity jump on the heattransfer fluid flow and entropy generation over a single rotatingdiskrdquo Journal of Heat Transfer vol 132 no 11 Article ID 1117032010
[29] THayatM RashidM Imtiaz andAAlsaedi ldquoMagnetohydro-dynamic (MHD) flow of Cu-water nanofluid due to a rotatingdisk with partial sliprdquo AIP Advances vol 5 no 6 Article ID067169 2015
[30] M Zakerullah and J A D Ackroyd ldquoLaminar natural convec-tion boundary-layers on Horizontal Circular disksrdquo Journal ofApplied Mathematics and Physics vol 30 pp 427ndash435 1979
[31] HHerwig ldquoThe effect of variable properties onmomentumandheat transfer in a tube with constant heat flux across the wallrdquoInternational Journal of Heat and Mass Transfer vol 28 no 2pp 423ndash431 1985
[32] H Herwig and K Klemp ldquoVariable property effects of fullydeveloped laminar flow in concentric annulirdquo Journal of HeatTransfer vol 110 no 2 pp 314ndash320 1988
[33] K AMaleque andMA Sattar ldquoSteady laminar convective flowwith variable properties due to a porous rotating diskrdquo Journalof Heat Transfer vol 127 no 12 pp 1406ndash1409 2005
[34] K A Maleque and M A Sattar ldquoThe effects of variable proper-ties and hall current on steady MHD laminar convective fluidflow due to a porous rotating diskrdquo International Journal of Heatand Mass Transfer vol 48 no 23-24 pp 4963ndash4972 2005
[35] E Osalusi and P Sibanda ldquoOn variable laminar convective flowproperties due to a porous rotating disk in a magnetic fieldrdquoRomanian Journal of Physics vol 9 no 10 pp 933ndash944 2006
[36] M M Rahman ldquoConvective Hydromagnetic slip flow withvariable properties due to a porous rotating diskrdquo The SultanQaboos University Journal for Science vol 15 pp 55ndash79 2010
[37] M M Rashidi S A Mohimanian Pour T Hayat and SObaidat ldquoAnalytic approximate solutions for steady flow over arotating disk in porousmediumwith heat transfer by homotopyanalysis methodrdquo Computers amp Fluids vol 54 no 1 pp 1ndash92012
[38] S Hussain F Ahmad M Shafique and S Hussain ldquoNumericalsolution for accelerated rotating disk in a viscous fluidrdquoAppliedMathematics vol 4 no 6 pp 899ndash902 2013
[39] M M Rashidi N Kavyani and S Abelman ldquoInvestigation ofentropy generation inMHDand slip flow over a rotating porousdisk with variable propertiesrdquo International Journal of Heat andMass Transfer vol 70 pp 892ndash917 2014
[40] N Freidoonimehr M M Rashidi S Abelman and G Loren-zini ldquoAnalytical modeling of MHD flow over a permeablerotating disk in the presence of Soret and Dufour effectsentropy analysisrdquo Entropy vol 18 no 5 article 131 2016
[41] M S Alam S M Chapal Hossain and M M Rahman ldquoTran-sient thermophoretic particle deposition on forced convectiveheat and mass transfer flow due to a rotating diskrdquo Ain ShamsEngineering Journal vol 7 no 1 pp 441ndash452 2016
[42] S Jayaraj ldquoThermophoresis in laminar flow over cold inclinedplates with variable propertiesrdquoHeat andMass Transfer vol 30no 3 pp 167ndash173 1995
[43] M M Rahman ldquoThermophoretic deposition of nanoparticlesdue to a permeable rotating disk effects of partial slip mag-netic field thermal radiation thermal-diffusion and diffusion-thermordquo International Journal of Mathematical ComputationalPhysical Electrical and Computer Engineering vol 7 no 5 2013
[44] I V Shevchuk Convective Heat and Mass Transfer in RotatingDisk Systems Springer Berlin Germany 2009
[45] C L Tien and D T Campbell ldquoHeat and mass transfer fromrotating conesrdquo Journal of FluidMechanics vol 17 no 1 pp 105ndash112 1963
[46] C J Elkins and J K Eaton ldquoHeat transfer in the rotatingdisk boundary layerrdquo Tech Rep TSD-103 Stanford UniversityDepartment of Mechanical EngineeringThermosciences Divi-sion Stanford Calif USA 1997
[47] G Karniadakis A Beskok and N Aluru Microflows Funda-mentals and Simulation Springer New York NY USA 2001
[48] N Kelson andA Desseaux ldquoNote on porous rotating disk flowrdquoAustralian amp New Zealand Industrial and Applied MathematicsJournal vol 42 pp 837ndash855 2000
[49] M Alam N Poddar M Rahman and K Vajravelu ldquoTransienthydromagnetic forced convective heat transfer slip flow due toa porous rotating disk with variable fluid propertiesrdquo AmericanJournal of Heat andMass Transfer vol 2 no 3 pp 165ndash189 2015
[15] B R Sharma and H Konwar ldquoEffect of chemical reaction onmass transfer due to a permeable rotating heated diskrdquo Interna-tional Journal of Computer Applications vol 119 no 21 pp 5ndash92015
[16] H Shahmohamadi and M Mohammadpour ldquoA series solutionfor three-dimensional navier-stokes equations of flow near aninfinite rotating diskrdquo World Journal of Mechanics vol 4 pp117ndash127 2014
[17] S Srinivas A S Reddy T R Ramamohan and A K ShuklaldquoThermal-diffusion and diffusion-thermo effects onMHD flowof viscous fluid between expanding or contracting rotatingporous disks with viscous dissipationrdquo Journal of the EgyptianMathematical Society vol 24 no 1 pp 100ndash107 2016
[18] M A Mansour ldquoRadiative and free-convection effects on theoscillatory flow past a vertical platerdquo Astrophysics and SpaceScience vol 166 no 2 pp 269ndash275 1990
[19] M A Hossain M A Alim and D A S Rees ldquoThe effectof radiation on free convection from a porous vertical platerdquoInternational Journal of Heat and Mass Transfer vol 42 no 1pp 181ndash191 1999
[20] A Raptis and C Perdikis ldquoMHD free convection flow by thepresence of radiationrdquo International Journal of Magnetohydro-dynamics Plasma and Space Research vol 9 pp 237ndash252 2000
[21] M A Hossain and H S Takhar ldquoRadiation effect on mixedconvection along a vertical plate with uniform surface temper-aturerdquoHeat and Mass Transfer vol 31 no 4 pp 243ndash248 1996
[22] A Devi and R U Devi ldquoEffects of thermal radiation onhydromagnetic flow due to a porous rotating disk with halleffectrdquo Journal of Applied Fluid Mechanics vol 5 no 2 pp 1ndash7 2012
[23] C Y Wang ldquoFlow due to a stretching boundary with partialslipmdashan exact solution of the Navier-Stokes equationsrdquo Chemi-cal Engineering Science vol 57 no 17 pp 3745ndash3747 2002
[24] E Osalusi ldquoEffects of thermal radiation on MHD and slip flowover a porous rotating disk with variable propertiesrdquo RomanianJournal of Physics vol 52 no 3-4 pp 217ndash229 2007
[25] A A Khidir ldquoViscous dissipation Ohmic heating and radiationeffects on MHD flow past a rotating disk embedded in aporous medium with variable propertiesrdquo Arabian Journal ofMathematics vol 2 no 3 pp 263ndash277 2013
[26] E M Sparrow G S Beavers and L Y Hung ldquoFlow about aporous-surfaced rotating diskrdquo International Journal of Heatand Mass Transfer vol 14 no 7 pp 993ndash996 1971
[27] M M Rashidi and N Freidooni Mehr ldquoEffects of velocity slipand temperature jump on the entropy generation in magne-tohydrodynamic flow over a porous rotating diskrdquo Journal ofMechanical Engineering vol 1 no 3 2012
[28] A Arikoglu G Komurgoz I Ozkol and A Y Gunes ldquoCom-bined effects of temperature and velocity jump on the heattransfer fluid flow and entropy generation over a single rotatingdiskrdquo Journal of Heat Transfer vol 132 no 11 Article ID 1117032010
[29] THayatM RashidM Imtiaz andAAlsaedi ldquoMagnetohydro-dynamic (MHD) flow of Cu-water nanofluid due to a rotatingdisk with partial sliprdquo AIP Advances vol 5 no 6 Article ID067169 2015
[30] M Zakerullah and J A D Ackroyd ldquoLaminar natural convec-tion boundary-layers on Horizontal Circular disksrdquo Journal ofApplied Mathematics and Physics vol 30 pp 427ndash435 1979
[31] HHerwig ldquoThe effect of variable properties onmomentumandheat transfer in a tube with constant heat flux across the wallrdquoInternational Journal of Heat and Mass Transfer vol 28 no 2pp 423ndash431 1985
[32] H Herwig and K Klemp ldquoVariable property effects of fullydeveloped laminar flow in concentric annulirdquo Journal of HeatTransfer vol 110 no 2 pp 314ndash320 1988
[33] K AMaleque andMA Sattar ldquoSteady laminar convective flowwith variable properties due to a porous rotating diskrdquo Journalof Heat Transfer vol 127 no 12 pp 1406ndash1409 2005
[34] K A Maleque and M A Sattar ldquoThe effects of variable proper-ties and hall current on steady MHD laminar convective fluidflow due to a porous rotating diskrdquo International Journal of Heatand Mass Transfer vol 48 no 23-24 pp 4963ndash4972 2005
[35] E Osalusi and P Sibanda ldquoOn variable laminar convective flowproperties due to a porous rotating disk in a magnetic fieldrdquoRomanian Journal of Physics vol 9 no 10 pp 933ndash944 2006
[36] M M Rahman ldquoConvective Hydromagnetic slip flow withvariable properties due to a porous rotating diskrdquo The SultanQaboos University Journal for Science vol 15 pp 55ndash79 2010
[37] M M Rashidi S A Mohimanian Pour T Hayat and SObaidat ldquoAnalytic approximate solutions for steady flow over arotating disk in porousmediumwith heat transfer by homotopyanalysis methodrdquo Computers amp Fluids vol 54 no 1 pp 1ndash92012
[38] S Hussain F Ahmad M Shafique and S Hussain ldquoNumericalsolution for accelerated rotating disk in a viscous fluidrdquoAppliedMathematics vol 4 no 6 pp 899ndash902 2013
[39] M M Rashidi N Kavyani and S Abelman ldquoInvestigation ofentropy generation inMHDand slip flow over a rotating porousdisk with variable propertiesrdquo International Journal of Heat andMass Transfer vol 70 pp 892ndash917 2014
[40] N Freidoonimehr M M Rashidi S Abelman and G Loren-zini ldquoAnalytical modeling of MHD flow over a permeablerotating disk in the presence of Soret and Dufour effectsentropy analysisrdquo Entropy vol 18 no 5 article 131 2016
[41] M S Alam S M Chapal Hossain and M M Rahman ldquoTran-sient thermophoretic particle deposition on forced convectiveheat and mass transfer flow due to a rotating diskrdquo Ain ShamsEngineering Journal vol 7 no 1 pp 441ndash452 2016
[42] S Jayaraj ldquoThermophoresis in laminar flow over cold inclinedplates with variable propertiesrdquoHeat andMass Transfer vol 30no 3 pp 167ndash173 1995
[43] M M Rahman ldquoThermophoretic deposition of nanoparticlesdue to a permeable rotating disk effects of partial slip mag-netic field thermal radiation thermal-diffusion and diffusion-thermordquo International Journal of Mathematical ComputationalPhysical Electrical and Computer Engineering vol 7 no 5 2013
[44] I V Shevchuk Convective Heat and Mass Transfer in RotatingDisk Systems Springer Berlin Germany 2009
[45] C L Tien and D T Campbell ldquoHeat and mass transfer fromrotating conesrdquo Journal of FluidMechanics vol 17 no 1 pp 105ndash112 1963
[46] C J Elkins and J K Eaton ldquoHeat transfer in the rotatingdisk boundary layerrdquo Tech Rep TSD-103 Stanford UniversityDepartment of Mechanical EngineeringThermosciences Divi-sion Stanford Calif USA 1997
[47] G Karniadakis A Beskok and N Aluru Microflows Funda-mentals and Simulation Springer New York NY USA 2001
[48] N Kelson andA Desseaux ldquoNote on porous rotating disk flowrdquoAustralian amp New Zealand Industrial and Applied MathematicsJournal vol 42 pp 837ndash855 2000
[49] M Alam N Poddar M Rahman and K Vajravelu ldquoTransienthydromagnetic forced convective heat transfer slip flow due toa porous rotating disk with variable fluid propertiesrdquo AmericanJournal of Heat andMass Transfer vol 2 no 3 pp 165ndash189 2015
