Effect of thermal radiation and heat generation on MHD flow past a uniformly heated vertical plate

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In this paper, the effect of heat generation and radiation parameters on MHD flow along a uniformly heated vertical flat plate in the presence of a magnetic field has been investigated numerically. The nonlinear partial differential equations,
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  Desalination and Water Treatment  16 (2010) 57–65   www.deswater.com April 1944-3994/1944-3986 © 2010 Desalination Publications. All rights reserveddoi no. 10.5004/dwt.2010.1086 Effect of thermal radiation and heat generation on MHDflow past a uniformly heated vertical plate Goutam Saha a, *, Tamanna Sultana  b , Sumon Saha c a Department of Mathematics, University of Dhaka, Dhaka-1000, BangladeshTel. +880 1 817 505874  ; email: ranamath06@gmail.com b Institute of Natural Science, United International University (UIU), Dhaka-1209, Bangladesh c Department of Mechanical Engineering, The University of Melbourne, Victoria 3010, Australia Received 8 March 2009; Accepted 13 December 2009 ABSTRACT In this paper, the effect of heat generation and radiation parameters on MHD flow along auniformly heated vertical flat plate in the presence of a magnetic field has been investigatednumerically. The nonlinear partial differential equations, governing the problem under consid-eration for this analysis are transferred to simultaneous nonlinear ordinary differential equa-tions of first order and those are further transformed into initial value problem by applyingMulti-segment integration technique. Finally, solutions are obtained by integrating the initialvalue problem using fourth order Runge-Kutta integration scheme. Rosseland approximationis used to describe the radiative heat flux in the energy equation. Comparison with previouslypublished work is performed and excellent agreement with the results is obtained. Numericalresults for the details of the temperature profiles are shown graphically with the variation of thegoverning parameters considering in the present problem.  Keywords :   Electrically conducting fluid; Porous medium; Radiation; Similarity solution. 1. Introduction The importance of the radiation effect on MHD flowand heat transfer problems has found increasing attentionin industries. At high operating temperature, radiationeffect can be quite significant. Many processes in engi-neering areas occur at high temperatures and knowledgeof radiation heat transfer becomes very important for thedesign of pertinent equipment. Nuclear power plants,gas turbines and the various propulsion devices for air-craft, missiles, satellites and space vehicles are examplesof such engineering areas. Most of the existing analyticalstudies for this problem are based on the constant physi-cal properties of the ambient fluid. However, it is knownthat these properties may change with temperature,especially fluid viscosity. To accurately predict the flowand heat transfer rates, it is necessary to take into accountthis variation of viscosity with temperature.Ostrach [1] presented the similarity solution of natu-ral convection along vertical isothermal plate. Kay [2]reported that thermal conductivity of liquids with lowPrandtl number varied linearly with temperature inrange of 0–400 ° F. Arunachalam and Rajappa [3] con-sidered forced convection flow of liquid metals (havinglow Prandtl number) with variable thermal conductiv-ity and derived explicit closed form of analytical solu-tion. Chaim [4] also studied heat transfer in fluid flow oflow Prandtl number with variable thermal conductivity.Carey and Mollendorf [5] observed the effect of temper-ature dependent viscosity on free convective fluid flow.Crepeau and Clarksean [6] discussed similarity solu-tion of natural convection with internal heat generation, *Corresponding author.  58 which decayed exponentially. Chamkha and Khaled [7]obtained similarity solution of natural convection on aninclined plate with internal heat generation or absorp-tion in presence of transverse magnetic field.The thermal radiation of a gray fluid, which is emit-ting and absorbing radiation in a non-scattering mediumhas been examined by Ali et al. [8], Ibrahim [9], Mansour[10], Hossain et al. [11] and Elbashbeshy and Dimian [12].In the aspect of convection and radiation, Viskanta andGrosh [13] considered the effects of thermal radiationon the temperature distribution and the heat transfer inan absorbing and emitting media flowing over a wedge by using the Rosseland diffusion approximation. Thisapproximation leads to a considerable simplification inthe expression for radiant flux. In Viskanta and Grosh[13] and Raptis [14], the temperature differences withinthe flow were assumed sufficiently small such that T 4  might be expressed as a linear function of temperature,i.e.T 4   ≈ 4(T ∞ ) 3 T − 3(T ∞ ) 4 . Hossain et al. [15] investigated thenatural convection–radiation interaction on a boundarylayer flow along a vertical plate with uniform suction.Yih [16] investigated the natural convection flow of anoptically dense viscous fluid over an isothermal trun-cated cone. Recently, Bataller [17] investigated radiationeffects in the laminar boundary layer about a flat-plate ina uniform stream of fluid. He found that as the value ofradiation parameter increases, a diminution in the ther-mal radiation’s effect occurs. Later, he studied radiationeffects for the Blasius and Sakiadis flows with a convec-tive surface boundary condition [18]. A comparison ofthese two flows was described in this work.Chen [19] performed an analysis to study the MHDnatural convection flow over a permeable inclined sur-face with variable wall temperature and concentration.The results showed that the velocity was decreased inthe presence of a magnetic field and with the increaseof the angle of inclination, the effect of buoyancy forcedecreased. Heat transfer rate was however increasedwhen the Prandtl number was increased. Duwairi [20]investigated the effect of viscous and Joule heating onforced convection flow from radiative isothermal sur-faces. He found that the heat transfer rate was decreasedwhen the radiation parameter was increased. Duwairiand Damseh [21] also studied the convection heat trans-fer problem with radiation effects from vertical surfacefor buoyancy aiding and opposing flows. They concludedthat increasing the conduction – radiation parameterdecreased the heat transfer rates for the buoyancy aidedflow and increased them for the buoyancy opposing flow.Ibrahim et al. [22] investigated similarity reductions forproblems of radiative and magnetic field effects on freeconvection and mass transfer flow past a semi-infiniteflat plate. They obtained new similarity reductions andfound an analytical solution for the uniform magneticfield by using Lie group method. They also presentedthe numerical results for the non-uniform magneticfield. Seddeek [23] investigated effects of radiation andvariable viscosity on a MHD free convection flow past asemi-infinite flat plate with an aligned magnetic field inthe case of unsteady flow. The effect of variable viscosityon hydromagnetic flow and heat transfer past a continu-ously moving porous boundary with radiation was fur-ther studied by Seddeek [24].In the present paper, investigation is carried out forthe thermal radiation interaction of the boundary layerflow of electrically conducting fluid past a uniformlyheated vertical plate embedded in a porous medium.The governing equations are converted into nonlinearsystem of coupled ordinary differential equations andsolved numerically using Multi-segment integrationtechnique. The normalized similarity solutions are thenobtained numerically for various parameters enteringinto the problem and discussed them from the physicalpoint of view. 2. Mathematical formulation of the problem Let us consider a steady, two-dimensional flow ofa viscous, incompressible and electrically conductingfluid of temperature T  ∞   past a semi-infinite heated ver-tical plate having constant temperature T  w (where, T  w   >   T  ∞ ). A magnetic field of uniform strength, B o is appliedperpendicular to the plate. The magnetic Reynoldsnumber is taken to be small enough so that the inducedmagnetic field can be neglected. The flow is assumedto be in the x -direction, which is taken along the platein the upward direction and  y -axis is normal to it. Theflow configuration and the coordinate system are shownin Fig. 1. In order to consider the effect of radiation, interms of the radiative heat flux, the Rosseland approxi-mation is incorporated in the energy equation. The radi-ative heat flux in the x -direction is considered negligiblein comparison to the  y -direction. Within the frameworkof the above-noted assumptions, it is considered that the B 0 T w TU(x)T ∞ yux υ Fig. 1. Flow configuration and coordinate system. G. Saha et al. / Desalination and Water Treatment 16 (2010) 57–65  G. Saha et al. / Desalination and Water Treatment 16 (2010) 57–65 59  boundary layer approximations hold and the govern-ing equations relevant to the problem in the presence ofradiation are given by ∂∂+∂∂= ux υ  y 0(1) uuxu y pxu yBuKUu ∂∂+∂∂= −∂∂+∂∂− − − ( ) υρνσρ ν 1 2202* (2) uT xT  ycT  yQcTT cq y  pppr ∂∂+∂∂=∂∂+ − −∂∂ ∞ υκ ρ ρ ρ 220 1()(3)where u and υ  are the velocity components along x and  y coordinates respectively, U ( x ) is the free stream veloc-ity, ν  = μ  / ρ is the kinematic viscosity, μ  is the coeffi-cient of dynamic viscosity, ρ is the mass density ofthe fluid, σ  is the electrical conductivity of the fluid, B o  is the magnetic induction, K * is the Darcy permeability, T  is the temperature of the fluid in the boundary layer, T  ∞ is the temperature of the fluid outside the boundarylayer, c  p is the specific heat of the fluid at constant pres-sure, κ  is the thermal conductivity and q r is the radiativeheat flux.It is assumed that the velocity of the free stream is inthe form of U(x) = a x + c x 2   (4) where a and c are constants.In the free stream u = U ( x ), Eq. (2) reduces to UUx pxBU ∂∂= −∂∂− 1 02 ρσρ .(5)Eliminating ∂∂  px between Eqs. (2) and (5), we obtain uuxu yu yUUxBUuKUu ∂∂+∂∂=∂∂+∂∂+ − − −υ νσρ ν 2202 ()() * (6)By using Rosseland approximation, q r [25] for radia-tion from an optically thick layer (Ali et al. [8]), it can bewritten as follows qT  y r = −∂∂ 43 4 σκ  ** , (7)where σ  * is the Stefan–Boltzmann constant and κ  * is themean absorption coefficient.Moreover, the temperature differences within theflow are assumed to be sufficiently small such that T 4  may be expressed as a linear function of temperature.This is accomplished by expanding T 4 in a Taylor’sseries about T ∞ and neglecting higher-order terms,thus TTTTTTTT  44334 443 ≅ + − = − ∞ ∞ ∞ ∞ ∞ ().(8)By using Eqs. (7) and (8), Eq. (3) gives uT xT  ycT  yQcTT T cT  y  ppp ∂∂+∂∂=∂∂+ − +∂∂ ∞∞ υκ ρ ρσρ κ  220322 163() ** .(9)The corresponding boundary conditions for theabove problem are given byat y0as y uTT uUxTT  w = = = =→ → → ∞⎫⎬⎪⎭⎪ ∞ 00,,(), υ (10)In order to obtain a solution of Eqs. (1), (6) and (9),the following transformations are introduced: η ν= a y  (11a) uaxfcxg =′+′ ()() η η 2  (11b) υ ν η η  ν = − − af cx g a ()()2 (11c) TTTTT cxaT  ww = + − +⎡⎣⎢⎤⎦⎥ ∞ ()()() 01 2 η η  (11d) Pc  p =ρνκ   (Prandtl number) (11e) N Ba =σρ 03 (Magnetic parameter) (11f)  MaK =ν *  (Darcy parameter) (11g) QQac  p = 0 ρ (Heat generation parameter) (11h) KT  = ∞ κ κ σ ** 4 3  (Radiation parameter) (11i) PnKPK =+ 334 (Radiative Prandtl number) (11j)where prime denotes differentiation with respect to η .In view of (11), Eq. (1) is satisfied identically andEqs. (6) and (9) reduce to ′′′+′′−′+ −′−′+ =  ffffNfMf  2 110()(12) ′′′+′′−′ ′+′′+ −′−′+ =  gfgfgfgNgMg 32130()(13) ′′+′+ + − = TPnfTTTQ 0000 10()()(14) ′′+ −′+′+′+ = TPnfTfTgTPnTQ 11101 0().(15)The corresponding boundary conditions (10) becomes  ffggTT  fgTT  =′= =′= = = =′=′= = = 00000001110 0101 ,,,,,,,,atas η   η→ ∞⎫⎬⎭ . (16)  G. Saha et al. / Desalination and Water Treatment 16 (2010) 57–65 60 The nonlinear system of coupled differential Eqs.(12)–(15) together with the boundary conditions (16)are solved numerically using Multi-segment integra-tion technique. First of all, higher order nonlinear dif-ferential Eqs. (12)–(15) are converted into simultaneouslinear or nonlinear differential equations of order firstand they are further transformed into initial valueproblem by applying Multi-segment integration tech-nique (Kalnins and Lestingi [26]). Once the boundaryvalue problem is reduced to initial value problem, it isthen solved using Runge-Kutta fourth order technique(Jain [28]). Now rewrite the governing Eqs. (12)–(15)into a set of first order ordinary differential equationsas follows: ⇒′′′′′′′′⎧⎨⎪⎪⎪⎪⎪⎪⎪⎩⎪⎪⎪⎪⎪⎪⎪⎫⎬⎪⎪⎪⎪⎪⎪⎪⎭⎪⎪⎪⎪⎪ dd f  f  f  g g gT T T T  η 0011 ⎪⎪⎪=′′′′ ( ) −′′+ −′ ( ) −′−′′′−′′+′ ′−′′+′  f  f  fffNfMf  g g fgfgfgN  2 1132  g gMgT PnQTfT T PnfTfTgTPnQ − ( ) +′−′− ( ) −′ { } ′− −′+′+′ ( ) − 131 0001110 T T  1 ⎧⎨⎪⎪⎪⎪⎪⎪⎪⎩⎪⎪⎪⎪⎪⎪⎪⎫⎬⎪⎪⎪⎪⎪⎪⎪⎭⎪⎪⎪⎪⎪⎪⎪ (17)The fundamental set of nonlinear equations (17)together with the boundary conditions (16) has to be integrated over a finite range of the independentvariable η . However, the numerical integration ofthese equations is not possible beyond a very limitedrange of η due to the loss of accuracy in solving forthe unknown initial values, as pointed out by Sepe-toski et al . [27]. Thus, the multi-segment integrationtechnique developed by Kalnins and Lestingi [26] has been used in this analysis. If the fundamental variables  fffgggTTTT  ,,,,,,,,, ′ ′′ ′ ′′′ ′ 0011  of Eqs. (17) are repre-sented in matrix notation by [w] in a standard form asfollows: dwdFwPQKNM ηη= ( ) ,;,,,,, (18)in which wwwherei iT  =⎡⎣⎤⎦≤ ≤ 110and FFFFFFFFFFFw =⎡⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎤⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥= 123456789102 wwwwwNwMwwwwwwwwwNwMw 32213225616253455 11321 − + − ( ) − −− + − + − ( ) + −−− ( ) − { } − − + + ( ) −⎧⎨⎪⎪⎪⎪⎪ 31 87181029110489 wPnQwwwwPnwwwwwwPnQw ⎪⎪⎪⎩⎪⎪⎪⎪⎪⎪⎪⎫⎬⎪⎪⎪⎪⎪⎪⎪⎭⎪⎪⎪⎪⎪⎪⎪ (19)The boundary conditions Eqs. (16) can be rearrangedin the following form as follows:  A w(0) + B w( η → ∞ ) = C,   (20) where  A = 10000000000100000000000100000000001000000000001000000000000100000000000000000000000000000000000000000 ⎡⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎤⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥ , B = 00000000000000000000000000000000000000000000000000000000000000100000000000010000000000010000000000010 ⎡⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎤⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥ ,  C =⎡⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎤⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥ 0000001110.Let us consider the initial value problem correspond-ing to boundary value problems, dW dGwWPQKNM ηη= ( ) ,,;,,,,(21)  G. Saha et al. / Desalination and Water Treatment 16 (2010) 57–65 61 with WIandWWwhereij ij 0110 1010 ( ) = = ≤ ≤ × [], , (22) GJwhereJ FFFFFFFFFFwwwww 0 123456789101234 ( ) = = [],,,,,,,,,,,,, 55678910 ,,,,, wwwww ⎛ ⎝ ⎜⎞ ⎠ ⎟  (23)where I  is the Identity matrix and  J  is the Jacobianmatrix.In which, GGGGGGGGGG  j j j j j j j j j j 12345678910 ⎡⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎤⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥=− + + + ( ) =  j j j jjjj j wwwwwwwwNMwww 121023221313256 2 ,,.., ()  j j jjjjjjj j wwwwwwwwwwwwNMww − + + + − + + + ()()()() 16162525343458 32 PPnQwwwwwwPnwwwwwwww  jjj j jjjj 71818102929110110 − + () { } − − + () + + (() + + () { } −⎧⎨⎪⎪⎪⎪⎪⎪⎪⎩⎪⎪⎪⎪⎪⎪⎪⎫⎬⎪⎪⎪⎪⎪⎪⎪⎭⎪⎪⎪⎪⎪ wwwwPnQw  jjj 48489 ⎪⎪⎪ (24) 3. Results and discussion To assess the accuracy of the present code, thegraphs of similarity temperatures T  0 ( η ) and T  1 ( η ) fordifferent radiation parameter have been plotted wherethe Prandtl number and magnetic parameter are takenfixed at 0.71 and 0.8 respectively. These graphs are alsocompared with that of Raptis et al. [29] for K = 3 and 30.Fig. 2 shows the comparison of the temperature profiles T  0   and T  1 for P = 0.71 and N  = 0.8 produced by the pres-ent code and that of Raptis et al. [29]. They obtained thesolution using Runge-Kutta shooting method whereasthe scheme exploited in the present paper is Multi-seg-ment integration technique. Infact, the results show aclose agreement and thus give an encouragement forthe use of the present code. Hence, the scheme used inthis paper is stable and accurate.The aim of this work is to determine the effects ofdifferent parameters on the normalized similarity tem-peratures T  0 ( η ) and T  1 ( η ). In the calculations, the val-ues of magnetic parameter (N), Darcy parameter (M) ,heat generation parameter (Q) , Prandtl number (P) andradiation parameter (K) are chosen arbitrarily. The effectof the Darcy parameter  M on the normalized similaritytemperatures T  0 ( η ) and T  1 ( η ) is shown in Fig. 3. Fromthis figure, it is observed that temperature profiles T  0 ( η )  increase with the increase of  M whereas T  1 ( η ) decreaseat the same time. As the value of  M increases, the resis-tance to the flow also increases, which means that thetemperature field approximates more closely to theequivalent conductive state. Fig. 4 shows the effect ofmagnetic field parameter (N) on the temperature pro-files. This figure reveals that the normalized similaritytemperature T  0 ( η ) shows no effect with the variation ofmagnetic field parameter but similarity temperature T  1 ( η ) increases with the increase of N  . This is due to thefact that the magnetic field tends to retard the velocityfield, which in turn induces the temperature field andthus results the increase of the temperature profiles. Themagnetic field can therefore be used to control the flowcharacteristics. In Fig. 5, the heat generation parameter (Q) is varied keeping all other parameters fixed. It isfound that the similarity temperatures T  0 ( η ) increasemonotonically and similarity temperatures T  1 ( η ) alsoincrease close to the plate as Q increases. However, aftera short distance from the plate, the profiles overlap anddecrease monotonically as Q increases. For differentvalues of Prandtl number (P) , significant changes ontemperature profiles are observed, which is presentedin Fig. 6. From this figure, it can be concluded that incase of cooling the plate, normalized similarity tem-perature T  0 ( η ) increases as P increases and similaritytemperatures T  1 ( η ) also increases close to the plate. It Fig. 2. Comparison of the temperature profiles T  0   and T  1 for P = 0.71 and N  = 0.8.
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