1. Introduction. Keywords Inclined magnetic, Unsteady, Porous medium, MHD, Radiation, Heat transfer and Suction

Transcription

1 American Journal of Fluid Dnamics 24, 4(3): 9- DOI:.5923/j.ajfd Numerical Solution for hermal Radiation Effect on Inclined Magnetic Field of Mhd Free Convective Heat ransfer Dissipative Fluid Flow Past a Moving Vertical Porous Plate with Variable Suction Amos S. Idowu,*, Abdulwaheed, Jimoh, Funmilao H. Oelami 2, Moses S. Dada Department of Mathematics, niversit of Ilorin, Ilorin, Nigeria 2 Department of Mathematics, Afe Babalola niversit, Ado-Ekiti, Nigeria Abstract An investigation was carried out on the numerical solution for thermal radiation effect on inclined magnetic field of magneto hdrodnamic (MHD) free convective heat transfer dissipative fluid flow past a moving vertical porous plate with variable suction. he dimensionless governing equations are formulated in (*, t*) co-ordinates sstem with appropriate boundar condition s and the radiative heat flux takes the Rosseland approximation. he equations are solved b using an implicit finite difference method of Crank-Nicolson tpe. he effects of various parameters on the velocit and temperature fields as well as the Coefficient of skin-friction and Nusselt number were presented graphicall and in tabulated forms. It was observed that, when the radiation parameter increases, the velocit and temperature increases in the boundar laer. he effect of increasing values of Hartman number (M) which resulted in decrease in velocit distribution, while, increase in temperature across the boundar laer because of the application of transfer magnetic field which resulted in a restrictive tpe of force (Lorenz force) similar to drag force which tends to resist the fluid and this reducing its velocit. his model finds applications in geophiscs, metallurgic and also in the design of high temperature industrial processing sstems. Kewords Inclined magnetic, nstead, Porous medium, MHD, Radiation, Heat transfer and Suction. Introduction he problem of fluid flow in an electromagnetic field has been studied for its importance in geophsics, metallurg and aerodnamic extrusion of plastic sheets and other engineering process such as in petroleum engineering, chemical engineering, composite or ceramic engineering and heat exchangers. Also, the effect of thermal radiation is significant in some industrial application such as glass production and furnace design and in space technolog application such as comical flight aerodnamics rocket, populsion sstem, plasma phsics which operate at high temperature. Vajravelu and Hadjinicolaou (993) studied the heat transfer characteristics in the laminar boundar laer of a viscous fluid over a stretching sheet with viscous dissipation or frictional heating and internal heat generation. Muthucumaraswam and Ganesan (2) investigated the effect of the chemical reaction and injection on flow characteristics in an unstead upward motion of an * Corresponding author: asidowu@gmail.com (Amos S. Idowu) Published online at Copright 24 Scientific & Academic Publishing. All Rights Reserved isothermal plate. Chamkha, akhar and Soundalgekar (2), also, observed the radiation effect on free convection flow past a semi-infinite vertical plate with mass transfer. Muthucumaraswam and Senthih (24) considered heat and mass transfer effect on moving vertical plate in the presence of thermal radiation. Kim and Fedorov (24) studied transient mixed radiative convection flow of a micro polar fluid past a moving semi-infinite vertical porous plate. Hossain et al. (24) investigated the problem of natural convection flown along a vertical wav surface with uniform surface temperature in the presence of heat generation/ absorption. Had, Mohammed and Mahd (26), gave attention to MHD Free convection flow along a vertical wav surface with heat generation or absorption effect. Gnaneshwara and Bhaskar (29) investigated the radiation and mass transfer effects on an unstead MHD free convection flow past a heated vertical porous plate with viscous dissipation. Alam, Rahman and Sattar (29), tried transient magnetohdrodnamic free convective heat and mass transfer flow with thermophoresis past a radiative inclined permeable plate in the presence of variable chemical reaction and temperature dependent viscosit. Ahmed, Sarma and Baruna (22), studied the Magnetic field effect on free convective oscillator flow between two vertical

2 92 Amos S. Idowu et al.: Numerical Solution for hermal Radiation Effect on Inclined Magnetic Field of Mhd Free Convective Heat ransfer Dissipative Fluid Flow Past a Moving Vertical Porous Plate with Variable Suction parallel plates with periodic plate temperature and dissipative heat. Rushi, Kumar and Sivaraj, (22), MHD Visco elastic fluid non-darc flow along a moving vertical cone. Idowu, Dada and Jimoh (23) investigated the heat and mass transfer of magnetohdrodnamic (MHD) and dissipative fluid flow pass a moving vertical porous plate with variable suction. he stud of heat generation or absorption effects in moving fluids is important in view of several phsical problems such as fluids undergoing exothermic or endothermic or transfer chemical reaction. In this direction Alam et al. (26) studied the problem of free convection heat and mass transfer flow past an inclined semi-infinite heated surface of an electricall conducting and stead viscous incompressible fluid in the presence of a magnetic field and heat generation. Md Abdus and Mohammed (26) considered the thermal radiation interaction with unstead MHD flow past a vertical porous plate immersed in a porous medium. he importance of radiation in the fluid led Muthucumaraswam and Chandrakala (26) to stud radiative heat and mass transfer effect on moving isothermal vertical plate in the presence of chemical reaction. In man Chemical Engineering processes, the chemical reaction do occur between a mass and fluid in which plate is moving. hese processes take place in numerous industrial applications such as polmer production, manufacturing of ceramics or glassware and food processing. In the light of the fact that, the combination of heat and mass transfer problems with chemical reaction are of importance in man processes, and have, therefore, received a considerable amount of attention in recent ears. In processes such as dring, evaporation at the surface of a water bod, energ transfer in a wet cooling tower and the flow in a desert cooler, heat and mass transfer occur simultaneousl. Possible applications of this tpe of flow can be found in man industries. For example, in the power industr, among the methods of generating electricit is one in which electrical energ is extracted directl from moving conducting fluid. Naving Kumar and Sandeep Gupta (28) investigated the effect of variable permeabilit on unstead two-dimensional free convective flow through a porous medium bounded b a vertical porous surface. Sandeep and Sugunamma (23) investigated the effect of inclined magnetic field on unstead free convective flow of dissipative fluid past a vertical plate. Sharma, Navin and Pooja (2) have studied the influence of chemical reaction on unstead MHD free convective flow and mass transfer through viscous incompressible fluid past a heated vertical plate immersed in porous medium in the presence of heat source. Mohammed (29) studied double-diffusive convection-radiation interaction on unstead MHD flow over a vertical moving porous plate with heat generation and Soret effects. Based on these investigations and works that have been reported in the field of stud, in particular, the stud of heat and mass transfer, heat radiation is of considerable importance in chemical and hdrometallurgical industries. Mass transfer process and evaporation of water from a pound to the atmosphere the diffusion of chemical impurities in lakes, rivers and ocean from natural or artificial sources. Magneto hdrodnamic mixed convection heat transfer flow in porous plate and non-porous media is of considerable interest in the technical field due to its frequent occurrence in industrial technolog and geothermal application, high temperature plasma application to nuclear fusion energ conversion, liquid metal fluid and MHD power generation sstems combined heat and mass transfer in natural convective flows on moving vertical porous plate. Das, Sarkar and Jana (22), investigated MHD Natural Convection Vertical parallel Plates with Oscillator Wall emperature. Srinvasa Rao, An and Babu. (2), studied the finite element analsis of radiation and mass transfer flow past semi-infinite moving vertical plate with viscous dissipation. Motivated b the above mentioned investigations and applications, we analzed the unstead MHD free convective, thermal radiation, inclined magnetic and dissipative fluid flow over a moving vertical porous plate with variable suction using the numerical approach of implicit finite difference method of Crank-Nicolson. 2. Mathematical Analsis Consider unstead two-dimensional unstead flow of an incompressible, electricall conducting viscous Boussinesq fluid in a cartesian ( x, ) coordinate sstem with a transversel applied magnetic field, the flow is assumed to be in the x-direction, which is taking along semi-infinite inclined plate and -axis normal to it. A magnetic field normal of uniform strength B is introduced normal to the direction of the flow. In the analsis we assume, that the magnetic Renold number is much lesser than unit so that the induced magnetic field is negleted in comparison to the applied magnetic field. It is also assumed that all fluid properties are constant except that of the influence of the densit variation and temperature in the bod force term. he surface is maintained at a constant temperature w which is higher than the constant temperature of the surrounding fluid. Since the plate is considered semi-infinite in the x-direction, all phsical variables are functions of and t onl. hen, under the Boussinesq s and boundar laer approximation s, also with the spirit of Gebhart (962), Singh and Sacheti (988) and Gebhart and Mollendorf (969) proposed the governing equations: Continuit equation vv = () Linear momentum equation uu tt + vv uu = υυ 2 uu 2 + gggg( ) + σσbb 2 SSSSSS 2 ψψ uu + vv ρρ KK uu (2)

3 American Journal of Fluid Dnamics 24, 4(3): 9-93 Energ equation is given as tt + vv = αα υυ ( uu ) 2 qq rr + σσbb CC PP ρρρρ PP ρρ uu 2 (3) he boundar conditions for the velocit, temperature and concentration fields are t*, u* =, * * for all * t*>, u* =, * * for *= (4) u* =, * * as * Where is dimensions coordinates, u* is velocitie component,, t* is dimensionless time, * is the dimensional temperature, g- the acceleration due to gravit, β - the volumetric coefficient of thermal expansion, β * is the volumetric coefficient of thermal expansion with concentration, ρ - the densit of the fluid, C p is the specific heat at constant pressure, ψ is the inclined angle, K* is the permeabilit of the porous medium, q r is the radiation heat flux, B - magnetic induction, ν- the kinematic viscosit and subscript denotes conditions in the free stream. For opticall thick fluid we have Modest (993) as follows the radiative heat flux term b using the Rosseland approximation is given b * * 4 * 4σ qr = (5) * * 3kr Where σ* is the Stefan-Boltzmann constant and k* r - the mean absorption co-efficent. It should be noted that b using the Rosseland approximation the present analsis is limited to opticall thick fluids. If temperature differences within the flow are sufficient small, then Equation (5) can be literalized b expanding * 4 into the alor series about *, which after neglecting higher order terms takes the form (6) qq rr = 6σσ 3KK (7) rr Substituting equation (6) and (7), into equation (3), gives tt + vv = αα σσ ss 3 2 3ρρCC PP kk ee 2 + υυ ( uu ) 2 QQ CC PP ( ) (8) Introducing these non-dimensional quantities uu = uu LL(GGGG) 2, = VV (GGGG) 4, tt = υυtt (GGGG) 2 υυ vvvv υυ 2, θθ = ww, PPPP = vvvvcc pp kk = vv αα, GGGG = gggggg( ww ) υυ 2 LL 3, EEEE = kkkk ee 2 υυ 2 GGGG LL 2 CC PP ( ww ), MM = σσbb 2 LL 2, RR = 4σσ 3 ss aaaaaa SSSS = νν ρρ(gggg) /2 DD (9) into the equations (2) and (8) with equation () identicall satisfied the following set of differential equations: ( + εεεεeennnn ) = 2 uu 2 +GGGGGG + GGGG MM 2 SSSSSS 2 ψψ + uu () kk ( + εεεεeennnn ) = + 4RR 2 θθ PPPP EEEE 2 + MMuu 2 () From (7) () we have that * w is wall dimensional temperature, * - the free stream temperature far awa from the plate, Re is the Renolds number, R is the radiation parameter, Pr is Prandtl number, is velocit, n is the frequenc, M is the Hartmann number, K is the permeabilit parameter, Gr is thermal Grashoff number, Ec is Eckert number, A is a real positive constant of suction velocit parameter, ε, and ε A< are small less than unit, i.e εa<<, V is a scale of suction velocit normal to the plate -the constant. he boundar conditions (4) are given b the following dimensionless form t*, u* =, θ* θ* for all * t*>, u* =, θ* θ* for *= (2) u* =, θ* θ* as * he dimensionless local value of skin friction (ττ) and Nusselt number (Nu) can be characterized in order b ττ = (3) = XX = NNNN = (4) he average of skin friction ( ττ ), average of Nusselt number (Nu) in dimensionless form can be written as (ττ) = Nu = - 3. Method of Solution θθ = XX = θθ dddd (5) dddd (6) In order to solve the non-linear coupled equation s (), () and (6) under the initial and boundar condition (2), an implicit finite difference scheme of crank-nicolson tpe has been emploed. he finite difference equations corresponding to equations () and () are discretized using the Crank-Nicolson Method as follows: nn+ nn jj jj tt (+ AAee nnnn ) jj nn+ + nn+ nn nn jj + jj + jj 4 = jj nn+ 2 jj nn+ + jj + nn+ jj nn 2 jj nn + jj + nn 2( ) 2

4 94 Amos S. Idowu et al.: Numerical Solution for hermal Radiation Effect on Inclined Magnetic Field of Mhd Free Convective Heat ransfer Dissipative Fluid Flow Past a Moving Vertical Porous Plate with Variable Suction + GGGG θθ jj nn + nn θθ jj MM 2 SSSSSS 2 ψψ + KK jj nn+ nn jj (7) θθ jj nn+ θθ jj nn = tt 2 (+ AAee nnnn ) θθ jj nn+ + θθ nn+ nn nn jj + θθ jj + θθ jj 4 PPPP +4RR (θθ 3 jj nn+ 2θθ nn+ jj + θθ nn+ nn jj + θθ jj 2θθ nn nn jj + θθ jj + 2( ) 2 +EEEE nn jj + nn jj 2 2Δ + nn jj + nn jj 2 2Δ With the following boundar conditions t*, u* =, θ = for all (8) t*>, u =, θ = for = (9) u =, θ = as * he subscript j and n denote the grid points along - and t-directions respectivel. he value of u and θθ are known at all grid points at t= from the initial conditions. he computations of the values of u and θθ at the (n+)th time using the values at previous time (n)th are carried out in this form: At all grid points, the values of θθ and u at t= from the initial conditions are known. herefore, the values of θθ at the next time step are calculated using the values alread known from the previous values. In this case, the finite difference Equations form a tri-diagonal sstem of equations where the values of θθ at ever nodal point at next time step are determined using the known values at the previous time. We therefore emplo homas algorithm to solve the tri-diagonal sstem of equations. In this wa, we were able to get the values of θθ at ever nodal point. he values of u are also computed at that particular time. Several values of u and θθ were obtained for the required time using this process. he 4 mesh size are Y =.6 and t =.. 4. Results and Discussions Numerical evaluation for the solutions of this problem is performed and the results are illustrated graphicall in Fig.-3, to the interesting features of significant parameters on velocit, temperature local skin friction and local Nusselt number. hroughout the computations we emplo A=.5, t=., n=. and ϵ=., while R, k r 2, Sc, Gr, Gc, M, Pr, η, and K were varied in order to account for their effects. Fig.2. shows the effect of radiation R on velocit. It was observed that as the value of R increases, the velocit increases with an increasing in the flow boundar laer thickness. hus, thermal radiation enhances the flow. he effect of radiation parameter R on the temperature profiles are presented in Fig. 6 it shows that, as the value of R increases the temperature profiles increases, with an increasing in the thermal boundar laer thickness. he velocit profiles for different values of Grashof number Gr were described in the fig. and fig.7. It is observed that an increasing in Gr leads to a rise in the values of velocit and temperature. Here the Grashof number represents the effect of the free convection currents. Phsicall, Gr> means heating of fluid of cooling of the boundar surface, Gr< means cooling of the fluid of heating of the boundar surface and Gr= corresponds to the absence of free convection current. In addition, the curves show that the peak value of velocit increases rapidl near the wall of the porous plate as Grashof number increases, and then decas to the relevant free stream velocit Gr=2 Gr=4 Gr=6 Gr= Figure. Effect of Gr on Velocit

5 American Journal of Fluid Dnamics 24, 4(3): K= K=.5 K=2 K= Figure 2. Effect of K on Velocit Pr=.6 Pr=.7 Pr=.7 Pr= Figure 3. Effect of Pr on Velocit ψ π/2 π/4 π/6 π/ Figure 4. Effect of Ψ on Velocit

6 96 Amos S. Idowu et al.: Numerical Solution for hermal Radiation Effect on Inclined Magnetic Field of Mhd Free Convective Heat ransfer Dissipative Fluid Flow Past a Moving Vertical Porous Plate with Variable Suction M=2 M=4 M=6 M= Figure 5. Effect of M on Velocit R=.2 R=.4 R=.6 R= Figure 6. Effect of R on emperature Gr=2.5 Gr=5 Gr=7.5 Gr= Figure 7. Effect of Gr on emperature

7 American Journal of Fluid Dnamics 24, 4(3): M=2 M=4 M=6 M= Figure 8. Effect of M on emperature Figure 9. Effect of Pr on emperature Ec= Ec=3 Ec=5 Ec= Figure. Effect of Ec on emperature

8 98 Amos S. Idowu et al.: Numerical Solution for hermal Radiation Effect on Inclined Magnetic Field of Mhd Free Convective Heat ransfer Dissipative Fluid Flow Past a Moving Vertical Porous Plate with Variable Suction Ec= Ec=3 Ec=5 Ec= Figure. Effect of Ec on Velocit.35.3 R= R=-2 R=3 R= Figure 2. Effect of R on Velocit π/2 π/4 π/6 π/ Figure 3. Effect of Ψ on emperature

9 American Journal of Fluid Dnamics 24, 4(3): 9-99 Fig.5 and Fig 8. shows that the effect of increasing values of Hartman number M parameter which results in decreasing velocit distribution, while increase in temperature across the boundar laer because of the application of transfer magnetic field it will result in a restrictive tpe of force(lorenz force) similar to drag force which tends to resist the fluid and this reducing its velocit. he velocit profiles across the boundar laer for different values of Prandtl number Pr were plotted in Fig.3 and Fig 9. he results shows that the effect of increasing values of Pr results in decreasing the velocit and temperature, while it peak up again at the value of.7 and 7. It was observed from Fig.4. that an increase in inclined angle ψ results in decreasing the thermal boundar laer thickness and more uniform temperature distribution across the boundar laer. Fig.3. show that the effect of inclined angle on the velocit profile. It shows that an increase in the inclined angle parameter increase the velocit profile. It was observed from fig.2. the velocit profiles for different values of the permeabilit K. Clearl, as K increases the value of velocit tends to increase. hese results could be ver useful in deciding the applicabilit of enhanced oil recover in reservoir engineering. Fig.. and Fig.. shows the effects of Eckert number on velocit and temperature respectivel, as Eckert number increases the velocit distribution across the boundar laer increases while, temperature distribution across the boundar laer decreases. able. Effect of R on Velocit n=., t= and A= ϵ Pr Ec R Gr M K Ψ C f π/ π/ π/ π/2.243 able 2. Effect of Ec on Velocit n=., t= and A= ϵ Pr Ec R Gr M K Ψ C f π/ π/ π/ π/2.388 able 3. Effect of Gr on Velocit n=., t= and A= ϵ Pr Ec R Gr M K Ψ C f π/ π/ π/ π/ able 4. Effect of M on Velocit n=., t= and A= ϵ Pr Ec R Gr M K Ψ C f π/ π/ π/ π/2.85 able 5. Effect of Pr on Velocit n=., t= and A= ϵ Pr Ec R Gr M K ψ C f π/ π/ π/ π/2.366 able 6. Effect of K on Velocit n=., t= and A= ϵ Pr Ec R Gr M K ψ C f π/ π/ π/ π/2.763 able 7. Effect of ψ on Velocit n=., t= and A= ϵ Pr Ec R Gr M K ψ C f π/ π/ π/ π/.2427 able 8. Effect of Gr on emperature n=., t= and A= π/ π/ π/ π/2.84 able 9. Effect of K on emperature n=., t= and A= π/ π/ π/ π/ able. Effect of ψ on emperature n=., t= and A= π/ π/ π/ π/ able. Effect of Pr on emperature n=., t= and A= π/ π/ π/ π/ able 2. Effect of M on emperature n=., t= and A= π/ π/ π/ π/2 3.82

10 Amos S. Idowu et al.: Numerical Solution for hermal Radiation Effect on Inclined Magnetic Field of Mhd Free Convective Heat ransfer Dissipative Fluid Flow Past a Moving Vertical Porous Plate with Variable Suction able 3. Effect of Ec on emperature n=., t= and A= π/ π/ π/ π/ able 4. Effect of R on emperature n=., t= and A= π/ π/ π/ π/2 4.6 able-()-(7) for velocit: he effects of the radiation parameter and inclined angle on the skin- friction coefficient. It is observed from this table that as radiation parameter, Grashof number, Eckert number, Porosit Permeabilit or inclined angle Parameter increases, the skin-friction coefficients increases. Also an increase in the Prandlt number Parameter and Magnetic field Parameter effect brings about a decrease in the skin-friction coefficient. able-(8)-(4) for temperature: As the Prandtl number, radiation and the magnetic field Parameters effect increases, the Nusselt number increases. An increase in inclined angle, Grashof number for heat transfer, Permeabilit parameter, and Eckert number resulted in the decrease in the Nusselt number. 5. Conclusions A numerical stud has been carried out to stud the numerical solution for thermal radiation effect on inclined magnetic field of magnetohdrodnamic (MHD) free convective heat transfer dissipative fluid flow past a moving vertical porous plate with variable suction. he fluid is gra absorbing-emitting but non-scattering medium and the radiative heat flux in the energ equation. A famil of governing partial differential equations was solved b an implicit finite difference scheme of Crank-Nicolson tpe, which is stable and convergent. he result were obtained for different values of radiation parameter R, thermal Grashof number Gr, inclined angle ψ, Eckert number Ec. Conclusion of this stud were as follows. A rise in the velocit leads to a fall in size of the temperature at the Grashof number Gr. As velocit and temperature increases the Porosit Parameter K also amplif. he velocit increases with an increase in the Prandtl number Pr. A rise in the inclined angle parameter ψ leads to an accelerated increase in velocit as well as the reduction in temperature. he size of fluid velocit and temperature reduces with the increase Eckert number Ec. he level of fluid temperature falls while velocit amplifies with the rise of Magnetic Parameter M. he velocit as well as temperature increases with an increase in radiation. It shows obviousl that the thermal radiation has a positive significant effect on both the velocit and temperature of the fluid through the medium which it flows. REFERENCES [] Ahmed, N. sarma, K. and Baruna, D.P. (22): Magnetic field effect on free convective oscillator flow between two vertical parallel plates with periodic plate temperature and dissipative heat. Applied Mathematical Sciences, 6(39), [2] A. S. Idowu, M.S. Dada and.a. Jimoh (23): Heat and mass transfer of magnetohdrodnamic (MHD) and dissipative fluid flow pass a moving vertical porous plate with variable suction. Mathematical heor and Modeling. ISSN: (Paper), ISSN: (online),3(3). [3] Alam M.S, Rahman M.M and sattar M.A, (29): ransient magnetohdrodnamic free convective heat and mass transfer flow with thermophoresis past a radiate inclined permeable plate in the presence of variable chemical reaction and temperature dependent viscosit, Nonlinear Analsis: Modelling and control (4): 3-2. [4] A.J Chamkha, akhar H.S. and soundalgekar V.M. (2): Radiation effect on free convection flow past a semi-infinite vertical plate with mass transfer, Chem. Engr. J.,.84: [5] B. Rushi Kumar and R. Sivaraj, (22): MHD Viscoelastic fluid non-darc flow along a moving vertical cone. Int. J. of Appl. Math. and Mech.8():69-8,22. [6] Carnahan B., Luther H.A., and Willkes J.O (969). Applied Numerical Method. John Wile and Sons, New York, SA. [7] F.M. Had, R.A. Mohammed, A. Mahd, (26): MHD Free convection flow along a vertical wav surface wih heat generation or absorption effect Int. Comm Heat Mass ransfer 33: [8] Gnaneshwara Redd M and Bhaskar Redd N (29): Radiation and mass transfer effect on an unstead MHD free convection flow past a heated vertical porous plate with viscous dissipation, Int. J. of Appl. Math and Mech, 6 (6): 96-. [9] Kim Y.J. and Fedorov A.G. (23): ransient Mixed Radiation Convection Flow of a Micro polar Fluid past a Moving, Semi-infinite vertical porous plate. Internation Journal of Heat and Mass ransfer. 46: [] K. Vajravelu, A. Hdjinicolaou, (993): Heat transfer in a viscous fluid over a streching sheet with viscous dissipation and internal heat generation, Int. Comm. Heat Mass ransfer 2, [] M. A. Hossain, M. M. Molla, L.S. Yaa, (24), Natural convection flow along a vertical wav surface temperature in the presence of heat generation /absorption, Int. J. hermal Science 43:

11 American Journal of Fluid Dnamics 24, 4(3): 9- [2] M.S. Alam, M.M. Rahman, M.A. Sattar. (26): MHD Free convective heat and mass transfer flow past an inclined semi-infinite heated surface of an electricall conducting and stead viscous incompressible fluid in the presence of a magnetic field and generation hamasat. Int. J. Sci. ech. (4): -8. [3] Md Abdus Samad and Mohammed Mansur Rahman. (26): hermal radiation interraction with unstead MHD flow past a vertical porous plate immersed in a porous medium. [4] Muthucumaraswam R and Chandrakala P (26): Radiative heat and mass transfer effect on moving isothermal vertical plate in the presence of chemical reaction, International Journal of Applied Mechanics and Engineering, : [5] Muthucumaraswam R and Senthih Kumar G (24): Heat and mass transfer effects on moving vertical plate in the presence of thermal radiation, heoretical Applied Mechanics, 3: [6] N. Sandeep, V. Sugunamma (23): Effect of inclined magnetic field on unstead free convective flow of dissipative fluid past a vertical plate.():6-23 ISSN: [7] Naving Kumar and Sandeep Gupta (28): Effect of variable permeabilit on unstead two-dimensional free convective flow through a porous bounded b a vertical porous surface, Asian J, Exp. Sci.. 22 (3): [8] P.R. Sharma, Navin Kumar and Pooja Sharma (2): Influence of Chemical Reaction and Radiation on nstead MHD Free Convective Flow and Mass ransfer through Viscous Incompressible Fluid Past a Heated Vertical Plate Immersed in Porous Medium in the Presence of Heat Source. [9] R. Muthucumaraswam, P. Ganesan, (2), Effect of the chemical reaction and injection on flow characteristics in an unstead motion of an isothermal plate, J. Appl. Mech. ech. Phs. 42, [2] R.A Mohamed (29): Double-Diffusive Convection- Radiation Interaction on nstead MHD Flow over a Vertical Moving Porous Plate with Heat Generation and Soret Effects, Applied Mathematics Science, (3) (3): [2] S. Das, B.C. Sarkar and R.N. Jana (22): MHD Natural Convection Vertical parallel Plates with Oscillator Wall emperature. J. Comp. and Math. Sci 3(4): [22] Sugunamma and Sandeep (2): nstead hdromagnetic free convective flow of a flux. Int. J. of mathematics and computer reseach. ():37-5. [23] V. Srinivasa Rao and L. Anand Babu (2): Finite element analsis of radiation and mass transfer flow past semi-infinite moving vertical plate with viscous dissipation. ARPN Journal of Engineering and Applied Sciences. (5).