Department of Mathematics, Usmanu Danfodiyo University, Sokoto, Nigeria. DOI: / Page

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1 IOSR Journal of Mathematics (IOSR-JM) e-issn: , p-issn: X. Volume, Issue Ver. II (Jan - Feb. 5), PP Finite difference solutions of magneto hdrodnamic free convective flow with constant suction and variable thermal conductivit in a Darc-Forchheimer porous medium Ime Jimm Uwanta, Halima Usman* Department of Mathematics, Usmanu Danfodio Universit, Sokoto, Nigeria Abstract: This paper presents the stud of the effects of variable thermal conductivit and Darc-Forchheimer on magnetohdrodnamic free convective flow in a vertical channel in the presence of constant suction. The resulting governing equations are non-dimensionalised, simplified and solved using Crank Nicolson tpe of finite difference method. To check the accurac of the numerical solution, stead-state solutions for velocit, temperature and concentration profiles are obtained b using perturbation method. Numerical results for the velocit, temperature and concentration profiles are illustrated graphicall while the skin friction, Nusselt number and Sherwood number are tabulated and discussed for some selected controlling thermo phsical parameters involved in the problem to show the behavior of the flow transport phenomena. It is found that the velocit and temperature increased due to increase in variable thermal conductivit parameter and there is decrease in concentration due to increase in Darc-Forchheimer number. It is also observed that the numerical and analtical solutions are found to be in excellent agreement. Kewords: Variable thermal conductivit, Porous medium, Magnetohdrodnamic, Chemical reaction, Darc- Forchheimer. I. Introduction There has been a great deal of interest generated in the area of heat and mass transfer of free convective flow in porous medium due to it is numerous applications in a variet of industrial processes as well as in man natural circumstances. The subject of free convective flow in porous media has attracted considerable attention in the last few decades. Examples of such technological applications are ground water flows, thermal insulation engineering, food processing, soil pollution, fibrous insulation, geothermal extraction to name just a few. A comprehensive bibliograph concerned with literature on convective flow in porous media can be found in the books of Nield and Bejan [] and Ingham and Pop [6]. Processes involving heat and mass transfer effect have long been recognized as important principle in chemical processing equipment. The effect of chemical reaction, heat and mass transfer along a wedge with heat source and concentration in the presence of suction or injection have been examine b Kandasam et al. [7]. Sattar [5] analzed the effect of free and forced convection boundar laer flow through a porous medium with large suction. Mohammed et al. [] investigated the effect of similarit solution for MHD flow through vertical porous plate with suction. In most of the studies on hdromagnetic heat transfer, thermal conductivit has been taken as constant. In metallurgical processing, the numerical value of thermal conductivit changes with temperature. Therefore the thermal conductivit is temperature dependent. In order to predict the accurac of heat and mass transfer flow, mathematical models most consider the variation of thermal conductivit with temperature. Man papers were published with problem of thermal conductivit under different phsical and geothermal conditions. Patowar and Dusmata [3] have presented the effects of thermal conductivit of micropolar fluid past a continuousl moving plate with suction or injection in presence of magnetic field. Abdou [] considered the effects of thermal radiation on unstead boundar laer flow with temperature dependent viscosit and thermal conductivit due to a stretching sheet in porous medium. Similarl, Chiam [3] analzed the effect of a variable thermal conductivit on the flow and heat transfer from a linearl stretching sheet. In another article, Seddek and Salema [8] have studied the effects of variable viscosit and thermal conductivit on an unstead two-dimensional laminar flow of a viscous incompressible conducting fluid past a vertical plate taking into account the effect of magnetic field in the presence of variable suction. Gitima [4] presented a numerical model on the effect of thermal radiation on unstead boundar laer flow with variable viscosit and variable thermal conductivit due to stretching sheet in a porous channel in the presence of magnetic field. Additionall Hossain et al. [5] described the stud of natural convection with variable viscosit and thermal conductivit from a vertical wav cone. Sharma and Singh [9] have presented a theoretical stud on the effects of variable thermal conductivit and heat source/sink on MHD flow near a stagnation point on a linearl stretching sheet. Ver recentl, Uwanta and Murtala [] investigated the stud of heat and mass transfer flow past an infinite vertical plate with variable thermal conductivit. Although considerable work has been reported on heat and mass transfer b free DOI:.979/ Page

2 Finite difference solutions of magneto hdrodnamic free convective flow with constant suction. convection in a porous medium, majorit of porous studies have been on Darc s law which states that the volume-averaged velocit is proportional to the pressure gradient which is valid onl for slow flows through porous media with low permeabilit. However, Darc s law fails when the velocities and inertial effects are greater. Inertial porous phenomena can be accounted through the addition of a velocit-squared term in the momentum equation and the resulting model is known as Darc-Forchheimer model Vafai and Tien []. The Darc-Forchheimer model described the effect of inertia as well as viscous forces in porous media. Man investigators have reported work on Darc and Darc-Forchheimer. Numerical simulation of chemical reaction and viscous dissipation effects on Darc-Forchheimer mixed convection in a fluid saturated media has been investigate b Mahd and Chamkha []. Salem [7] has presented the radiation and mass transfer effects in Darc-Forchheimer mixed convection from a vertical plate embedded in a fluid saturated porous medium. Lai [9] studied mixed convection heat and mass transfer from a vertical wall in a Darcian fluid saturated porous medium. Additionall, Kishan et al. [8] analzed MHD free convective heat and mass transfer from a vertical surface embedded in a porous medium. A detailed numerical stud has been carried out on the natural convection in a porous square cavit with an isoflux and isothermal discrete heater placed at the left wall b Saeid and Pop [6] using Darc law. Rawat et al. [4] have investigated a theoretical stud on free convection MHD micropolar flow, heat and species diffusion between vertical plates enclosing a non-darcian porous medium with variable thermal conductivit and internal heat generation/absorption effects. Inspite of all these studies, the aim of this work is to investigate the numerical solutions to the sstem of equations for magnetohdrodnamic free convective flow past a finite vertical channel with constant suction and thermal conductivit in a porous medium using finite difference method. II. Mathematical analsis Consider an unstead laminar one-dimensional boundar laer heat and mass transfer flow in an electricall conducting fluid with thermal conductivit, chemical reaction and radiation in a Darc-Forchheimer porous medium. Fluid suction or injection is imposed at the boundar of the clinder. The x - axis is taken along the plate in the verticall upward direction and the - axis is taken normal to the plate. A uniform transverse magnetic field B is applied normal to the flow direction. The magnetic Renolds number is assumed to be small so that the assumed magnetic field can be neglected. It is also assumed that the effects of joule heating and viscous dissipation are neglected and the external electric field due to the effect of charge polarization and the hall effect of MHD are neglected. Under these assumptions, the governing boundar laer equations of continuit, momentum, energ and diffusion equations under Boussinesq s approximations could be written as follows: v () u t - v u = n u - æ s B r + ö ç b è K u - n u + gb(t -T ø K ) + gb * C -C ( ) () T T - v t = k é +a(t -T rc p ) T ù ê ú- q r (3) ë û rc p C C - v t = D C - R* (C -C ) (4) The corresponding initial and boundar conditions are prescribed as follows: t, u, T T, C C for all t, u, T T, C C at w w w w (5) u, T T, C C at = H From continuit equation, it is clear that the suction velocit is either a constant or a function of time. Hence, on integrating equation (), the suction velocit normal to the plate is assumed in the form, where is a scale of suction velocit which is non-zero positive constant. The negative sign indicates that the suction is towards the plate and corresponds to stead suction velocit normal at the surface. DOI:.979/ Page

3 Finite difference solutions of magneto hdrodnamic free convective flow with constant suction. The fourth and fifth terms on the right hand side of equation () denote the thermal and concentration buoanc effects respectivel. The last term of equation (3) represents the radiative heat flux term, while the last term of equation (4) is the chemical reaction term. u and v are the Darcian velocit components in the x - and - directions respectivel, t is the time, is the kinematic viscosit, g is the acceleration due to gravit, is the coefficient of volume expansion, the densit of the fluid, is the scalar electrical conductivit, is the volumetric coefficient of expansion with concentration, pressure, K is the Darc permeabilit of the porous medium, * C p is the specific heat capacit at constant b is the empirical constant, k is the dimensionless thermal conductivit of the ambient fluid, is a constant depending on the nature of the fluid, * v is the constant R is the dimensionless chemical reaction, D is the coefficient of molecular diffusivit, suction parameter, strength. q r is the radiative heat flux in the - direction, B is the magnetic induction of constant T and T are the temperature of the fluid inside the thermal boundar laer and the fluid temperature in the free stream respectivel, while C and C are the corresponding concentrations. To obtain the solutions of equations (), (3) and (4) subject to the conditions (5) in non-dimensional forms, we introduce the following non-dimensional quantities: u = u,t = t u u H, = H,q = T -T T -T, w C = C -C C -C,Pr = u rc p,sc = u k w D, Da = K u H v, ( ) u M = s B H,Gr = H gb T -T w ru ( ),Gc = H gb * C w -C l = a ( T -T ), g = n H,K u r = R* H,R = 6as HT 3,Fs = b u u k u H v Appling these non-dimensionless quantities (6), the set of equations (), (3), (4), and (5) reduces to the following: u u u Fs M u u Gr GcC t Da Da Pr Pr Pr R t C C C KC r t Sc with the following initial and boundar conditions t u,, C, for all, t u,, C, at, u,, C at. Where Pr is the Prandtl number, Sc is the Schmidt number, M is the Magnetic field parameter, Gr is the thermal Grashof number, Gc is the Solutal or mass Grashof number, is the variable thermal conductivit, is the variable suction parameter, R is the Radiation parameter, u, (7) (8) (9) () (6) K is the chemical reaction parameter, Da is the Darc number, Fs is the Darc-Forchheimer number, t is the dimensionless time while u and v are dimensionless velocit components in x - and - directions respectivel. r DOI:.979/ Page

4 Finite difference solutions of magneto hdrodnamic free convective flow with constant suction. III. Analtical solutions In order to check the accurac of the present numerical scheme of this model, there is need to compare numerical solutions with the analtical solutions. Since the governing equations are highl coupled and nonlinear, it is, therefore, of interest to reduce the governing equations of the present problem due to its nonlinearit into a form that can be solved analticall. At stead state the governing equations of the problem becomes: u u u Gr GcC Da Pr R C C Sc K rscc The boundar conditions are u,, C, at, u,, C at. (4) To find the approximate solution to equations ()-(3) subject to equation (4), we emploed a regular perturbation method b taking a power series expansion and assume solution of the form: u u Ru ( R ) R R ( ) C C RC R ( ) () () (3) (5) Using equation (5) into equations ()-(4) and equating the coefficient of like powers of R, the solutions to the governing equations are obtained as: s s Pr h h u F9 e F e F F e F3 e F4 e s s Pr Pr R F5 e F6 e F7 F8 e F9 F e (6) Pr Pr F Fe R F3 F4e F5 F6 e (7) h h C F e F e (8) 7 8 IV. Numerical solution procedure To solve the sstem of transformed coupled non-linear partial differential equations (7)-(9), under the initial and boundar conditions (), an implicit finite difference scheme of Crank-Nicolson tpe which is unconditionall stable Carnahan et al, [] has been emploed. The equivalent finite difference approximations corresponding to equations (7)-(9) are given as follows: æ ç è u i, j+ - u i, j Dt D ö - g ø D u - u i+, j i-, j ( ) = ( u ( ) i+, j+ - u i, j+ + u i-, j+ + u i+, j - u i, j + u i-, j ) _ æ M + Fs ö ç u è Da i, j ø ( ) - ( Da u i, j ) +Gr ( q i, j ) +Gc( C i, j ) (9) DOI:.979/ Page

5 Finite difference solutions of magneto hdrodnamic free convective flow with constant suction. æq i, j+ -q ö i, j ç è Dt - g ø D q -q i+, j i-, j æ ç è H Pr D C i, j+ -C i, j Dt ( ) = ( ( ) q - q +q +q - q +q i+, j+ i, j+ i-, j+ i+, j i, j i-, j ) + l ( ( ) q -q i+, j i, j ) - R ( Pr q i, j ) Pr D ö - g ø D C -C i+, j i-, j Sc D ( ) = ( ( ) C - C +C +C - C +C i+, j+ i, j+ i-, j+ i+, j i, j i-, j ) _ K r ( C i, j ) () () The initial and boundar conditions take the following forms u i, j =, q i, j =, C i, j = u, j =, q, j =, C, j = u H, j =, q H, j =, C H, j = where H corresponds to. The index i corresponds to space and j corresponds to time t. () t are the mesh sizes along - and direction and time t-direction respectivel. The analtical solutions obtained in this work are used to check on the accurac and effectiveness of the numerical scheme. Computations are carried out for different values of phsical parameters involved in the problem. The skin friction coefficient, Nusselt number and Sherwood number are important phsical parameters for this tpe of boundar laer flow, which respectivel are given b: u C C f, Nu, Sh (3) V. Results and discussion Numerical computations have been carried out for various values of thermo phsical parameters involved in this work such as thermal Grashof number ( Gr ), solutal Grashof number ( Gc ), Magnetic parameter ( M ), Darc number Da, suction parameter ( ), radiation parameter ( R ), chemical reaction parameter ( K ), Darc-Forchheimer number Fs, variable thermal conductivit ( ), Prandtl number ( Pr ) r and Schmidt number ( Sc ) using an implicit method of Crank-Nicolson tpe. In order to illustrate the results graphicall, the numerical values are plotted in Figs. (-5). Values of local skin friction C f, Nusselt number Nu and Sherwood number Sh are presented in Tables and for various values of phsical parameters. Solutions are obtained for fluids with Prandtl number (Pr =.7,. and 7.) corresponding to air, salt water and water respectivel. The values of Schmidt number are chosen in such a wa as the represent the diffusing chemical species of most common interest in air and are taken for (Sc =..6,.78,.6) corresponding to hdrogen, water, NH3 and Propl benzene respectivel. Default values of the phsical parameters for velocit, temperature and concentration profiles are specified as follows: DOI:.979/ Page

6 Finite difference solutions of magneto hdrodnamic free convective flow with constant suction. Gr = 5.,Gc = 5., M =., K r =.,Pr =.7,Sc =., R = 3., Da =., Fs = 3.,g = 5.,l =.5. All graphs therefore correspond to these values unless specificall indicated on the appropriate graphs. The effects of various thermo phsical parameters on the fluid velocit are illustrated in Figs.- at timet.. The influence of thermal Grashof number and solutal Grashof number are presented in Figs. and. As expected, it is observed that there is a rise in the velocit profiles due to the enhancement of thermal and mass buoanc force. The velocit distribution increases rapidl near the porous plate and then decrease smoothl to the free stream velocit. Fig. 3 illustrates the effect of magnetic parameter on the velocit profile. It is interesting to note that increase in M leads to decreasing the flow velocit. This is because the presence of magnetic field in an electricall conducting fluid tends to produce a bod force, which reduces the velocit of the fluid. Fig. 4 depicts tpical velocit distribution across the boundar laer for various values of Darc number. It is seen that the dimensionless velocit increases with increase in Darc number. This is due to the fact that larger values of Da parameter correspond to higher permeabilit porous media, which implies that, less porous fiber resistance to the flow and therefore acceleration in transport. The effect of Darc-Forchheimer parameter is demonstrated in Fig. 5. It is noticed that an increase in Fs parameter increases the resistance to the flow and so a decrease in the velocit profile. Figs. 6-8 graphicall illustrate the behavior of Prandtl number, suction and Schmidt number respectivel. It is observed that the momentum boundar laer decreases with the increase in Pr this is due to the fact that convection current becomes weak and hence results in a decrease in the velocit. Also, it is seen that the velocit decreases with an increase in g parameter. This is because suction parameter decelerates fluid particles through porous wall there b reducing the growth of the fluid boundar laers. Figure 8 illustrates the effect of Sc on the velocit profiles. Clearl, it is observed that the velocit distribution decreases with an increase in Sc. This is due to the fact that as Sc increases, the concentration buoanc effect decreases ielding a reduction in the velocit profiles. The influence of variable thermal conductivit parameter, radiation parameter and dimensionless time are displaed in Figs. 9- respectivel. It is observed that as and t increases the momentum boundar laer increases, while an increase in radiation parameter obviousl decreases the velocit within the boundar laer. This is because larger values of R correspond to an increased dominance of conduction over radiation, thereb decreasing the buoanc force and the thickness of the momentum boundar laer. The influences of various thermo phsical parameters on the temperature distribution are respectivel presented in Figs. -7 at time t.. Fig. depicts the behavior of Prandtl number that is inversel proportional to the thermal diffusivit of the working fluid on temperature distribution. It is noticed that the thermal boundar laer decreases with increasing Pr. This is due to the fact that as values of Pr increases, the thermal diffusivit of the fluid increases which results in a corresponding decrease in the temperature within the fluid. The temperature profile decreases with increasing suction parameter in Fig.3. This is because suction decelerates the fluid properties through porous wall thereb reducing the thermal boundar laer. The effect of variable thermal conductivit parameter on temperature profile is shown in Fig. 4. Here it is observed that the temperature profile decreases with increase in the variable thermal conductivit parameter. Fig. 5 described the influence of radiation parameter on the temperature profile. An increase in R leads to increasing the temperature boundar laer. This is obvious because the effect of radiation is to increase the temperature distribution in the thermal boundar laer. Figs.6 and 7 graphicall illustrate the effects of Darc- Forchheimer parameter and dimensionless time. It is observed that there is no significant change in the temperature profile as Darc-Forchheimer parameter is increased, but there is a rise in the thermal boundar laer as the time increases. Figs. 8- illustrate the concentration profiles for various values of Schmidt number, chemical reaction parameter, suction parameter and Darc-Forchheimer parameter respectivel at time t.4. A decrease in concentration profile with increasing these parameters is observed from these figures. This is due to the fact that an increase of Schmidt number means decrease in molecular diffusivit, while suction parameter stabilizes the boundar laer growth and therefore the concentration profile decreased. The effect of dimensionless time on the concentration profile is presented in Fig.. It is seen that the concentration profile increases with increase in the dimensionless time. The validit of the present model has been verified b comparing the numerical solutions and the analtical solutions through Fig. 3-5 for velocit, temperature and concentration profiles respectivel.these results are presented to illustrate the accurac of the numerical solution. It is observed that the agreement between the results is excellent because the curves corresponding to analtical and numerical solutions coincide with one another. This has established confidence in the numerical results reported in this paper. DOI:.979/ Page

7 Vel(U) Finite difference solutions of magneto hdrodnamic free convective flow with constant suction..8.6 Gr = 5,, 5, Fig.. Effect of Gr on the velocit profiles Fig.. Effect of Gc on the velocit profiles Fig.3. Effect of M on the velocit profiles DOI:.979/ Page

8 Finite difference solutions of magneto hdrodnamic free convective flow with constant suction. Fig.4. Effect of Da on the velocit profiles Fig.5. Effect of Fs on the velocit profiles Fig.6. Effect of Pr on the velocit profiles DOI:.979/ Page

9 Finite difference solutions of magneto hdrodnamic free convective flow with constant suction. Fig.7. Effect of ( = S) on the velocit profiles Fig.8. Effect of Sc on the velocit profiles Fig.9. Effect of ( = V) on the velocit profiles DOI:.979/ Page

10 Tem(T) Vel(U) Vel(U) Finite difference solutions of magneto hdrodnamic free convective flow with constant suction... R =, 5,, Fig.. Effect of R on the velocit profiles.5.4 t =.,.,.3, Fig.. Effect of t on the velocit profiles.8 Pr =.7,., 3., Fig.. Effect of Pr on the temperature profiles DOI:.979/ Page

11 Tem(T) Tem(T) Finite difference solutions of magneto hdrodnamic free convective flow with constant suction..8.6 S =, 5,, Fig.3. Effect of ( = S) on the temperature profiles Fig.4. Effect of ( = V) on the temperature profiles.8 R =, 5,, Fig.5. Effect of R on the temperature profiles DOI:.979/ Page

12 Con(C) Tem(T) Tem(T) Finite difference solutions of magneto hdrodnamic free convective flow with constant suction..8.6 Fs = 5,, 5, Fig.6. Effect of Fs on the temperature profiles.8 t =.,.,.3, Fig.7. Effect of t on the temperature profiles.8 Sc =.,.6,.78, Fig.8. Effect of Sc on the concentration profiles DOI:.979/ Page

13 Con(C) Con(C) Con(C) Finite difference solutions of magneto hdrodnamic free convective flow with constant suction..8 Kr =., 5.,, Fig.9. Effect of K on the concentration profiles r.8 S = 5,, 5, Fig.. Effect of ( = S) on the concentration profiles.8.6 Fs = 5,, 5, Fig.. Effect of Fs on the concentration profiles DOI:.979/ Page

14 Tem(T) Vel(U) Con(C) Finite difference solutions of magneto hdrodnamic free convective flow with constant suction..8 t =.,.,.3, Fig.. Effect of t on the concentration profiles Numerical solution Analtical solution Fig.3. Comparison of numerical and analtical solutions for velocit profiles.8 Numerical solution Analtical solution Fig.4. Comparison of numerical and analtical solutions for temperature profiles DOI:.979/ Page

15 Con(C) Finite difference solutions of magneto hdrodnamic free convective flow with constant suction..8 Numerical solution Analtical solution Fig.5. Comparison of numerical and analtical solutions for concentration profiles Table elucidates the effect of thermal conductivit parameter on skin friction and Nusselt number respectivel. It is observed that an increase in causes a reduction in the skin friction coefficient for Pr.7,7. respectivel. Whereas heat transfer rate increase as expected with increase in andpr. The effects of various thermo phsical parameters on Sherwood number are shown in Table. It is observed that the Sherwood number increases with an increase in Schmidt number and chemical reaction parameter respectivel, but it is noticed that there is no significant change in the Sherwood number when the thermal Grashof number, thermal conductivit parameter and Darc-Forchheimer parameter are increased. Table : Skin friction ( C ) and Nusselt number f DOI:.979/ Page Nu for different values of thermal conductivit ( l =V ) and Prandtl number with., Gr.5, Gc, M, K, R., Sc.6, Fs 5, Da.5. C f C f Pr.7 Pr 7. Pr.7 Pr Gr Table : Sherwood number ( Sh ) for different values of ( Gr, Sc, Kr,, Fs ) with., Gc, M, R., Da.5, Pr.7. Sc K Fs Sh r VI. Conclusion A finite difference solution of MHD free convective flow in a vertical channel with constant suction and thermal conductivit in a Darc-Forchheimer porous medium has been analzed both analticall and numericall. The resulting governing equations of the problem are non-dimensionalised, simplified and solved b Crank Nicolson tpe of implicit finite difference method. It reveals that the velocit increased due to increases in either of the Darc parameter or thermal conductivit parameter and decreased as the magnetic parameter, Darc- Forchheimer parameter, Prandtl number, and suction parameter were increased. An increased in the fluid temperature and the thermal boundar laer thickness is a function of an increased in thermal conductivit parameter and radiation parameter. In addition, the concentration decreased as either of the Schmidt number or r Nu Nu

16 Finite difference solutions of magneto hdrodnamic free convective flow with constant suction. chemical reaction parameter or suction or Darc-Forchheimer parameters was increased. The thermal conductivit and Prandtl number decreases the skin friction coefficient, whereas reverse trend is seen in the rate of heat transfer increasing with increase in thermal conductivit and Prandtl number. Also, the Schmidt number and chemical reaction parameter increases the Sherwood number. It is hoped that solutions presented in this work for various thermo phsical effects will serve as a tool for understanding more general problems in heat and mass transfer in metallic materials processing and also in geophsical processes. Acknowledgment The author Halima Usman is thankful to Usmanu Danfodio Universit, Sokoto for financial support. Nomenclature C Concentration C Specific heat at constant pressure p D - Mass diffusivit g Acceleration due to gravit Gr Grashof number Gc Solutal Grashof number Nu Nusselt number Pr Prandtl number Sc Schmidt number R - Radiation parameter Kr - Chemical reaction parameter T Temperature C -Skin friction f Sh - Sherwood number u, v velocities in the x and -direction respectivel x, Cartesian coordinates along the plate and normal to it respectivel B - Magnetic field of constant strength M Magnetic field parameter Da -Darc number Fs -Forchheimer number Greek letters β* - Coefficient of expansion with concentration β - Coefficient of thermal expansion ρ- Densit of fluid θ- Dimensionless temperature υ- Kinematic viscosit - suction parameter - Variable thermal conductivit - electrical conductivit of the fluid References []. M. Abdou, Effect of radiation with temperature dependent viscosit and thermal conductivit on unstead stretching sheet through porous medium, Non-linear Analsis: Modeling and Control, 5(3), (), []. B. Carnahan, H. A. Luther and J. O. Wilkes, Applied numerical methods, John Wile and Sons, New York, (996). [3]. T. C. Chiam. Heat transfer in a fluid with variable thermal conductivit over a linearl stretching sheet, Acta Mechanic, 9, (998), [4]. P. Gitima, Effects of variable viscosit and thermal conductivit of micropolar fluid in a porous channel in presence of magnetic field, International Journal for Basic Sciences and Social Sciences, (3), (), [5]. M. A. Hossain, M. S. Munir and I. Pop, Natural convection with variable viscosit and thermal conductivit from a vertical wav cone, International Journal of Thermal Sciences, 4, (), [6]. D. B. Ingham and I. Pop, Transport phenomena in porous media, Elsevier, Oxford, UK. (5). [7]. R. Kandasam, K. Perisam and P. K. Sivagnana, Effects of chemical reaction, heat and mass transfer along a wedge with heat source and concentration in the presence of suction or injection, International Journal of Heat and Mass Transfer, 48, (5,) DOI:.979/ Page

17 Finite difference solutions of magneto hdrodnamic free convective flow with constant suction. [8]. N. Kishan, M. Srinivas, R. C. Srinivas, MHD effects on free convective heat and mass transfer in a doubl stratified non-darc porous medium, International Journal of Engineering Sciences and Technolog, 3(), (), [9]. F. C. Lai, Coupled heat and mass transfer b mixed convection from a vertical plate in a saturated porous medium, International Communication Heat and Mass Transfer, 8, (99), []. Mahd and Chamkha, Chemical reaction and viscous dissipation effects on Darc-Forchheimer model convection in a fluid saturated media, International Journal of Numerical Methods for Heat and Mass Transfer Fluid Flow, (8), (), []. F. Mohammad, O. Masatiro, S. Abdus and A. Mohamud, Similarit solution for MHD flow through vertical porous plate with suction, Journal of Computational and Applied Mechanics, 6(), (5), 5-5. []. D. A. Nield and A. Bejan, Convection in porous media, third edition, Springer, New York, (6). [3]. G. Patowar and K. Dusmata, Effects of variable viscosit and thermal conductivit of micropolar fluid past a continuousl moving plate with suction or injection in presence of magnetic field, International Journal of Mathematics Trend and Technolog, (), (), [4]. S. Rawat, R. Bhargava and B. O. Anwar, A finite element stud of the transport phenomena MHD micropolar flow in a Darc- Forchheimer porous medium, Proceedings of the World Congress on Engineering and Computer Science, Oct 4-6, (7). [5]. M. A. Sattar, Free and forced convection boundar laer flow through a porous medium with large suction, International Journal of Energ Research, 7, (993), -7. [6]. N. H. Saied and I. Pop, Natural convection from discrete heater in a square cavit filled with a porous medium, Journal of Porous Media, 8(), (5), [7]. A. M. Salem, Coupled heat and mass transfer in Darc-Forchheimer mixed convection from a vertical flat plate embedded in a fluid-saturated porous medium under the effects of radiation and viscous dissipation, Journal of the Korean Phsical Societ, 48(3), (6), [8]. M. A. Seddek and F. A. Salema, The effects of temperature dependent viscosit and thermal conductivit on unstead MHD convective heat transfer past a semi-infinite vertical porous moving plate with variable suction, Computational Material Science, 4(), (7), [9]. P. R. Sharma and G. Singh, Effects of variable thermal conductivit and heat source/sink on MHD flow near a stagnation point on linearl stretching sheet, Journal of Applied Fluid Mechanics, (), (9), 3-. []. I. J. Uwanta and S. Murtala, Heat and Mass Transfer flow past an infinite vertical plate with variable thermal conductivit, International Journal of Applied Information Sstem, 6(), (3), 6-4. []. K. Vafai and C. L. Tien, Boundar and inertia effects on flow and heat transfer in a porous media, International Journal of Heat Mass Transfer, 4 (98) DOI:.979/ Page