Dhaka University of Engineering and Technology, (DUET), Gazipur-1700, Bangladesh 2 Department of Mathematics

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1 ANALYSIS OF MHD FREE CONVECTION FLOW ALONG A VERTICAL POROUS PLATE EMBEDDED IN A POROUS MEDIUM WITH MAGNETIC FIELD AND HEAT GENERATION M. U. Ahammad, Md. Obayedullah and M. M. Rahman Department of Mathematics Dhaka University of Engineering and Technology, (DUET), Gazipur-700, Bangladesh Department of Mathematics Bangladesh University of Engineering and Technology, (BUET), Dhaka-000, Bangladesh main3737@gmail.com ABSTRACT The problem of two-dimensional steady laminar MHD free convection flow over a vertical porous flat plate immersed in porous medium has been studied numerically in the presence of magnetic field and heat generation effect. The resulting momentum, energy and concentration equations have been transformed into a system of ordinary differential equations by using a suitable similarity transformation. The obtained equations are then solved numerically by applying the Nachtsheim-Swigert shooting iteration technique along with the sixth-order Runge-Kutta integration scheme. A parametric study is presented graphically to point up the influence of different physical parameters involving the problem. Finally, the numerical data for the local skin-friction coefficient (Cf x ), the local Nusselt number (Nu x ) and the local Sherwood numbers (Sh x ) also have been tabulated. It is found that the pertinent parameters play a significant role on velocity, temperature and concentration fields. Keywords: Vertical plate; Porous medium; Magnetic field; Heat generation. Nomenclature b Stretching rate f w Dimensionless suction velocity B 0 Magnetic field intensity g Acceleration due to gravity C C w C C p Concentration of the fluid Fluid concentration at the surface Fluid concentration in the free stream Specific heat at constant pressure k Gr M Nu Darcy permeability constant Grashof number Magnetic field parameter Nusselt number Da Darcy number Pr Prandtl number D m Molecular diffusivity Q 0 Volumetric rate of heat generation F Magnetic force Q Heat generating Fs Local Forchheimer number Re parameter Reynolds number Sc Schmidt number φ Dimensionless concentration Sh Sherwood number λ Thermal conductivity of T Fluid temperature Similarity variable T w Fluid temperature at the surface ν Kinematic viscosity T Fluid temperature in free stream θ Dimensionless temperature u, v Fluid velocity components in the ρ Density of the fluid x and y-direction respectively x, y Cartesian coordinates along σ Electrical conductivity the plate and normal to it respectively α Thermal diffusivity Mass buoyancy parameter β Coefficient of thermal expansion w Condition at surface γ Temperature buoyancy parameter Condition at infinity.0 INTRODUCTION The study of free convection flows, which occur in nature have become a potential topics in recent years due to its applications in geophysical and industrial fields such as chemical engineering process or drying process. In many metallurgical processes involving cooling of continuous strips, the rate of cooling can be controlled by the use of electrically conducting fluids and the use of the magnetic fields. The practical importance of heat generation or absorption effects is followed in some physical problems such fluids undergoing exothermic or endothermic chemical reactions. Raptis and Perdikis (006) investigated viscous flow for a non-linearly stretching sheet with chemical reaction and magnetic field. Samad et al. (00) studied natural convection flow through a porous medium considering the effect of magnetic field with thermal radiation, viscous dissipation and 0

2 variable suction. The influence of magnetic field on heat and mass transfer by natural convection from vertical surfaces in porous media with Soret and Dufour effects has been carried out by Postelnicu (004). Ahammad and Mollah (0) performed the MHD free convection flow and mass transfer problem over a stretching sheet considering Dufour & Soret effects with magnetic field. In the presence of a magnetic field natural convection for an electrically conducting fluid was analyzed by Lykoudis (96). Gupta (96) discussed steady and transient free convection of an electrically conducting fluid from a vertical plate in the presence of magnetic field. Ravikumar et al. (03) carried out a steady free convective and mass transfer flow of an electrically conducting viscous fluid through a porous medium bounded by two vertical plates. Dufour and Soret effects on free-forced convection flow past a vertical porous plate immersed in a porous medium for a hydrogen-air mixture as the non-chemical reacting fluid pair was reported by Alam et al. (006). Further Alam et al. (007) took into account the viscous dissipation effects on MHD natural convection boundary layer flow over a sphere of an electrically conducting fluid in the presence of heat generation. They reported that for rising values of heat generation parameter, the skin-friction coefficient increases whereas the Nusselt number decreases significantly within the boundary layer. Sattar and Kalim (996) presented unsteady free convection interaction in a boundary layer flow past a vertical porous plate with thermal radiation. Kairi and Murthy (03) investigated the Soret effect with the influence of variable viscosity on natural convection from a melting vertical surface in a non-darcy porous medium saturated with Newtonian fluid of variable viscosity. Seddeek and Salama (007) carried out the effects of temperature dependent viscosity and thermal conductivity with variable suction on unsteady MHD convective heat transfer past a vertical moving porous plate. Considering the internal heat generation/absorption and suction/blowing effects the flow and heat transfer of a fluid through a porous medium over a stretching surface was investigated by Cortell (005). Kafoussis (990) calculated local similarity solution for mixed convective and mass transfer flow past a semi-infinite vertical plate. Alam et al. (0) analyzed the effects of variable chemical reaction and variable electric conductivity on free convective flow with heat and mass transfer over a stretching sheet considering Dufour and Soret effects. Convection heat transfer of non-newtonian power-law fluids past a power-law stretched sheet with surface heat flux under the influence of magnetic field and suction/injection was performed by Chen (008). Anghel et al. (000) studied Dufour and Soret effects on free convection boundary layer over a vertical surface embedded in a porous medium. Alam et al. (006) investigated the mixed convection and mass transfer flow past a vertical porous plate in a porous medium in the presence of heat generation and thermal diffusion. Rahman et al. (00) presented the effects of Reynolds and Prandtl numbers on MHD mixed convection in a lid-driven cavity along with joule heating and a centered heat conducting circular block. Ahammad et al. (0) investigated the influence of inlet and outlet port in a ventilated cavity containing a heat generating square block. They found that the location of inlet and exit plays a significant role on both the flow and thermal fields. Alam et al. (0) studied the thermophoretic particle deposition on unsteady hydromagnetic radiative heat and mass transfer flow along an infinite inclined permeable surface with viscous dissipation and joule heating. Very recently Esmaeil Khaje et al. (03) analyzed the effect of heat generation on free convection boundarylayer flow over an arbitrarily impermeable inclined surface in a saturated porous medium. The present study investigates the effects of magnetic field and heat generation on MHD free convection flow over a vertical porous plate embedded in a porous medium.. 0 MATHEMATICAL ANALYSIS Assume a steady two-dimensional heat transfer flow of a viscous and incompressible electrically conducting fluid along a vertical stretching permeable sheet in a porous medium with heat generation. The flow is taken in the x-direction, which is along the plate in the upward direction while the y-axis is taken to be normal to the plate. A strong magnetic field is applied in the y- direction. The applied magnetic field of strength B 0 produces magnetic force B 0 F in x-direction, where is the electrical conductivity. Here we have considered that the induced magnetic field is

3 negligible. In the beginning, the plate and the fluid are taken into account at the same temperature T while C is the concentration all over the place in the fluid. The surface of the plate is maintained at a uniform constant temperature T w ( > T ) and concentration C w ( > C ), where T and C respectively are the corresponding values sufficiently far away from the flat surface. Using the Darcy-Forchhemier model together with the Boussnesq s and the usual boundary-layer approximations the current problem is governed by the continuity, momentum, energy and concentration equations respectively are given by: u v 0 x y () u u u Bu 0 u v g( T T ) x y y b u u k k () T T T cp u v Q0 T T x y y (3) C C C u v Dm x y y (4) where u, v are the fluid velocity components along the x and y directions respectively, is the kinematic viscosity, g is the acceleration due to gravity, is the density, k is the Darcy permeability constant, is the volumetric coefficient of thermal expansion, σ is the electrical conductivity, B0 is the uniform magnetic field strength, b is the stretching rate, λ is the thermal conductivity of fluid, cp is the specific heat at constant pressure, T is the fluid temperature inside the boundary layer, T is the fluid temperature in the free-stream, C is the concentration of the fluid within the boundary layer, Q0 is the volumetric rate of heat generation, Dm is the molecular diffusivity of the species concentration. The boundary conditions for the model are given as: u bx, v vw x, T Tw, C Cw at y 0, u 0, T T, C C as y. (5) where b is a constant called stretching rate and vw(x) represents the permeability of the porous surface where its sign indicates suction ( 0 ) or injection ( 0 ). Now, the following appropriate similarity variables [see Acharya et al. (999)] are introduced: u, v y x / / ( ), / b xf b y T T, = C C Tw T Cw C (6) Then equations ()-(4) yields the following equations ' ' f ff f Mf f Da Re Fs Da f 0 Prf PrQ 0 Scf 0 (7) (8) (9) where the dimensionless parameters are defined as follows: Gr Re B0 M is the magnetic field parameter, b is the temperature buoyancy parameter, is the Darcy number, x Da k x uw x Re is the Reynolds v b number, Fs is the Forchhemier number, x vpc p Pr is the Prandtl number, Q0 Q bpc p is the v heat generating parameter. Sc is the Schmidt number. D m The boundary conditions (5) becomes f fw, f,, at = 0 f 0, 0, 0 as (0) where fw = - vw /(b)/ is the dimensionless wall mass transfer coefficient such that fw 0 indicates wall suction and fw 0 indicates wall injection. For the present problem the local skin-friction coefficient, the local Nusselt number and the local Sherwood number are the parameters of engineering interest which are given respectively as below: Cf x x f Re 0

4 Nu x Re x Sh x Re x 0 0 Applying the boundary conditions (0), the system of equations (7) to (9) have been solved numerically by using the Nachtsheim-Swigert (965) shooting iteration technique with sixth-order Runge-Kutta integration scheme. Various groups of the parameters, γ, fw, M, Da, Fs, Q, Re Pr, Sc, were considered in different phases. In all the computations the step size = 0.0 was selected that satisfied a convergence criterion of 06 in almost all of different phases mentioned above. velocity, temperature and concentration decreases with increase of Darcy number and suction parameter. The imposition of the wall suction (fw>0) have the tendency to reduce both the momentum and thermal boundary layer thickness and thus the reduction in both the velocity and temperature profiles. Figures (a) to (c) display the effects of Forchhemier number and Prandtl number on the velocity, temperature and concentration profiles. Figure (a) and Figure (b) indicate that the velocity profiles and the temperature profiles decrease with the increase of Forchhemier number and Prandtl number. But it is seen from Figure (c) that the concentration field increase as Forchhemier and Prandtl number increase. For verifying the accuracy of our code, the present result has been compared with Kafoussias (990), when M = Df = Sr = 0 (see Table-). This table shows an excellent agreement between them. Table Comparison of local skin-friction coefficient (Cf x ) and local Nusselt number (Nu x ) with Kafoussias (990) for M = Df = Sr = 0 γ Kafoussias 990 (Cf x ) Present (Cf x ) Kafoussias 990 (Nu x ) Present (Nu x ) Fig. (a) velocity profiles for different values of Da and fw 3.0 RESULTS AND DISCUSSION The numerical calculations have been carried out for different values of the suction parameter (f w ), magnetic field parameter (M), Forchhemier number (Fs), heat source parameter (Q), local Darcy number (Da), Reynolds number (Re) in the form of non-dimensional velocity, temperature and concentration profiles. The value of Prandtl number (Pr) is taken to be 0.7 (air), Schmidt number (Sc) is chosen 0. (hydrogen). Due to free convection problem a positive large value of buoyancy parameter γ = 0 is chosen. The numerical results are displayed in Figures. to 5 and Tables to 4. The effects of local Darcy number and suction parameter on the velocity, temperature and concentration fields are shown in Figures (a) to (c) respectively. From these figures we observe that the Fig. (b) temperature profiles for different values of Da and fw 3

5 Fig. (c) concentration profiles for different values of Da and fw Fig. (c) concentration profiles for different values of Fs and Pr Fig. (a) velocity profiles for different values of Fs and Pr Fig. 3(a) velocity profiles for different values of γ and Q Fig. (b) temperature profiles for different values of Fs and Pr Fig. 3(b) temperature profiles for different values of γ and Q 4

6 Fig.3(c) concentration profiles for different values of γ and Q Fig. 4(c) concentration profiles for different values of M and Sc Fig. 5(a) velocity profiles for different values of Re Fig. 4(a) velocity profiles for different values of M and Sc Fig. 5(b) temperature profiles for different values of Re Fig. 4(b) temperature profiles for different values of M and Sc 5

7 show that the temperature and concentration profiles increase for rising values of Reynolds number. Table shows the numerical values of local skinfriction coefficient, local Nusselt number and local sherwood number for some values of the parameters M & Q. It is evident from this table that for fixed M and rising values of Q; all of Cfx, Nux and Shx increase. On the other hand it is found that Cfx, Nux and Shx decreases while M increases and Q=0.75. Fig. 5(c) concentration profiles for different values of Re The combined influences of buoyancy parameter and heat generation parameter on the fluid velocity, temperature and concentration profiles respectively are exposed in Figures 3(a) to 3(c). It is clearly seen that the velocity increases in Figure 3(a) but the temperature and concentration decreases in Figures 3(b) and 3(c) as the buoyancy force with heat generation parameter increases. This happened because the buoyancy parameter induces a favorable pressure gradient that enhances the fluid flow and heat transfer in the boundary layer. Also it is noticed that the buoyancy effect dominates the heat generation effect. Figures 4(a) to 4(c) depict the behavior of magnetic field parameter and Schmidt number on the velocity, temperature and concentration profiles. It is seen that infigure 4(a) the velocity profiles reduces for higher values of magnetic parameter and Schmidt number. But in Figures 4(b) and 4(c) the temperature and concentration fields increase with the increase of magnetic parameter. The presence of magnetic field in an electrically conducting fluid tends to create a body force against the flow. This type of resistive force tends to slow down the motion of the fluid in the boundary layer which, in turn, reduces the rate of heat convection in the flow and this appears in increasing the flow temperature. Finally, Figures 5(a) to 5(c) are aimed to display the velocity, temperature and concentration profiles for different values of Reynolds number. It is noticed from Figure 5(a) that velocity decreases sharply up to η =.9 with the increase of Reynolds number and after η =.9 the higher values of Reynolds number gives higher velocity. On the other hand, Figures 5(b) and 5(c) The numerical values of local skin-friction coefficient, local Nusselt number and local sherwood number for several values of the parameters Re & Pr are tabulated in Table-3. It is apparent from this table that for the growing values of Pr and at the flat value of Re; Cfx, and Shx reduce whereas Nux increases. Besides that, it follows that the values of Cfx, Nux and Shx shrink while Re increases and Pr is kept at 0.7. Table 4 gives an idea about the numerical values of local skin-friction coefficient, local Nusselt number and local sherwood number for some selected values of the parameters Da & Fs. The fact noted from this table is that for fixed Da and rising values of Fs; Cfx, and Shx increase, but Nux decreasse. On the other hand it is seen that Cfx decreases and Nux, Shx increases while Da enhances and Fs =0.. Table Effects of M and Q on the local skin-friction coefficient (Cfx), local Nusselt number (Nux) and local Sherwood number (Nux) for γ = 0, Pr =0.7, Sc = 0., r =.0, K=.0 and n =.0. M Q Cfx Nux Shx Table 3 Effects of Re and Pr on the local skin-friction coefficient (Cfx), local Nusselt number (Nux) and local Sherwood number (Shx) for γ = 0, Pr =0.7, Sc = 0., r =.0, K=.0 and n =.0. Re Pr Cfx Nux Shx

8 Table 4 Effects of Da and Fs on the local skin-friction coefficient (Cfx), local Nusselt number (Nux) and local Sherwood number (Shx) for γ = 0, Pr =0.7, Sc = 0., r =.0, K=.0 and n =.0. Da Fs Cfx Nux Shx CONCLUSION In this paper we have studied the magnetic field and heat generation effects numerically on a laminar steady MHD free convection flow over a vertical porous plate embedded in a porous medium. From the present study the followings can be summarized: Velocity boundary layer growth can be found by increasing heat generation parameter and by enhancing magnetic field parameter boundary layer thickness can be reduced. Temperature profiles can be elevated by increasing magnetic field parameter and for more cooling buoyancy parameter should be increased. Concentration fields have a significant effect of both magnetic field and heat generation parameters. REFERENCES Acharya M., Singh L.P. and Dash G.C., 999. Heat and mass transfer over an accelerating surface with heat source in the presence of blowing, Int. J. Eng. Sci., 37: 89. Ahammad M. U. and Shirazul Hoque Mollah Md., 0. Numerical study of MHD free convection flow and mass transfer over a stretching sheet considering Dufour & Soret effects in the presence of magnetic Field, International Journal of Engineering & Technology IJET-IJENS, (5): 4-. Ahammad M.U., Rahman M.M. and Rahman M.L., 0. Effect of inlet and outlet position in a ventilated cavity with a heat generating square block, Engineering e-transaction 7(): Alam M.A., Alim Md. and Chowdhury M.K., 007. Viscous Dissipation Effects on MHD Natural Convection Flow over a Sphere in the Presence of Heat Generation, Nonlinear Analysis: Modelling and Control, (4): Alam M. S. and Rahman M. M., 006. Dufour and Soret Effects on Mixed Convection Flow Past a Vertical Porous Flat Plate with Variable Suction, Nonlinear Analysis: Modelling and Control, (): 3. Alam M.S., Rahman M.M. and Samad M.A., 006. Numerical study of the combined free-forced convection and mass transfer flow past a vertical porous plate in a porous medium with heat generation and thermal diffusion, Nonlinear Anal. Model. Control, (4): Alam M.S. and Ahammad M.U., 0. Effects of variable chemical reaction and variable electric conductivity on free convective heat and mass transfer flow along an inclined stretching sheet with variable heat and mass fluxes under the influence of Dufour and Soret effects, Nonlinear Analysis: Modelling and Control, 6(): 6. Alam M.S. and Rahman M.M., 0. Thermophoretic particle deposition on unsteady hydromagnetic radiative heat and mass transfer flow along an infinite inclined permeable surface with viscous dissipation and joule heating, Engineering e-transaction 7(): 6-6. Anghel M., Takhar H.S. and Pop I., 000. Dufour and Soret effects on free convection boundary layer over a vertical surface embedded in a porous medium, Studia Universitatis Babes-Bolyai, Mathematica, XL, V:. Chen C.H., 008. Effects of magnetic field and suction/injection on convection heat transfer of non- Newtonian power-law fluids past a power-law stretched sheet with surface heat flux, Int. J.Therm. Sci., 47(7): Cortell R, 005. Flow and heat transfer of a fluid through a porous medium over a stretching surface with internal heat generation/absorption and suction/blowing, Fluid Dyn. Res., 37: Esmaeil Khaje, Kayhani M. H. and Sadi M., 03. Effect of heat generation on natural convection from an impermeable inclined surface embedded in a porous medium, Journal of Porous Media, 6(5): Gupta A. S., 96. Steady and transient free convection of an electrically conducting fluid from a vertical plate in the presence of magnetic field, Appl. Sci. Res., 9A: Kafoussis N.G., 990. Local similarity solution for combined free-forced convective and mass transfer flow past a semi-infinite vertical plate, Int. J. Energy Res., 4: Kairi R. R. and Murthy P.V.S.N., 03. Soret effect on free convection from a melting vertical surface in a non-darcy porous medium, Journal of Porous Media, 6(): Lykoudis P. S., 96. Natural convection of an electrically conducting fluid in the presence of a magnetic field, Int. J. Heat Mass Transfer, 5: Nachtsheim P. R. and Swigert P., 965. Satisfaction of the asymptotic boundary conditions in numerical 7

9 solution of the system of non-linear equations of boundary layer type, NASA TND Postelnicu A., 004. Influence of a magnetic field on heat and mass transfer by natural convection from vertical surfaces in porous media considering Soret and Dufour effects, Int. J. Heat Mass Transfer, 47: Rahman M.M., Billah M.M., Mamun M.A.H., Saidur R. and Hasanuzzaman M., 00. Reynolds and Prandtl numbers effects on mhd mixed convection in a liddriven cavity along with joule heating and a centered heat conducting circular block, International journal of mechanical and materials engineering, 5(): Raptis A. and Perdikis C., 006. Viscous flow over a non-linearly stretching sheet in the presence of a chemical reaction and magnetic field, International Journal of Non-linear Mechanics, 4(4): Ravikumar V., Raju M.C., Raju G.S.S. and Chamkha A.J., 03. MHD double diffusive and chemically reactive flow through porous medium bounded by two vertical plates, International Journal of Energy & Technology, 5(7): 8. Samad M.A., Karim M. E. and Mohammad D., 00. Free convection flow through a porous medium with thermal radiation, viscous dissipation and variable suction in presence of magnetic field, The Bangladesh Journal of Scientific Research, 3():