UNSTEADY MHD FORCED CONVECTION FLOW AND MASS TRANSFER ALONG A VERTICAL STRETCHING SHEET WITH HEAT SOURCE / SINK AND VARIABLE FLUID PROPERTIES

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1 International Research Journal of Engineering and echnology (IRJE) e-issn: Volume: Issue: 3 June-15.irjet.net p-issn: UNSEADY MHD FORCED CONVECION FLOW AND MASS RANSFER ALONG A VERICAL SRECHING SHEE WIH HEA SOURCE / SINK AND VARIABLE FLUID PROPERIES P. R. Sharma 1 Manisha Sharma and R. S. Yadav 3 1. Professor, Department of Mathematics, University of Rajasthan, Jaipur-34, India.. Research Scholar, Department of Mathematics, University of Rajasthan, Jaipur-34, India. 3. Assistant Professor, Department of Mathematics, University of Rajasthan, Jaipur-34, India *** Abstract- In this paper, unsteady magnetohydrodynamic forced convection flo of a viscous incompressible, electrically conducting fluid and mass transfer along a vertical porous stretching sheet is investigated, in the presence of heat source /sink ith variable viscosity and thermal conductivity. he governing coupled non-linear partial differential equations are reduced to ordinary differential equations using similarity transformation and solved numerically using the Runge-Kutta fourth order method along ith shooting technique. he effects of various flo parameters on the velocity, temperature and concentration distributions are analyzed and presented graphically. Skin-friction coefficient, Nusselt number and Sherood number are derived at the sheet, discussed numerically and their numerical values for various values of physical parameters are presented through tables. Key Words : MHD, variable viscosity, variable thermal conductivity, stretching sheet, heat source / sink. 1. Introduction he magnetohydrodynamics heat and mass transfer flo in the boundary layer induced by a moving surface in a fluid finds important applications in chemical engineering and meteorology. MHD thermal boundary layer flo ith variable fluid properties has received a great deal of attention due to its important roles and ide applications in geophysics and thermal insulation engineering. Erikson et al. (8) studied heat and mass transfer on a moving continuous plate ith suction and injection. Gehart and Pera (9) observed nature of vertical natural convection flos resulting from the combined buoyancy effects of thermal and mass diffusion. Chakrabarti and Gupta (4) investigated hydromagnetic flo and heat transfer over stretching sheet. Apelblat (1) presented mass transfer ith a chemical reaction of first order ith effects of aial diffusion. Forced convection over a flat plate submersed in a porous medium ith variable viscosity is investigated by Ling and Dybbs (16). Chen and Char (5) discussed heat transfer of a continuous stretching surface ith suction or bloing. Lai and Kulacki (17) analyzed effects of variable viscosity on convective heat transfer along a vertical surface in a saturated porous medium. Das et al. (7) studied effects of mass transfer on flo past an impulsively started infinite vertical plate ith constant heat flu and chemical reaction. Hossain and akhar (1) obtained radiation effect on mied convection along a vertical plate ith uniform surface temperature. Heat and mass transfer in the boundary layers on an eponentially stretching continuous surface as considered by Magyari and Keller (18). Hossain et al. (11) discussed the effect of radiation on free convection flo of fluid ith variable viscosity from a porous vertical plate. Heat and mass transfer on a laminar flo along a semi-infinite horizontal plate ith temperature dependent viscosity and chemical reaction as investigated by Ghay and Seddek (1). Seddeek and Salama (6) analyzed the effects of temperature dependent viscosity and thermal conductivity on unsteady MHD convective heat transfer past a semi infinite vertical porous moving plate ith variable suction. Mukhopadhyay and Layek () found effects of thermal radiation and variable fluid viscosity on free convection flo and heat transfer past a porous stretching surface. Mukhopadhyay (1) presented unsteady boundary layer flo and heat transfer past a porous stretching sheet in presence of variable viscosity and thermal diffusivity. Olajuon (5) analyzed convection heat and mass transfer in an electrical conducting poer la flo over a heated vertical porous plate. Rahman and Salahuddin (4) 15, IRJE.NE- All Rights Reserved Page 131

2 International Research Journal of Engineering and echnology (IRJE) e-issn: Volume: Issue: 3 June-15.irjet.net p-issn: studied hydromagnetic heat and mass transfer flo over an inclined heated surface ith variable viscosity and electric conductivity. Dual solutions in boundary layer flo stagnation-point flo and mass transfer ith chemical reaction past a stretching/ shrinking sheet as studied by Bhattacharyya (3). Hunsain et al. (13) analyzed heat and mass transfer in unsteady boundary layer flo through porous media ith variable viscosity and thermal diffusivity. Makinde (19) discussed effects of variable viscosity on boundary layer over a permeable flat plate ith radiation and a convective surface boundary condition. Nadeem et al. (3) observed MHD three dimensional casson fluid flo past a porous linear stretching sheet. Conjugated forced convection heat transfer from a heated flat plate of finite thickness and temperature dependent thermal conductivity as analyzed by Mohammed and Nourazar (). Chen (6) studied mied convection unsteady stagnation-point flo toards a stretching sheet ith slip effects. he objective of the paper is to investigate effect of variable viscosity and thermal conductivity on unsteady MHD forced convection and mass transfer flo of a viscous incompressible, electrically conducting fluid along a porous stretching vertical sheet in the presence of heat source/sink.. Formulation of the Problem he -ais is oriented about the vertical plate in the upard direction and y-ais is normal to the plate. Unsteady to dimensional incompressible viscous fluid flos on a heated vertical porous stretching plate in the region y is considered. he sheet is stretching in its on plane ith velocity U, a t 1t.... (1) a is the stretching parameter and is the unsteadiness parameter and both have dimensions of time -1. he temperature, that of the ambient medium and, t of the sheet is different from C t is concentration distribution near the sheet and both vary ith time t and that distance along the sheet. It is assumed that the eternal electric field is zero and Hall effects are negligible. It is also assumed that the induced magnetic field is negligibly small. he level of concentration of foreign mass is assumed to be lo, so that the Soret and Dufour effects are negligible. he fluid velocity and thermal conductivity are assumed to vary linearly ith temperature. he system influenced by an eternal transverse magnetic field of strength B defined as B t B t 1/ ( ) (1 )....() he volumetric rate of heat generation/absorption is given as Q t Q t 1 ( ) (1 ).... (3) Under above assumptions, the governing equations of continuity, momentum, energy and concentration are given by u v, y...(4) * u u u 1 u * u u v g t y y y y * B g C C u,... 5 * 1 * u v t y CP y y y Q,... 6 C p C C C C u v D, t y y...(7) here u and v are the velocity components along the and y directions respectively, is the density of 15, IRJE.NE- All Rights Reserved Page 13

3 International Research Journal of Engineering and echnology (IRJE) e-issn: Volume: Issue: 3 June-15.irjet.net p-issn: the fluid, g is the gravitational acceleration, is the thermal epansion coefficient, is the concentration epansion coefficient, is the electrical conductivity, is fluid temperature inside the thermal boundary layer, C is the species concentration in boundary layer, is the temperature far aay from the sheet, C is the species concentration far aay from the sheet. C p is the specific heat at constant pressure, is the variable thermal conductivity, D is the mass diffusion coefficient. Variation of the viscosity and thermal conductivity ith temperature are assumed to be of the form given belo b b1, 1 d,... (8)... (9) here is the constant value of coefficient of viscosity far aay from the plate, b, b1 are constants is the conductivity of the fluid at temperature, d is the parameter that depends on nature of the fluid. he corresponding boundary conditions are given by y : u U (, t), v v ( t), (, t), C C (, t) y : u,, C C. * 1/ a...(1) Here, v t V is the suction velocity, 1t a, 1 3/ t t is the temperature of * a the sheet, C, 1 3/ t C t is the * concentration distribution near the sheet, V is the Cross- * flo velocity of the fluid and / is the kinematic viscosity. It is implicitly assumed that the mathematical problem is defined only for. 3. Method of Solution Introducing the similarity variable, dimensionless functions f, and, and physical parameters as given belo 1/ 1/ * a 1/ a * 1 t y, f ( ) (, y, t), 1t ( ) a ( ), 1t ( C C ) a ( ), C C 1t C C U 3/ 3/ B (1 t) A / a, b1, M, a g g C C Gr, Gm, U * * Q 1t Cp S, Pr, Sc ; ac D p...(11) here (, y, t) is the physical stream function. Stream function assures mass conservation automatically. he velocity components are obtained as u a t f y 1 1,... 1 * 1/ v a 1 t 1/ f.... (13) 15, IRJE.NE- All Rights Reserved Page 133

4 International Research Journal of Engineering and echnology (IRJE) e-issn: Volume: Issue: 3 June-15.irjet.net p-issn: Substituting (11) into (5) to (7), e obtain d / f fs A f ( / ) f ( f ) ff b f f Mf Gr Gm, d Pr A 3 / 3 / / f f Sc A here prime indicates differentiation ith respect to, A is the dimensionless measure of the unsteadiness, is the temperature-dependent viscosity parameter, M is the magnetic parameter, Gr is the Grashof number, Gm is the modified Grashof number, S is the heat generation/absorption parameter, Pr is the Prandtl number and Sc is the Schmidt number. he corresponding boundary conditions are reduced to : f ( ) V, f ( ) 1, ( ) 1, ( ) 1; : f ( ), ( ), ( )....(17) In order to obtain numerical solution of the equations (14) to (16) under the boundary condition (17) the problem is transformed into a system of first order equations as given by f f, f f, f f, f f, f, f, f, f, f, f, f f f Mf Grf Gmf f f (18) f A f / f3 / b f4,... (19) f 5 APr 3 / f4 / f5 df5 Sf f f f f df Pr / () f 7 Sc A 3/ f6 / f7 f f6 f1 f 7....(1) he corresponding boundary conditions are reduced to : f V, f 1, f 1, f 1; : f, f, f () o solve eq. (19), () and (1) ith boundary conditions (), as an initial value problem e need the values of f, f and f i e f and ,, but no such values are given. he initial guess values for f, and are chosen and using the fourth order Runge-Kutta method, the values are obtained. We compare the calculated values of f, and at a finite value of ith the given boundary conditions f values f, and,, and adjust the to give a better approimation for the solution. he step-size is taken as.1. he process is repeated until e obtain results correct up to the desired accuracy level of the criterion of convergence. 4. Skin friction Coefficient he skin friction coefficient at the sheet is defined as C f U 1/ Re f,...(3) 5 1 as 15, IRJE.NE- All Rights Reserved Page 134

5 International Research Journal of Engineering and echnology (IRJE) e-issn: Volume: Issue: 3 June-15.irjet.net p-issn: u y here and Re y U is the Reynolds number. * 5. Nusselt Number is the shear stress at the sheet he rate of heat transfer in terms of Nusselt number at the surface of sheet is given by Nu q here q y y the sheet. 6. Sherood Number, 1/ Re (),... (4) is the rate of heat transfer at he rate of mass transfer in terms of Sherood number at the surface of sheet is given by Sh m D C C 1/ Re (),...(5) C here m D, is the rate of mass transfer at y y the sheet. 7. Results and Discussion In order to investigate the behavior of velocity, temperature, species concentration, skin-friction coefficient at the sheet, rate of heat transfer in terms of Nusselt Number at the sheet and rate of mass transfer in terms of Sherood Number at the sheet, a comprehensive numerical computation is carried out for various values of parameters that describe the flo characteristics and the results are reported in terms of graphs and tables, discussed numerically and eplained physically. Figures 1 and, respectively represent that fluid velocity increase due to increase in Grashof number or modified Grashof number. It is noted from figure 3 that fluid velocity decreases ith increase in Hartmann number. Figure 4 illustrates that fluid velocity decreases near the plate ith an increase in temperature dependent viscosity parameter but the reverse behavior is seen aay from the plate. It is observed from figure 5 that fluid velocity increases due to increase in unsteadiness parameter. Figure 6 depicts that fluid velocity decreases ith an increase in cross flo velocity of fluid. Figures 7 reveals that fluid velocity increases due to increase in parameter d. It is observed from Figure 8 that fluid temperature decreases due to increase in Prandtl number. Figure 9 represents that fluid temperature increases due to increase in parameter d. Figures 1 and 11, respectively sho that fluid temperature decreases due to increase in unsteadiness parameter or cross flo velocity of fluid. Figure 1 illustrates that fluid temperature increases due to increase in temperature dependent viscosity.it is seen from figure 13 that fluid temperature increases ith heat source hile decreases ith heat sink. Figure 14 that species concentration decreases due to increase in Schmidt number. Figures 15 and 16 respectively illustrates that species concentration decreases due to increase in unsteadiness parameters or cross flo velocity of fluid. Figure 17 shos that concentration profiles increase near the plate ith an increase in temperature dependent viscosity parameter but the reverse behavior is seen aay from the plate. able 1 depicts that Skin-friction coefficient at the sheet increases due to increase in Grashof number, modified Grashof number or parameter d, hile it decreases due to increase in Hartmann number, temperature dependent viscosity parameter, unsteadiness parameter or cross flo velocity of fluid. able shos that Nusselt number at the sheet increases due to increase in cross flo velocity of fluid, unsteadiness parameter, heat sink or Prandtl number, hile it decreases due to increase in temperature dependent viscosity parameter, parameter d or heat source. able 3 illustrates that Sherood number at the sheet increases due to increase in cross flo velocity of fluid, unsteadiness parameter, Schimdt number hile it decreases due to increase in temperature dependent viscosity parameter. 8. Conclusions he findings of the numerical results can be summarized as follos: 1. Grashof number, modified Grashof number, Unsteadiness parameter or parameter d accelerate fluid velocity, hereas Hartmann 15, IRJE.NE- All Rights Reserved Page 135

6 International Research Journal of Engineering and echnology (IRJE) e-issn: Volume: Issue: 3 June-15.irjet.net p-issn: number or cross flo velocity of fluid retards fluid velocity.. Increase in temperature dependent viscosity parameter decreases fluid velocity near the sheet but increases far aay from the sheet. 3. Increase in temperature dependent viscosity parameter or parameter d lead to increases in fluid temperature. 4. Prandtl number, unsteadiness parameter or cross flo velocity of fluid retard fluid temperature. 5. Heat source tends to enhance fluid temperature hereas heat sink has reverse effect on it. 6. Species concentration decreases due to increase in Schmidt number, cross flo velocity of fluid retards fluid velocity or unsteadiness parameter. Figure Velocity profiles versus for different values of Gm hen Gr.1, A.1, M 1, 1, d, Pr 3, S.5, Sc.3, and V Increase in temperature dependent viscosity parameter increases species concentration near the sheet but have reverse effect far aay from the sheet. Figure 1 Velocity profiles versus for different Valuesof Gr hen A.1, Gm.1, M 1, 1, d, Pr 3, S.5, Sc.3 and V 1. Figure 3 Velocity profiles versus for different valuesof M hen Gr.1, Gm.1, A.1, 1, d, Pr 3, S.5, Sc.3and V 1. 15, IRJE.NE- All Rights Reserved Page 136

7 International Research Journal of Engineering and echnology (IRJE) e-issn: Volume: Issue: 3 June-15.irjet.net p-issn: Figure 4 Velocity profiles versus for different valuesof hengr.1, Gm.1, A.1, M 1, d, Pr 3, S.5, Sc.3 and V 1. Figure 6 Velocity profiles versus for different valuesofvhen Gr.1, Gm.1, A.1, M 1, d, Pr 3, S.5, Sc.3 and 1. Figure 5 Velocity profiles versus for different valuesof A hen Gr.1, Gm.1, 1, M 1, d, Pr 3, S.5, Sc.3 and V 1. Figure7 Velocity profiles versus for different values of d hen Gr.1, Gm.1, A.1, M 1, 1, Pr 3, S.5, Sc.3 and V 1. 15, IRJE.NE- All Rights Reserved Page 137

8 International Research Journal of Engineering and echnology (IRJE) e-issn: Volume: Issue: 3 June-15.irjet.net p-issn: Figure8 emperature profiles versus for different values of Pr hen Gr.1, Gm.1, A.1, M 1, d, 1, S.5, Sc.3 and V 1. Figure1 emperature profiles versus for different values of A hen Gr.1, Gm.1, d, M 1, 1, Pr 3, S.5, Sc.3 and V 1. Figure 9 emperature profiles versus for different values of d hen Gr.1, Gm.1, A.1, M 1, 1, Pr 3, S.5, Sc.3 and V 1. Figure 11 emperature profiles versus for different values of Vhen Gr.1, Gm.1, A.1, M 1, 1, Pr 3, S.5, Sc.3 and d. 15, IRJE.NE- All Rights Reserved Page 138

9 International Research Journal of Engineering and echnology (IRJE) e-issn: Volume: Issue: 3 June-15.irjet.net p-issn: Figure14 Concentration profiles versus for different values of Sc hen Gr.1, Gm.1, A.1, M 1, 1, Pr 3, S.5, d and V 1. Figure1 emperature profiles versus for different values of hen Gr.1, Gm.1, A.1, M 1, d, Pr 3, S.5, Sc.3 and V 1. Figure 15 Concentration profiles versus for different values of A hen Gr.1, Gm.1, Sc.3, M 1, 1, Pr 3, S.5, d and V 1. Figure13 emperature profiles versus for different values of S hen Gr.1, Gm.1, A.1, M 1, 1, Pr 3, d, Sc.3 and V 1. Figure 16 Concentration profiles versus for different values of Vhen Gr.1, Gm.1, A.1, M 1, 1, Pr 3, S.5, d and Sc.3. 15, IRJE.NE- All Rights Reserved Page 139

10 International Research Journal of Engineering and echnology (IRJE) e-issn: Volume: Issue: 3 June-15.irjet.net p-issn: V d A S Pr Figure 17 Concentration profiles versus for different values of hen Gr.1, Gm.1, A.1, M 1, Sc.3, Pr 3, S.5, d and V 1. able-1: Numerical values of skin friction coefficient at the sheet for various values of physical parameters. Gr Gm M A V d f able -: Numerical values of Nusselt number at the sheet for various values of physical Parameters able -3: Numerical values of Sherood number at the sheet for various values of physical parameters. V A Sc References Apelblat A., Mass transfer ith a chemical reaction of the first order. Effects of aial diffusion. he Chemical Engineering Journal, Vol. 3, 198, pp Bansal, J. L., Viscous Fluid Dynamics. Oford & IBH Pub. Co., Ne Delhi, India Bhattacharya, K. Dual solutions in boundary layer flo stagnation-point flo and mass transfer ith chemical reaction past a stretching / shrinking sheet. Int. Commun Heat Mass ransf, Vol. 38, 11, pp , IRJE.NE- All Rights Reserved Page 133

11 International Research Journal of Engineering and echnology (IRJE) e-issn: Volume: Issue: 3 June-15.irjet.net p-issn: Chakrabarti, A. and Gupta A. S., Hydromagnetic flo and heat transfer over stretching sheet. Quarterly Journal of Mechanics and Applied Mathematics,Vol. 37, 1979, pp Chen, C. K. and Char, M. I., Heat transfer of a continuous stretching surface ith suction or bloing. J. Math. Anal. Appl., Vol. 135, 1988, pp Chen, H., Mied convection unsteady stagnationpoint flo toards a stretching sheet ith slip effects. Mathematical problems in Engineering, Vol. 14, Article ID , 7 pages. 13. Hunsain S. ; Mehmood, A. and Ali, A., Heat and mass transfer analysis in unsteady boundary layer flo through porous media ith variable viscosity and thermal diffusivity. Journal of Appl. Mechanics and ech. physics, Vol. 53, 1, pp Jain, M. K., Numerical Solution of Differential Equations. Ne Age Int. Pub., Ne Delhi. 15. Jain, M. K. ; Iyengar, S. R. and Jain, R. K., Numerical Methods for Scientific and Engineering Computation, Wiley Eastern Ltd., Ne Delhi, India Das U. N. ; Deka R. and Soundalgekar, V. M., Effects of mass transfer on flo past an impulsively started infinite vertical plate ith constant heat flu and chemical reaction. Forschung im Ingenieuresen, Vol. 6, 1994, pp Erikson L. E. ; Fan L.. and Fo, V. G., Heat and Mass transfer on a moving continuous plate ith suction and injection. Ind. Eng. Chem. Fundamental, Vol ,pp Gehart, B. and Pera, L., he nature of vertical natural convection flos resulting from the combined buoyancy effects of thermal and mass diffusion. International Journal of Heat Mass ransfer, Vol. 14, 1971, pp Ghay, A. Y. and Seddek, M. A., Chebyshev finite difference method for the effects of chemical reaction. Heat and Mass transfer on laminar flo along a semi-infinite horizontal plate ith temperature dependent viscosity. Chaos Solitons Fractas, Vol. 19, 4, pp Hossain, M. A.; Khanafer, K. and Vafai, K., he effect of radiation on free convection flo of fluid ith variable viscosity from a porous vertical plate. Int. J. herm. Sci., Vol. 4, 1, pp Hossain, M. A. and akhar, H. S., Radiation effect on mied convection along a vertical plate ith uniform surface temperature. Int. J. Heat mass ransfer, Vol. 31, 1996, pp Ling, J. X. and Dybbs, A. Forced convection over a flat plate submersed in a porous medium: Variable viscosity case. ASME Winter Annual meeting, Boston, 1987, pp Lai, F. C. and Kulacki, F. A., he effects of variable viscosity on convective heat transfer along a vertical surface in a saturated porous medium. Int. J. Heat Mass transfer, Vol. 33, 199, pp Magyari, E. and Keller, B., Heat and mass transfer in the boundary layers on an eponentially stretching continuous surface. J. Phys. D Appl Phys, Vol. 3, 1999, pp Makinde, O. D., Effects of variable viscosity on boundary layer over a permeable flat plate ith radiation and a convective surface boundary condition. Journal of Mechanical Science and echnology, 6(5), 1, Mohammed, R. H. and Nourazar, S., Conjugated forced convection heat transfer from a heated flat plate of finite thickness and temperature dependent thermal conductivity. Heat ransfer Engineering, Vol. 35, 14, pp Mukhopadhyay, S., Unsteady boundary layer flo and heat transfer past a porous stretching sheet in presence of variable viscosity and thermal diffusivity. Int. J. of Heat and Mass ransfer, Vol. 5, 9, pp Mukhopadhyay, S. and Layek, G.C., Effects of thermal radiation and variable fluid viscosity on free convection flo and heat transfer past a 15, IRJE.NE- All Rights Reserved Page 1331

12 International Research Journal of Engineering and echnology (IRJE) e-issn: Volume: Issue: 3 June-15.irjet.net p-issn: porous stretching surface. Int. J. of Heat and Mass ransfer, Vol. 51, 8, pp Nadeem, S. ; Rizan, H. ; Noreen, S. A. and Khan, Z. H., MHD three dimensional casson fluid toards past a porous linearly stretching sheet. Aleandria Engineering Journal, 13, Vol. 5, pp Olajuon, B., Convection heat and mass transfer in an electrical conducting poer la flo over a heated vertical porous plate. International Journal for Computational Methods in Engineering Mechanics, Vol. 11, 1, pp Rahman, M. M. and Salahuddin K. M., Study of hydromagnetic heat and mass transfer flo over an inclined heated surface ith variable viscosity and electrical conductivity. Comm. Non-lnear Sci. Numer. Simulat., Vol. 15, 1, pp Seddeek, M.A. and Salama, F.A., he effects of temperature dependent viscosity and thermal conductivity on unsteady MHD convective heat transfer past a semi-infinite vertical porous moving plate ith variable suction. Compt. Mater. Sci., Vol. 4, 7, pp Spurk J. H. and Aksel N., Fluid Mechanics, Second ed. Springer, Germany, 8. 15, IRJE.NE- All Rights Reserved Page 133