Indian Journal of Pure & Applied Physics Vol. 48, March 2010, pp. 157-165
Radiation and mass transfer effects on unsteady MHD free convective fluid flow embedded in a porous medium with heat generation/absorption B Shanker1, B Prabhakar Reddy2* & J Anand Rao3 1
Department of Mathematics, University College of Science, Osmania University, Hyderabad 500 007 2
Department of Mathematics (PG), AV P G Centre, Gaganmahal, Domalguda, Hyderabad 500 029
3
Department of Mathematics, University College of Science, Osmania University, Hyderabad 500 007 E-mail:
[email protected] Received 27 July 2009; revised 13 November 2009; accepted 4 January 2010
The numerical solution of unsteady two-dimensional, laminar, boundary layer flow of a viscous, incompressible, electrically conducting fluid along a semi-infinite vertical plate in the presence of thermal and concentration buoyancy effects under the influence of uniform magnetic field applied normal to the flow, has been obtained. A time-dependent suction is assumed and the radiative flux is described using the differential approximation for radiation. Asymptotic series expansion about a small parameter ε is performed to obtain the flow fields. The Galerkin finite element method is used to solve the equations governing the flow. The results obtained are discussed for cooling (Gr > 0) and heating (Gr < 0) of the plate with the help of graphs and tables to observe the effects of various parameters such as Pr, Sc, M, Q, K, Nr, kr, A, Gm and Gr. It has been found that when the thermal and solutal Grashof numbers increase, the thermal and concentration buoyancy effects are enhanced and the fluid velocity increases. Furthermore, when the Prandtl and Schmidt numbers increase, the thermal and concentration levels decrease resulting in reduced fluid velocity. These results are in good agreement with earlier results. Keywords: Buoyancy effects, Radiative heat flux, Heat generation/absorption, MHD, Galerkin finite element method
1 Introduction The problem of fluid flow in an electromagnetic field has been studied for its importance in geophysics, metallurgy and aerodynamic extrusion of plastic sheets and other engineering processes such as in petroleum engineering, chemical engineering, composite or ceramic engineering and heat exchangers. In the last five years, many investigations dealing with heat flow and mass transfer over a vertical porous plate with variable suction, heat absorption/generation or Hall current have been reported. Singh1 studied MHD free convection and mass transfer flows with Hall current, viscous dissipation, Joule heating and thermal diffusion. Azzam2 presented radiation effects on the MHD mixed free-fixed convective flow past a semi-infinite moving vertical plate for high temperature differences. Chamkha3 presented thermal radiation and buoyancy effects on hydro-magnetic flow over an accelerating permeable surface with heat source or sink. Chen4 studied heat and mass transfer with variable wall temperature and concentration. Cookey et al.5 investigated the influence of viscous dissipation and radiation on unsteady MHD free-convection flow past an infinite heated vertical plate in a porous
medium with time dependent suction. Hayat and Abbas6 presented the radiation effects on MHD flow in a porous space. Ogulu et al.7 studied unsteady MHD free convective flow of a compressible fluid past a moving vertical plate in the presence of radiative heat transfer. Ogulu and Mbeledogu8 studied heat and mass transfer of an unsteady MHD natural convection flow of a rotating fluid past a vertical porous flat plate in the presence of radiative heat transfer. Pilani and Ganesan9 presented finite difference analysis of unsteady natural convection MHD flow past an inclined plate with variable surface heat and mass flux. Recently, Prakash et al.10 have studied MHD free convection and mass transfer flow of a micro-polar thermally radiating and reacting fluid with time dependent suction. Raptis11 discussed the flow of a micro polar fluid past a continuously moving plate in the presence of radiation. Sunitha et al.12 presented radiation and mass transfer effects on MHD free convection flow past an impulsively started isothermal vertical plate with dissipation. In the present paper, effect of thermal radiation, time-dependent suction and chemical reaction on the two-dimensional flow of an incompressible Boussinesq fluid in the presence of uniform magnetic
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field applied normal to the flow has been studied. The problem is governed by the system of coupled nonlinear partial differential equations whose exact solutions are difficult to obtain, if possible. So, Galerkin finite element method has been adopted for its solution, which is more economical from computational point of view.
All the variables are defined in the nomenclature. It is assumed that the temperature differences within the flow are sufficiently small so that T4 can be expanded in a Taylor series about the free stream temperature T∞ so that after rejecting the higher order terms:
2 Theory
The energy equation after substitution of Eqs (5) and (6) can now be written as:
T 4 ≈ 4T∞3T − 3T∞4
...(6)
2.1 Mathematical formulation of the problem
Consider the problem of unsteady two-dimensional, laminar, boundary layer flow of a viscous, incompressible, electrically conducting fluid along a semi-infinite vertical plate in the presence of thermal and concentration buoyancy effects. Time dependent suction is considered normal to the flow. The x′-axis is taken along the plate in the direction of the flow and y′-axis normal to it. Further, due to the semiinfinite plane surface assumption the flow variables are the functions of normal distances y′ and t′ only. A uniform magnetic field is applied normal to the direction of the flow. Now, under the usual Boussinesq’s approximation, the governing equations of the flow problem are:
∂T ∂T k ∂ 2T 16σ∗T∞3 ∂ 2T + v′ = + + Q0 (T − T∞ ) ∂t ′ ∂y ′ ρcp ∂y ′2 3ρcp k ∗ ∂y ′2 ...(7) From Eq. (1) one can see that the suction is a function of time only, so it is assumed in the form5: v′ = −V0 (1 + εAe n ′t ′ )
where A is the suction parameter and εA 0). It is observed that an increase in Pr, Sc, kr and M leads to decrease in the velocity field while an increase in Q, Nr, K, Gr and
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Fig. 4 — Effect of Pr on velocity field u for cooling of the plate when Gr = 5, Gm = 5, Sc = 0.22, Q = 1.0, Nr = 0.5, kr = 0.5, M = 0.5, K = 1.0, A = 0.3, ε = 0.02, n = 0.5, t =1.0
Fig. 5 — Effect of Sc on velocity field u for cooling of the plate when Gr = 5, Gm = 5, Pr = 0.71, Q = 1.0, Nr = 0.5, kr = 0.5, M = 0.5, K = 1.0, A = 0.3, ε = 0.02, n = 0.5, t =1.0
Fig. 2 — Effect of Q on temperature field θ when Pr = 0.71, Nr = 0.5, A = 0.3
Fig. 3 — Concentration profiles
Fig. 6 — Effect of Q on velocity field u for cooling of the plate when Gr = 5, Gm = 5, Pr = 0.71, Sc = 0.22, Nr = 0.5, kr = 0.5, M = 0.5, K = 1.0, A = 0.3, ε = 0.02, n = 0.5, t =1.0
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Fig. 7 — Effect of Nr and A on velocity field u for cooling of the plate when Gr = 5, Gm = 5, Pr = 0.71, Sc = 0.22, Q = 1.0, kr = 0.5, M = 0.5, K = 1.0, ε = 0.02, n = 0.5, t =1.0
Fig. 8 — Effect of kr and A on velocity field u for cooling of the plate when Gr = 5, Gm = 5, Pr = 0.71, Sc = 0.22, Q = 1.0, Nr = 0.5, M = 0.5, K = 1.0, ε = 0.02, n = 0.5, t =1.0
Fig. 9 — Effect of M and K on velocity field u for cooling of the plate when Gr = 5, Gm = 5, Pr = 0.71, Sc = 0.22, Q = 1.0, Nr = 0.5, kr = 0.5, A = 0.3, ε = 0.02, n = 0.5, t =1.0
Fig. 10 — Effect of Gr on velocity field u for cooling of the plate when Gm = 5, Pr = 0.71, Sc = 0.22, Q = 1.0, Nr = 0.5, kr = 0.5, M = 0.5, K = 1.0, A = 0.3, ε = 0.02, n = 0.5, t =1.0
Fig. 11 — Effect of Gm on velocity field u for cooling of the plate when Gr = 5, Pr = 0.71, Sc = 0.22, Q = 1.0, Nr = 0.5, kr = 0.5, M = 0.5, K = 1.0, A = 0.3, ε = 0.02, n = 0.5, t =1.0
Gm leads to increase in the velocity field. From Figs 7 and 8, it is also observed that an increase in the suction parameter A leads to decrease in the velocity field. A comparison of velocity field curves due to cooling of the plate shows that velocity increases rapidly near the plate and after attaining a maximum value, it decreases as y increases. Figures 12 to19 show the effects of material parameters Pr, Sc, Q, Nr, kr, A, M, K, Gr and Gm, when the plate is heated by free-convection currents (Gr < 0). It is observed that an increase in Pr, M, Gr and Gm leads to increase in the velocity field while an increase in Sc, Q, Nr, kr and K leads to decrease in the velocity field. From Figs 15 and 16, it can be seen that an increase in the suction parameter A leads to increase in the velocity field. Table 1 presents numerical values of the skinfriction coefficient (τ) for variations in Pr, Sc, Q, Nr, kr, A, M, K, Gr and Gm, for cooling of the plate
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Fig. 12 — Effect of Pr on velocity field u for heating of the plate when Gr = -5.0, Gm = 5, Sc = 0.22, Q = 1.0, Nr = 0.5, kr = 0.5, M = 0.5, K = 1.0, A = 0.3, ε = 0.02, n = 0.5, t =1.0
Fig. 15 — Effect of Nr and A on velocity field u for heating of the plate when Gr = −5.0, Gm = 5, Pr = 0.71, Sc = 0.22, Q = 1.0 kr = 0.5, M = 0.5, K = 1.0, ε = 0.02, n = 0.5, t =1.0
Fig. 13 — Effect of Sc on velocity field ufor heating of the plate when Gr = -5.0, Gm = 5, Pr = 0.71, Q = 1.0, Nr = 0.5, kr = 0.5, M = 0.5, K = 1.0, A = 0.3, ε = 0.02, n = 0.5, t =1.0
Fig. 16 — Effect of kr and A on velocity field u for heating of the plate when Gr = -5.0, Gm = 5, Pr = 0.71, Sc = 0.22, Q = 1.0 Nr = 0.5, M = 0.5, K = 1.0, ε = 0.02, n = 0.5, t =1.0
Fig. 14 — Effect of Q on velocity field u for heating of the plate when Gr = -5.0, Gm = 5, Pr = 0.71, Sc = 0.22, Nr = 0.5, kr = 0.5, M = 0.5, K = 1.0, A = 0.3, ε = 0.02, n = 0.5, t =1.0
Fig. 17 — Effect of M and K on velocity field u for heating of the plate when Gr = −5.0, Gm = 5, Pr = 0.71, Sc = 0.22, Q = 1.0, Nr = 0.5, kr = 0.5, A = 0.3, ε = 0.02, n = 0.5, t =1.0
(Gr < 0). It is observed that, an increase in Pr, Sc, kr and M leads to decrease in the value of skin-friction coefficient while an increase in Q, Nr, K, Gr and Gm leads to increase in the value of skin-friction coefficient.
Table 2 presents numerical values of the skinfriction coefficient (τ) for variations in Pr, Sc, Q, Nr, kr, A, M, K, Gr and Gm for heating of the plate (Gr < 0). It is observed that, an increase in Pr, M, Gr and Gm leads to increase in the value of skin-friction
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Fig. 18 — Effect Gr on velocity field u for heating of the plate when Gm = 5, Pr = 0.71, Sc = 0.22, Q = 1.0, Nr = 0.5, kr = 0.5, M = 0.5, K = 1.0, A = 0.3, ε = 0.02, n = 0.5, t =1.0
coefficient while an increase in Sc, Q, Nr, kr and K leads to decrease in the value of skin-friction coefficient. Table 3 presents numerical values of heat transfer coefficient in terms of Nusselt number (Nu) for different values of the Prandtl number Pr, heat source parameter Q and thermal radiation Nr, respectively. It is observed that, an increase in the Prandtl number leads to decrease in the value of heat transfer coefficient while an increase in the heat source parameter or thermal radiation leads to increase in the value of heat transfer coefficient. Table 4 presents numerical values of mass transfer coefficient in terms of Sherwood number (Sh) for different values of Schmidt numberSc and chemical Table 3 — Heat transfer coefficient in terms of Nusselt number (Nu) Pr
Q
Nr
Nu
0.71 3.00 0.71 0.71
1.0 1.0 2.0 1.0
0.5 0.5 0.5 1.0
−0.375516 −1.237810 0.016614 −0.288008
Table 4 — Mass transfer coefficient in terms of Sherwood number (Sh)
Fig. 19 — Effect of Gm on velocity field u for heating of the plate when Gr = −5, Pr = 0.71, Sc = 0.22, Q = 1.0, Nr = 0.5, kr = 0.5, M = 0.5, K = 1.0, A = 0.3, ε = 0.02, n = 0.5, t =1.0
Sc
kr
Sh
0.22 0.60 0.22
0.5 0.5 1.0
−0.443266 −0.800992 −0.558000
Table 1 — Skin-friction coefficient (τ) for cooling of the plate (Gr > 0) Pr
Sc
Q
Nr
kr
M
K
Gr
Gm
τ
0.71 7.00 0.71 0.71 0.71 0.71 0.71 0.71 0.71 0.71
0.22 0.22 0.60 0.22 0.22 0.22 0.22 0.22 0.22 0.22
1.0 1.0 1.0 1.5 1.0 1.0 1.0 1.0 1.0 1.0
0.5 0.5 0.5 0.5 1.0 0.5 0.5 0.5 0.5 0.5
0.5 0.5 0.5 0.5 0.5 1.0 0.5 0.5 0.5 0.5
0.5 0.5 0.5 0.5 0.5 0.5 1.0 0.5 0.5 0.5
1.0 1.0 1.0 1.0 1.0 1.0 1.0 2.0 1.0 1.0
5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 10.0 5.0
5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 10.0
5.404040 1.312256 4.525942 8.510313 5.738986 4.956634 4.369522 6.881610 9.514360 7.935074
Table 2 — Skin-friction coefficient (τ) for heating of the plate (Gr < 0) Pr
Sc
Q
Nr
kr
M
K
Gr
Gm
τ
0.71 7.00 0.71 0.71 0.71 0.71 0.71 0.71 0.71 0.71
0.22 0.22 0.60 0.22 0.22 0.22 0.22 0.22 0.22 0.22
1.0 1.0 1.0 1.5 1.0 1.0 1.0 1.0 1.0 1.0
0.5 0.5 0.5 0.5 1.0 0.5 0.5 0.5 0.5 0.5
0.5 0.5 0.5 0.5 0.5 1.0 0.5 0.5 0.5 0.5
0.5 0.5 0.5 0.5 0.5 0.5 1.0 0.5 0.5 0.5
1.0 1.0 1.0 1.0 1.0 1.0 1.0 2.0 1.0 1.0
-5.0 -5.0 -5.0 -5.0 -5.0 -5.0 -5.0 -5.0 -10.0 -5.0
5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 10.0
−2.816585 1.275202 −3.694690 −5.922676 −3.151528 −3.263996 −2.678056 −2.998256 −6.926902 0.287558
SHANKER et al.: RADIATION AND MASS TRANSFER EFFECTS ON FLUID FLOW
reaction rate constant kr, respectively. It is observed that, an increase in the Schmidt number or chemical reaction rate constant leads to decrease in the value of mass transfer coefficient. 4 Conclusions The problem of two-dimensional fluid flow in the presence of thermal and concentration buoyancy effects under the influence of uniform magnetic field applied normal to the flow is formulated and solved numerically. A Galerkin finite element method is adopted to solve the equations governing the flow. The results obtained are compared with previous works and found to be in good agreement. It is found that increasing the Prandtl number results in a decrease in the temperature and temperature boundary layer while increasing the thermal radiation leads to a rise in the temperature and temperature boundary layer. An increase in the Schmidt number or chemical reaction rate constant decrease in the concentration and concentration boundary layer. Also, it has been observed that the suction has a little or no effect on the temperature and concentration. The value of the skin-friction coefficient (τ) decrease as increase in the Prandtl number Pr and Schmidt number Sc. The value of the skin-friction coefficient is maximum for cooling of the plate than in case of heating of the plate. Also, it is observed that Nusselt number and Sherwood numbers decrease with increase in the Prandtl number Pr and Schmidt number Sc. These observations are in good agreement with results from literature. Acknowledgement The authors wish to express their sincere thanks to the editor and referee for careful reading of the manuscript and valuable comments. Nomenclature u,v T q′r x,y t
velocity components dimensional temperature radiative flux vector cartesian coordinates time
g β β* ρ cp v M k U0 Sc Pr k 2r ε Gr Gm A n D Nr K Q
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acceleration due to gravity coefficient of volume expansion due to temperature coefficient of volume expansion due to concentration Density specific heat at constant pressure kinematic viscosity magnetic parameter thermal conductively mean velocity Schmidt number Prandtl number chemical reaction rate constant small reference parameter
