FLUID DYNAMICS MHD FLOW PAST AN IMPULSIVELY STARTED INFINITE VERTICAL PLATE IN PRESENCE OF THERMAL RADIATION M. K. MAZUMDAR, R. K. DEKA Department of Mathematics, Gauhati University Guwahat-781 014, Assam, India E-mail: muninmazumdar@yahoo.com, rkdgu@yahoo.com Received January 10, 007 The effect of thermal radiation past an impulsively started infinite vertical plate under the influence of transverse magnetic field has been discussed. The fluid is assumed to be gray, emitting-absorbing but non-scattering medium and the optically thick radiation limit is considered. The dimensionless governing equations are solved using Laplace transformation technique. The velocity and temperature profiles are shown on graphs. The variation of skin-friction is also shown in a table. Key words: radiation, skin-friction, temperature, MHD. 000 Mathematics Subject Classification: 76B, 76W05. 1. INTRODUCTION In recent years, the subject of magnetohydrodynamics has attracted the attention of many authors in view not only of its own interest, but also of its application to problems in geophysics, astrophysics and engineering. At the high temperatures attained in some engineering devices, gas, for example, can be ionized and so becomes an electrical conductor. The ionized gas or plasma can be made to interact with the magnetic field and alter heat transfer and friction characteristic. Since some fluids can also emit and absorb thermal radiation, it is of interest to study the effect of magnetic field on the temperature distribution and heat transfer when the fluid is not only an electrical conductor but also when it is capable of emitting and absorbing thermal radiation. This is of interest because heat transfer by thermal radiation is becoming of greater importance when we are concerned with space applications and higher operating temperatures. In this paper, we consider one of the simplest problems of this type in which an electrically conducting, radiating, viscous, incompressible fluid past an impulsively started infinite vertical plate under the action of a constant pressure gradient is subjected to an external magnetic field of constant strength in the direction perpendicular to the plate and to the direction of flow. Rom. Journ. Phys., Vol. 5, Nos. 5 7, P. 565 573, Bucharest, 007
566 M. K. Mazumdar, R. K. Deka Several investigations have been carried out on problem of heat transfer by radiation as an important application of space and temperature related problems. Greif, Habib and Lin [1] obtained an exact solution for the problem of laminar convective flow in a vertical heated channel in the optically thin limit. In the optically thin limit, the fluid does not absorb its own emitted radiation which means that there is no self absorption but the fluid does absorb radiation emitted by the boundaries. Viskanta [] investigated the forced convective flow in a horizontal channel permeated by uniform vertical magnetic fluid taking radiation into account. He studied the effects of magnetic field and radiation on the temperature distribution and the rate of heat transfer in the flow. Later Gupta and Gupta [3] studied the effect of radiation on the combined free and forced convection of an electrically conducting fluid flowing inside an open-ended vertical channel in the presence of a uniform transverse magnetic field for the case of optically thin limit. They found that radiation tends to increase the rate of heat transfer to the fluid there by reducing the effect of natural convection. Soundalgekar and Takhar [4] first, studied the effect of radiation on the natural convection flow of a gas past a semi-infinite plate using the Cogly-Vincentine- Gilles equilibrium model (Cogly et al. [5]). For the same gas Takhar et al. [6] investigated the effects of radiation on the MHD free convection flow past a semi-infinite vertical plate. Later, Hossain and Takhar [7] analyzed the effect of radiation using the Rosseland diffusion approximation which leads to nonsimilar solution for the forced and free convection of an optically dense viscous incompressible fluid past a heated vertical plate with uniform free stream and uniform surface temperature, while Hossain et al. [8] studies the effect of radiation on free convection from a porous vertical plates. Muthucumaraswamy and Kumar [9] studied the thermal radiation effects on moving infinite vertical plate in the presence of variable temperature and mass diffusion. It is the purpose of this paper to examine quantitatively the effect of magnetic field of an optically thin viscous incompressible fluid from an impulsively started vertical plate in the presence of thermal radiation. The temperature and velocity distributions are evaluated numerically for some values of the radiation parameter, magnetic field parameter Prandtl number and time.. MATHEMATICAL ANALYSIS We consider an electrically conducting, radiating, viscous, incompressible fluid past an impulsively started infinite vertical plate. Initially, the plate and the surrounding gas are at the same temperature T. At time t > 0, the plate temperature is slightly increased to Tw T > 0 and the magnetic field of strength B 0 is assumed to be applied in a direction perpendicular to the vertical plate, so that there exists free convection current in the vicinity of the plate when
3 MHD flow 567 the plate is given an impulsive motion with a velocity u 0. The x -axis is taken along the plate in the vertically upward direction and the y -axis is taken normal to the plate, in the direction of the applied magnetic field. Then the fully developed flow of a radiating gas is governed by the following set of equations: u u σbu 0 =ν + gβ ( T T ) t y ρ (1) T T qr ρ C p = k () t y y The initial and boundary conditions are u = 0, T = T, for all y, t 0 u = u0, T = Tw at y = 0 (3) t > 0 u = 0, T T as y Here u is the velocity in the x -direction, ρ the density, g the acceleration due to gravity, β the coefficient of thermal expansion, T the temperature of the fluid near the plate, C p the specific heat at constant pressure, K the thermal conductivity, q r the radiative flux, ν the kinematic viscosity and σ the electrical conductivity. In the optically thick limit, the fluid does not absorb its own emitted radiation that is there is no self absorption, but it does absorb radiation emitted by the boundaries. It has been shown by Cogly et al. [5], that in the optically thick limit for a non-gray gas near equilibrium, that qr debλ = 4( T T ) K dλ= 4I ( T T ) y (4) λw 1 dt 0 w where K λw is the absorption coefficient, e bλ is the Planck function and the subscript w refers to values at the wall. On introducing the following non-dimensional quantities, u Gy u0 Gt u0 T T u=, y =, t =, θ=, u0 υ υ Tw T σb0 υ 4I C 1υ μ p gβυ( Tw T ) M =, F =, Pr =, G = ρgu K 3 0 Ku0G u 0 where Pr the Prandtl number, F the radiation parameter, M the magnetic field parameter, G the Grashof number and θ the dimensionless temperature. (5)
568 M. K. Mazumdar, R. K. Deka 4 Then in view of (5), equations (1) and () reduce to u = u +θ Mu t y Pr θ = θ Fθ t y with following boundary conditions u= 0, θ= 0, for all y, t 0 u= 1, θ= 1 at y = 0 t > 0 u 0, θ 0 as y (6) (7) (8) We now solve equations (6) and (7) subject to the initial and boundary conditions (8) by the usual Laplace transformation technique. The solutions are given by 1 y F y Pr y F y Pr e F erfc t e erfc F θ= t + + ( 1 ) 1 1 y M y y M y u e = + erfc Mt e erfc Mt a + + t t 1 ct y b y y b y e e erfc bt e erfc bt a + + t t 1 y F y Pr F y F y Pr e erfc t e erfc F t a + + + 1 ct y b y Pr e e erfc b + t + y b y Pr b a e erfc + t (9) (10) where a= F M, b= F MPr, c= a. 1 Pr 1 Pr To understand the physical meaning of the problem, we have computed the expression for u and θ for different values of radiation parameter F, magnetic field parameter M, Prandtl number Pr = 0.71 and time t. The purpose of the calculations given here is to assess the effects of the parameters F, M and t upon the nature of the flow and transport. The temperature profiles are shown in Figs. 1 and for air Pr = 0.71. The effect of thermal radiation parameter is important in temperature profiles. We observe from Figs. 1 and, which the temperature decreases with increasing F and
5 MHD flow 569 Fig. 1 Temperature profiles for different values of F. Fig. Temperatures profiles for different values of t.
570 M. K. Mazumdar, R. K. Deka 6 Fig. 3 Velocity profiles for different values of F. Fig. 4 Velocity profiles for different values of t.
7 MHD flow 571 Fig. 5 Velocity profiles for different values of M. increases with increasing t. The velocity profiles are shown in Figs. 3, 4 and 5 for air Pr = 0.71. It is observed that the velocity decreases with increasing F and M, but increases with increasing time t. 3. SKIN-FRICTION Knowing the velocity field, we now study the changes in the skin-friction, which is given in non-dimensional form as τ= du dy y = 0 Then from equations (10) and (11), we have 1 ( 1 ) Merf 1 1 1 ( Mt Mt ct bt ) e e e berf ( bt) τ= + + + a πt a πt b F b Pr t Pr 1 Pr t berf t e e Pr Ferf F t Pr + πt a πt Pr (11) (1) The numerical values of skin-friction τ are presented in Table 1 for different values of radiation parameter F, magnetic field parameter M, Prandtl
57 M. K. Mazumdar, R. K. Deka 8 number Pr = 0.71 and time t. It is inferred from this table, skin-friction decreases with increasing values of the radiation parameter F. This shows that the wall shear stress decreases with increasing radiation parameter. It is also observed that the skin-friction decreases with increasing values of magnetic field parameter M. It is interesting to note that, the skin-friction decreases with increasing values of the time t. Table 1 Values of skin-friction for air Pr = 0.71 t F M τ 0. 0.0 0.5.363048 0. 0. 0.5 1.376334 0. 0.6 0.5 1.16459 0. 1.0 0.5 1.141745 0. 0. 0.0 5.368017 0. 0. 1.0 0.495390 0. 0..0 0.001345 0.4 0. 0.5 0.893055 0.6 0. 0.5 0.640809 0.8 0. 0.5 0.473178 4. CONCLUSIONS i. The temperature decreases with increasing F and increases with increasing t. ii. The velocity decreases with increasing F and M, but increases with increasing time t. iii. The wall shear stress decreases with increasing radiation parameter F and time t. REFERENCES 1. R. Greif, I. S. Habib, J. C. Lin, Laminar convection of a radiating gas in a vertical channel, Journal of Fluid Dynamics, 46, 513 50, 1971.. R. Viskanta, Effect of transverse magnetic field on heat transfer to an electrically conducting and thermal radiating fluid flowing in a parallel plate channel, ZAMP, 14, 353 361, 1963. 3. P. S. Gupta, A. S. Gupta, Radiation effect on hydromantic convection in a vertical channel, Int. J. Heat Mass Transfer, 17, 1437 144, 1974. 4. V. M. Soundalgekar, H. S. Takhar, Radiative convective flow past a semi infinite vertical plate, Modelling Measure and Cont., 51, 31 40, 199. 5. A. C. Cogly, W. C. Vincentine, S. E. Gilles, Differential approximation for radiative transfer in a non-gray gas near equilibrium, AIAA Journal, 6, 551 555, 1968.
9 MHD flow 573 6. H. S. Takhar, R. S. R. Gorla, V. M. Soundalgekar, Radiation effects on MHD free convection flow of a radiating gas past a semi infinite vertical plate, Int. J. Numerical Methods Heat Fluid Flow, 6, 77 83, 1996. 7. M. A. Hossain, H. S. Takhar, Radiative effect on mixed convection along a vertical plate with uniform surface temperature, Int. J. Heat Mass Transfer, 31, 43 48, 1996. 8. M. A. Hossain, M. A. Alim, D. A. S. Rees, Effect of radiation on free convection from a porous vertical plate, Int. J. Heat Mass Transfer, 4, 181 191, 1999. 9. R. Muthcuumaraswamy, G. Senthil Kumar, Heat and mass transfer effects on moving vertical plate in the presence of thermal radiation, Theoret. Appl. Mach., 31(1), 35 46, 004.