Applied Mathematical Sciences, Vol. 5, 2011, no. 3, 149-157 MHD Flow Past an Impulsively Started Vertical Plate with Variable Temperature and Mass Diffusion U. S. Rajput and Surendra Kumar Department of Mathematics and Astronomy University of Lucknow Lucknow, India rajputsurendralko@gmail.com Abstract MHD flow past an impulsively started vertical plate with variable temperature and mass diffusion is studied here. The fluid considered is gray, absorbing-emitting radiation but a non-scattering medium. The governing equations involved in the present analysis are solved by the Laplace-transform technique. The velocity and skin friction are studied for different parameters like Schmidt number, Thermal Grashof number, mass Grashof number, Prandtl number and magnetic field parameter. Mathematics Subject Classification: 76W05, 80A20 Keywords: MHD, mass diffusion 1 Introduction Study of MHD flow with heat and mass transfer play an important role in chemical, mechanical and biological Sciences. Some important applications are cooling of nuclear reactors, liquid metals fluid, power generation system and aero dynamics. The response of laminar skin friction and heat transfer to fluctuations in the stream velocity was studied by Lighthill [6]. Free convection effects on the oscillating flow past an infinite vertical porous plate with constant suction - I, was studied by Soundalgekar [11] which was further improved by Vajravelu et al. [13]. Further researches in these areas were done by Gupta et al.[1], jaiswal et al.[4]and Soundalgekar et al. [12] by taking different models. Some effects like radiation and mass transfer on MHD flow were studied by Muthucumaraswamy et al. [7-8] and Prasad et al. [9]. Radiation effects on mixed convection along a vertical plate with uniform surface temperature were studied by Hossain and Takhar [3]. Mass transfer effects on the flow past an
150 U. S. Rajput and S. Kumar exponentially accelerated vertical plate with constant heat flux was studied by Jha, Prasad and Rai [5]. On the other hand, Radiation and free convection flow past a moving plate was considered by Raptis and Perdikis [10]. We are considering the effects on MHD flow past an impulsively started vertical plate with variable temperature and mass diffusion. The results are shown with the help of graphs (Fig-1 to Fig-6) and table-1. 2 Mathematical Analysis In this paper we have considered the flow of unsteady viscous incompressible fluid. The x - axis is taken along the plate in the upward direction and y - axis is taken normal to the plate. Initially the fluid and plate are at the same temperature. A transverse magnetic field B 0, of uniform strength is applied normal to the plate. The viscous dissipation and induced magnetic field has been neglected due to its small effect. Initially, the fluid and plate are at the same temperature T and concentration C in the stationary condition. At time t > 0, temperature of the plate is raised to T w and the concentration level near the plate is raised linearly with respect to time. The flow modal is as under: u = gβ (T T t )+gβ (C C )+ν 2 u σb2 y 2 0 u,... (1) ρ C t = D 2 C y 2,... (2) ρc p T t = k 2 T y 2.... (3) Here the symbols have the meaning as described below: C p - specific heat at constant pressure; T - temperature of the fluid near the plate; C - species concentration in the fluid; T - temperature of the fluid far away from the plate; C - concentration in the fluid far away from the plate; t - time; ρ - fluid density; g - acceleration due to gravity; β - volumetric coefficient of thermal expansion; β - volumetric coefficient of concentration expansion; ν - kinematics viscosity; B 0 - external magnetic field; σ - Stefan-Boltzmann constant; u - velocity of the fluid in the x - direction; k - thermal conductivity of the fluid; y - coordinate axis normal to the plate and D - mass diffusion coefficient. The following boundary conditions have been assumed:
MHD flow past an impulsively started vertical plate 151 t 0 : u = 0, T = T, C = C for all the values of y t > 0 : u = u 0, T = T + (T w T ) u2 0 t, ν C = C + (C w C ) u2 0 t at y = 0 ν u 0, T T, C C as y 0 Introducing the following non- dimensional quantities:... (4) ȳ = yu 0, ū = u ν u 0, θ = P r = µcp k G m = gβ ν (C w C ) u 3 0 (T T ), S (T w T ) c = ν, µ = ρν, D, M = σb2 0 ν, ρu 2 0, G r = gβν(tw T ) u 3 0, C = (C C ), (C w C ) t = tu2 0 ν... (5) with symbols: ū - dimensionless velocity; P r - Prandtl number; M - magnetic field parameter; ȳ - dimensionless coordinate axis normal to the plate; θ - dimensionless temperature; G r - thermal Grashof number; G m - mass Grashof number; S c - Schmidt number; C - dimensionless concentration and µ - coefficient of viscosity. Equations (1), (2) and (3) leads to ū = G t rθ + G m C + 2 ū Mū,... (6) ȳ 2 C t = 1 S c 2 C ȳ 2,... (7) and θ = 1 2 θ t P r,... (8) ȳ 2 with the following boundary conditions: t 0 : ū = 0, θ = 0, C = 0 for all the values of ȳ t > 0 : ū = 1, θ = t, C = t, at ȳ = 0, ū 0, θ 0, C 0 as ȳ.... (9) Dropping bars in the above equations, we have u = G t rθ+g m C+ 2 u Mu,... (10) y 2 C t = 1 S c 2 C y 2,... (11) and θ t = 1 P r 2 θ y 2,... (12)
152 U. S. Rajput and S. Kumar with the following boundary conditions: t 0 : u = 0, θ = 0, C = 0 for all the values of y t > 0 : u = 1, θ = t, C = t, at y = 0, u 0, θ 0, C 0 as y.... (13) The dimensionless governing equations (10) to (12), subject to the boundary conditions (13), are solved by the usual Laplace - transform technique with some help from the article An algorithm for generating some inverse Laplace transforms of exponential form by Hetnarski, R.B. [2]. The solutions are derived as below: u = G 1 [e ym erfc ( η Mt ) + e ym erfc ( η + Mt ) ] + ( ) ( ) G r+g m M [ t y 4 M e y M erfc ( η Mt ) + ( ) t + y 4 M e y M erfc ( η + Mt ) ] tgr [(1 + M 2η2 P r ) erfc ( η ) P r 2η π P r e η2 P r ] tgm [(1 + M 2η2 S c ) erfc ( η ) S c 2η π S c e η2 S c ] + G 2 [e y ap r erfc ( η P r at ) + e y ap r erfc ( η P r + at ) { eat 2 e y M+a erfc ( η (M + a)t ) + e ym+a erfc ( η + + G 3 [e y bs c erfc ( η S c bt ) + e y bs c erfc ( η S c + bt ) ebt 2 { e y M+b erfc ( η (M + b)t ) + e y M+b erfc ( η + (M + a)t )} ] (M + b)t )} ] G 2 erfc ( η P r ) G3 erfc ( η S c ),... (14) C = t [ (1 + 2η 2 P r ) erfc ( η ) P r 2η ] π P r e η2 P r,... (15) θ = t [ (1 + 2η 2 S c ) erfc ( η ) S c 2η ] π S c e η2 S c.... (16) For making the solution concise, the following symbols have been used above: G 1 = 1 + G 2 + G 3, G 2 = a = Gr, G a 2 (P r 1) 3 = M, b = M and η = y (P r 1) (S c 1) 2 t Gm, b 2 (S c 1) 3 Skin Friction (τ) We now study skin friction from velocity field. It is given in non-dimensional form as :
MHD flow past an impulsively started vertical plate 153 τ = ( ) u = ( ) 1 y y=0 2 u.... (17) t η η=0 Therefore using equation (14), we get τ: τ = 2G 1 [ Merf ( ) Mt + e Mt πt ] 2tGr [ tpr ] 2tGm [ tsc ] M π M π + ( ) ( ) ( ) G r+g m M 2t M + 1 2 M erf Mt + 2 te Mt π ] G 2 [e at (M + a)erf ( (M + a)t ) + e Mt πt 2 ap r erf( at) 2 Pr πt e at + Pr ] πt G 3 [e bt (M + b)erf ( (M + b)t ) + e Mt πt 2 bs c erf( bt) 2 Sc πt e bt + Sc ]. πt 4 Result and Discussion The velocity profile for different parameters M, G m, G r, P r, S c and t is shown by figures (1) to (6). From figure-1, it is clear that the velocity increases when G m is increased (figure-2). it is observed in figure-3 that the velocity decrease with increase in the time. When Schmidt number is increased the velocity increases (figure-4). The effects of Magnetic field is shown in figure-5 (i.e. when M is increased the velocity near the plate is decreased). Figure-6 shows the velocity profile for different values of Prandtl number. The values of skin friction are tabulated in table-1 for diffrent parameters. Table 1: Skin Friction
154 U. S. Rajput and S. Kumar Figure 1: Velocity Profile Figure 2: Velocity Profile Figure 3: Velocity Profile
MHD flow past an impulsively started vertical plate 155 Figure 4: Velocity Profile Figure 5: Velocity Profile Figure 6: Velocity Profile
156 U. S. Rajput and S. Kumar 5 Conclusion In this paper a theoretical analysis has been done to study the effect of MHD flow past an impulsively started vertical plate with variable temperature and mass diffusion. Solutions for the model have been derived by using Laplace - transform techenique. Some conclusions of the study are as below : Velocity increases with the increase in Prandtl number, mass Grashof number and Schmidt number. Velocity decreases when thermal Grashof number, magnetic field and time is increased. Skin friction increases when mass Grashof number, thermal Grashof number, Prandtl number, Schmidt number and time increases. Skin friction decrease when magnetic field parameter is increased. ACKNOWLEDGEMENTS. We acknowledge the U.G.C. (University Grant Commission) and thank for providing financial support for the research work. We are also thankful to different software/hardware companies (Mathematica, Intel, Microsoft, Matlab etc.) for developing the techniques that help in the computation and editing. References [1] A.S.Gupta, I. Pop and V.M. Soundalgekar, Free convection effects on the flow past an accelerated vertical plate in an incompressible dissipative fluid, Rev. Roum. Sci. Techn. -Mec. Apl, 24 (1979), 561-568. [2] R.B. Hetnarski, An algorithm for generating some inverse Laplace transforms of exponential form, ZAMP, 26 (1975), 249-253. [3] M.A. Hossain and H. S. Takhar, Radiation effect on mixed convection along a vertical plate with uniform surface temperature, Heat and Mass Transfer,31 (1996), 243-248. [4] B.S. Jaiswal and V.M. Soundalgekar, Oscillating plate temperature effects on a flow past an infinite porous plate with constant suction and embedded in a porous medium, Heat and Mass Transfer, 37 (2001), 125-131. [5] B.K Jha, R. Prasad and S. Rai, Mass transfer effects on the flow past an exponentially accelerated vertical plate with constant heat flux, Astrophysics and Space Science,181(1991), 125-134.
MHD flow past an impulsively started vertical plate 157 [6] M.J. Lighthill,The response of laminar skin friction and heat transfer to fluctuations in the stream velocity, Proc. R. Soc., A, 224 (1954), 1-23. [7] R. Muthucumaraswamy and Janakiraman, MHD and Radiation effects on moving isothermal vertical plate with variable mass diffusion, Theoret. Appl. Mech., 33, no. 1 (2006), 17-29. [8] R. Muthucumaraswamy, K.E. Sathappan and R. Natarajan, Mass transfer effects on exponentially accelerated isothermal vertical plate, Int. J. of Appl. Math. and Mech., 4, no. 6 (2008), 19-25. [9] V.R. Prasad, N.B. Reddyn and R. Muthucumaraswamy, Radiation and mass transfer effects on two- dimensional flow past an impulsively started infinite vertical plate,int.j. Thermal Sci., 46, no. 12 (2007), 1251-1258. [10] A. Raptis and C. Perdikis, Radiation and free convection flow past a moving plate,int. J. of App. Mech. and Engg., 4 (1999), 817-821. [11] V. M. Soundalgekar, Free convection effects on the oscillatory flow an infinite, vertical porous, plate with constant suction I,Proc. R. Soc., A, 333 (1973), 25-36. [12] V.M. Soundalgekar and H.S. Takhar, Radiation effects on free convection flow past a semi-infinite vertical plate, Modeling, Measurement and Control,B51 (1993), 31-40. [13] K.Vajravelu and K. S. Sastri,Correction to Free convection effects on the oscillatory flow an infinite, vertical porous, plate with constant suction I,Proc. R. Soc., A,51 (1977), 31-40. Received: August, 2010