The three-dimensional flow of a non-newtonian fluid over a stretching flat surface through a porous medium with surface convective conditions

Transcription

1 Global Journal of Pure and Applied Mathematics. ISSN Volume 13, Number 6 (217), pp Research India Publications The three-dimensional flow of a non-newtonian fluid over a stretching flat surface through a porous medium with surface convective conditions K. Kalyani, K. Sreelakshmi and G. Sarojamma * Department of Applied Mathematics, Sri Padmavati Mahila Visvavidyalayam, Tirupati, 51752, A.P, India * Corresponding Author Abstract The aim of this paper is to explore the interaction of Lorentz force, thermal radiation and heat generation on the heat and mass transfer characteristics of the flow of an incompressible Casson fluid on a surface which is stretched linearly in two lateral directions. The boundary layer equations governing the flow, heat and mass transfer are transformed by applying similarity transformation into a set of coupled non-linear ordinary differential equations. The numerical solutions of these equations for velocity, temperature and concentration are obtained using the popular Runge-Kutta-Fehlberg method along with shooting technique. Parametric analysis of velocities, temperature and concentration for different variations in these parameters is performed graphically. The results indicate that the non-newtonian rheology of the fluid and Lorentz force suppress the fluid velocities. Increasing stretching ratio parameter is seen to diminish the x component of velocity while the other component in the y-direction has an opposite effect. Temperature is strongly influenced by variations in the thermal radiation parameter and heat source parameter. Species concentration is depreciated for higher values of chemical

2 2194 K. Kalyani, K. Sreelakshmi and G. Sarojamma reaction parameter and for smaller values of molecular diffusion. The components of surface drag coefficient in both the directions are increased for increasing values of Casson parameter. A comparison of our numerical analysis with the results of already published works is made which indicates very close agreement between the two. Keywords: Three-dimensional flow, Casson fluid, thermal radiation, convective boundary conditions INTRODUCTION Study of Non-Newtonian fluids is of pragmatic technical significance as they are abundantly used in chemical and oil industry, organic and bio sciences etc. Due to the rheological property of the non-newtonian fluid, the Navier-Stokes equations cannot represent completely the flow of a non-newtonian fluid flow. Hence, it is necessary to supplement the constitutive relation between shear stress and shear rate to describe the non-newtonian fluid flow accurately. The existence of several non-newtonian fluids warrants an exclusive rheological model for each fluid. Casson fluid is one such non- Newtonian fluid whose constitutive relation between shear stress and rate of deformation is described in terms of its yield stress and plastic dynamic viscosity. Radiative heat transfer is encountered frequently in engineering and technological processes that are operated at very high temperatures. Study of thermal radiation is essential to control the rate of heat transfer. Abundant investigations are available [1-5] in literature on the effect of thermal radiation. The effect of surface convective conditions in heat transfer processes is seen to cause high temperatures which are practically seen in nuclear plants, gas turbines, thermal storage units etc. Makinde and Olanrewaju [6] examined the effect of buoyancy forces on the flow past a vertical plate with convective boundary conditions. Murthy et al. [7] studied the flow, heat and mass transfer in a thermally stratified non-darcy porous medium saturated with a nanofluid under convective boundary condition. Ahmad and Khan [8] investigated the heat and mass transfer of a MHD viscous flow over a moving wedge, considering the effects of viscous dissipation, heat source/sink and convective boundary condition. Studies on two dimensional boundary layer flows induced by stretching surfaces are extensively discussed with different fluids under various conditions due to its significant applications in manufacturing industry. However, studies on three-

3 The three-dimensional flow of a non-newtonian fluid over a stretching flat 2195 dimensional boundary flows induced due to stretching of sheet in two directions are limited. Wang [9] initiated the three-dimensional boundary layer flow of a viscous fluid over flat surface which is stretched with a linear velocity in two lateral directions. Following this investigation a few studies on three dimensional flows in certain non-newtonian fluids such as Maxwell fluid, Eyring-Powell fluid, viscoelastic fluid etc are made [1-12]. Nadeem et al. [13] analyzed the Casson fluid flow on a permeable sheet induced due to the stretching of the sheet in x and y directions subjected transverse magnetic field. In a subsequent study, Nadeem et al. [14] extended the analysis to a Casson nanofluid over a linearly stretching sheet considering surface convective conditions. Shehzad et al. [15] explored the flow characteristics considering the effect of heat generation/absorption on flow of a Casson fluid over a linearly stretching surface. Sharada and Shankar [16] examined the three dimensional steady, laminar, incompressible MHD mixed convection flow of a Casson fluid over an exponential stretching sheet taking the effect of heat generation. Using Keller-box method numerical solutions are obtained. Butt et al. [17] investigated the three-dimensional flow of a magnetohydrodynamic Casson fluid over an unsteady stretching surface immersed in a porous medium. In all the above the investigations the effect of thermal radiation is not studied. In this paper we analyze the influence of radiative heat transfer on steady three dimensional flow of an electrical conducting chemically reactive Casson fluid induced by an elastic surface embedded in a porous medium by stretching the surface in two lateral directions under convective boundary conditions on temperature and concentration. MATHEMATICAL FORMULATION We consider a highly elastic membrane immersed in a Casson fluid through a saturated porous medium which is continuously stretched in the x and y- directions. The geometrical configuration of the present flow is shown in figure 1. A constant magnetic field B is applied in a direction normal to the fluid flow. The magnetic Reynolds number is assumed to be small and thus the induced magnetic field is neglected. The fluid velocities on the surface are given by U w = ax, V w = by and w = along the xy-plane, where a and b are constants. The fluid has no lateral motions at z. The effects of thermal radiation and heat generation are taken into consideration. The thermal and solutal convective boundary conditions are reckoned.

4 2196 K. Kalyani, K. Sreelakshmi and G. Sarojamma Figure 1. Physical model and coordinate system The constitutive equation of the Casson fluid can be written as [18] τ ij = { 2 (μ B + P y 2π ) e ij, π > π c 2 (μ B + P y 2π c ) e ij, π < π c where τ ij is the (i, j) th component of the stress tensor, μ B is the plastic dynamic viscosity of the non-newtonian fluid, P y is the yield stress of the fluid, π is the product of the component of deformation rate with itself, namely, π = e ij e ij, and e ij is the (i, j) th component of deformation rate, and π c is the critical value of π depends on non-newtonian model. The steady boundary layer equations governing the flow, heat and diffusion transport can be expressed as: (1) u x + v y + w z = (2) u u u + v x y u + w = ν (1 + 1 ) 2 u σb2 u ν u (3) z β z 2 ρ k 1 u v v + v x y v + w = ν (1 + 1 ) 2 v σb2 v ν v (4) z β z 2 ρ k 1 u T T T + v + w = x y z + 16σ T3 2 T + Q z 2 3ρc p k z 2 k 2 T ρc p ρc p (T T ) (5)

5 The three-dimensional flow of a non-newtonian fluid over a stretching flat 2197 u C C C + v + w = D 2 C k x y z z (C C 2 ) (6) where u, v and w are the fluid velocity components along x, y and z-directions respectively, ν is the kinematic viscosity, ρ is the density of the fluid, B is magnetic induction, β = μ B 2π c /P y is the Casson parameter, k 1 is the permeability of the porous medium, T and T are respectively the fluid and ambient temperatures, k is the mean absorption coefficient, σ is the electrical conductivity, c p is the specific heat at constant pressure, σ is the Stefen-Boltzman constant, k is the thermal conductivity of the medium, Q is the uniform volumetric heat generation and absorption, C and C are respectively the fluid and ambient concentrations, D is the mass diffusivity and k is the chemical reaction. The pertinent boundary conditions of equations of the problem are as follows: u = U w (x) = ax, v = V w (x) = by, w =, k T z = h f(t f T), D C z = h s(c f C) at z = u, v, T T, C C as z (7) where h f is the convective heat transfer coefficient and T f is the convective fluid temperature below the sheet, h s is the convective mass transfer coefficient and C f is the convective fluid concentration. METHOD OF SOLUTION Introducing the following similarity transformations u = axf (η), v = ayg (η), w = aν (f(η) + g(η)) η = z a ν, θ(η) = T T, φ(η) = C C (8) T f T C f C We see that u, v and w satisfy equation (2), and equations (3) (6) take the form (1 + 1 β ) f (f ) 2 + (f + g)f (M 2 + k p )f = (9) (1 + 1 β ) g (g ) 2 + (f + g)g (M 2 + k p )g = (1) ( Nr) θ + Pr ((f + g)θ + Qθ) = (11) φ + Sc ((f + g)φ γφ) = (12)

6 2198 K. Kalyani, K. Sreelakshmi and G. Sarojamma where,m 2 = σb 2 ρa is magnetic field parameter, k p = ν ak 1 is porosity parameter, c = b/a is stretching ratio parameter, Pr = ρc p ν k is Prandtl number, Nr = 4σ T 3 kk is thermal radiation parameter, Q = Q ρc p a is heat source/sink parameter,sc = ν Dis Schmidt number, γ = k a is chemical reaction parameter, Bi 1 = h f k ν a is heat transfer Biot number and Bi 2 = h s D ν a with the boundary conditions f =, g =, f = 1, g = c, mass transfer Biot number. θ = Bi 1 (1 θ()), φ = Bi 2 (1 φ()), at η = (13) f, g, θ, φ, as η (14) Skin friction, heat and mass transfer coefficients, are significant in estimating the surface drag, rate of heat transfer and mass transfer rate on the surface. The local skin friction coefficients in the x and y directions are given by: C fx = 2τ wx ρu w 2, C fy = 2c2 τ wy ρu w 2, (15) where τ w is the wall shear stress. Thus, the wall shear stresses along the x and y directions are denoted by τ wx = μ (1 + 1 ) β ( u) z w respectively and τ wy = μ (1 + 1 ) β ( v), z w C fx = 2 (1 + 1 ) (Re β x) 1 2 f (), C fy = 2 (1 + 1 ) β c1/2 (Re y ) 1 2 g () (16) The local heat transfer coefficient in terms of the local Nusselt number is expressed as: Nu x = x( T z ) w = (Re (T f T ) x) 1 2 θ (), (17) Similarly, the local mass transfer coefficient in terms of the local Sherwood number is expressed as: Sh x = x( C z ) w = (Re (C f C ) x) 1 2 φ (), (18) where Re x = U wx and Re ν y = V wy are the local Reynolds numbers, μ is the ν coefficient of viscosity, C fx and C fy are the local skin friction coefficients along x and y directions (i.e., C fx and C fy are the local skin friction coefficients for the primary and secondary flows, respectively), Nu x is the local Nusselt number and Sh x is the local Sherwood number.

7 The three-dimensional flow of a non-newtonian fluid over a stretching flat 2199 The ordinary differential equations (9) (12) are coupled and highly non linear and exact analytical solutions cannot be determined. Hence, these equations with the boundary conditions (13) and (14) are solved using Runge-Kutta-Fehlberg method together with shooting technique and obtained numerical solutions. In order to confirm the accuracy of our numerical procedure, we compared our results, viz, surface shear stress and surface temperature gradient [f (), g (), θ ()] with those evaluated by Wang [9], Lakshmisha et al. [19] and Rajeswari et al. [2] in the absence of magnetic field, porous medium, thermal radiation, heat source/sink and diffusion equation with prescribed wall temperature and concentration (M = k p = Nr = Q = Sc = γ =, Bi 1, Bi 2 and β )for different values of stretching parameter when Pr =.7. Table 1 shows that our results are in excellent agreement. Table 1. Comparison of various results (M = k p = Nr = Q = Sc = γ =, Bi 1, Bi 2, Pr =.7 and β ). c Wang [9] Lakshmisha et al. [19] Rajeswari et al. [2] Present Results. f () g () θ () f () g () θ () f () g () θ () f () g () θ () f () g () θ ()

8 22 K. Kalyani, K. Sreelakshmi and G. Sarojamma RESULTS AND DISCUSSION Our focus is to make a parametric analysis of all governing parameters on the velocities, temperature and concentration distributions. The values of skin friction coefficients in the x and y directions, local Nusselt number and Sherwood number are tabulated and discussed for different variations of the physical parameters. The variation of the non- Newtonian rheology of the fluid represented through the Casson parameter on velocities (f and g ) in the x and y directions is highlighted in figure 2. It can be seen that velocities in the two directions are diminished due to the suppressing nature of the non- Newtonian nature of the fluid i.e. increasing in plastic dynamic viscosity. Figure 3 is the plots of velocities in the x and y directions, illustrating the influence of magnetic field. The presence of magnetic field suppresses the velocities in both directions. This reduction is due to the resistive nature of Lorentz force arising due to the magnetic field acts a decelerating force in diminishing the velocity of the flow. Consequently thinner momentum boundary layers in both the directions are formed. Higher values of M correspond to stronger Lorentz force and the corresponding velocities are further decreased. Figure 4 shows that the stretching ratio parameter (c) has an appreciable influence on velocity component in the y direction. When the parameter c = the problem reduces to flow in the neighbourhood of the two dimensional stagnation point and when c = 1 the problem represents an axisymmetric flow. The stretching ratio parameter has an increasing influence on the y-component of velocity for increasing values of c while it has a decreasing effect on the corresponding x-component. The effect of c on the x- component of velocity is relatively smaller compared to its component in y-direction. An increase in c implies that the velocity in the y-direction dominates the velocity in x-direction and hence the velocity in the y-direction enhances with a decrease in the x component. Figure 5 depicts that the velocities in both directions reduce as the porous parameter (k p ) increases. This is due to the Darcy resistance offered by the porous medium. Higher values of porous parameter correspond to greater resistance to flow and hence velocities are diminished. This is in conformity with the fact that porosity stabilizes the growth of the boundary layer. Figure 6 shows the variation of Prandtl number (Pr) on temperature. It is seen that increasing values of Prandtl number show a drop in temperature. Physically an increase in Pr amountss to smaller values of the thermal conductivity and hence the temperature tends to decrease, i.e. the temperature of fluids with higher Prandtl number drops rapidly. Figure 7 reveals that increasing values of Biot number (Bi 1 )elevate the temperature. Physically an increase in Biot number amounts to an enhancement in the heat transfer coefficient and higher values of heat transfer coefficient boost the thermal energy to the fluid and thus higher temperatures are

9 The three-dimensional flow of a non-newtonian fluid over a stretching flat 221 obtained. Figure 8 illustrates that as thermal radiation parameter (Nr) increases, the thermal boundary layer thickness is found to increase resulting in higher temperatures. Figure 9 depicts the influence of heat generation/absorption. In the presence of heat source in the fluid, thermal energy is released and as a consequence the fluid gets heated up giving higher temperatures. Increasing values of the heat source parameter would cause a rise in the temperature further. In the case of a sink ( Q < )energy is absorbed due to which the fluid gets cooled and hence the temperature of the fluid decreases. Increase in the value of sink parameter causes further cooling of the fluid resulting in reduction in the temperature. Figure 1 is the plots of species concentration for a variation of Schmidt number. Physically, larger Schmidt number implies that the molecular diffusivity of the fluid is smaller and hence the concentration decreases. Figure 11 presents the effect of solutal Biot number (Bi 2 ) which is the ratio of convective mass transfer to molecular diffusion. When a solutal gradient is applied on the surface the concentration at the surface varies predominantly. It is observed that as the solutal Biot number increases from to, there is a threefold enhancement in the mass concentration. Figure 12 shows the effect of chemical reaction parameter (γ) on species concentration. In the presence of the chemical reaction parameter the species concentration decreases asymptotically along η. 1.9 f '() g'().8 f '() / g'().7.6 M = k p = c = ; Pr = 1.8; Nr = ; Q = ; Sc =.8; = ; Bi 1 = Bi 2 = =, 1., 1.5, Figure 2. Velocity profiles [f (η) and g (η)] for different values of β

10 222 K. Kalyani, K. Sreelakshmi and G. Sarojamma 1.9 f '() g'().8 f '() / g'().7.6 = k p = c = ; Pr = 1.8; Nr = ; Q = ; Sc =.8; = ; Bi 1 = Bi 2 = M =.,,.7, Figure 3. Velocity profiles [f (η) and g (η)] for different values of M 1.9 f '() g'().8.7 f '() / g'().6 M = = k p = ; Pr = 1.8; Nr = ; Q = ; Sc =.8; = ; Bi 1 = Bi 2 = c =,,.6, Figure 4. Velocity profiles [f (η) and g (η)] for different values of c

11 The three-dimensional flow of a non-newtonian fluid over a stretching flat f '() g'().8 f '() / g'().7.6 M = = c = ; Pr = 1.8; Nr = ; Q = ; Sc =.8; = ; Bi 1 = Bi 2 = k p =.,, 1., Figure 5. Velocity profiles [f (η) and g (η)] for different values of k p 5 5 M = = k p = c = ; Nr = ; Q = ; Sc =.8; = ; Bi 1 = Bi 2 = Pr = 1.8 Pr = 3. Pr = 5. Pr = 7. () Figure 6. Temperature profiles for different values of Pr

12 224 K. Kalyani, K. Sreelakshmi and G. Sarojamma () Bi 1 = Bi 1 = Bi 1 = 1. Bi 1 = 2. Bi 1 = M = = k p = c = ; Pr = 1.8; Nr = ; Q = ; Sc =.8; = ; Bi 2 = Figure 7. Temperature profiles for different values of Bi M = = k p = c = ; Pr = 1.8; Q = ; Sc =.8; = ; Bi 1 = Bi 2 = Nr = Nr = Nr =.7 Nr = 1. () Figure 8. Temperature profiles for different values of Nr

13 The three-dimensional flow of a non-newtonian fluid over a stretching flat Q = - Q = - Q =. Q = Q = () 5 M = = k p = c = ; Pr = 1.8; Nr = ; Sc =.8; = ; Bi 1 = Bi 2 = Figure 9. Temperature profiles for different values of Q 5 Sc = Sc = 1. Sc = 1.5 Sc = 2. 5 M = = k p = c = ; Pr = 1.8; Nr = ; Q = ; = ; Bi 1 = Bi 2 = () Figure 1. Concentration profiles for different values of Sc

14 226 K. Kalyani, K. Sreelakshmi and G. Sarojamma 5 Bi 2 = Bi 2 = Bi 2 = 5 Bi 2 = () 5 M = = k p = c = ; Pr = 1.8; Nr = ; Q = ; Sc =.8; = ; Bi 1 = Figure 11. Concentration profiles for different values of Bi 2 5 = = =.7 = 1. 5 M = = k p = c = ; Pr = 1.8; Nr = ; Q = ; Sc =.8; Bi 1 = Bi 2 = () Figure 12. Concentration profiles for different values of γ

15 The three-dimensional flow of a non-newtonian fluid over a stretching flat 227 In Table 2 the values of the surface skin friction, Nusselt number and Sherwood number obtained for different variations of the governing parameters are tabulated. The surface stresses in both directions are negative. Casson parameter is seen to increase the skin friction coefficients in the x and y directions while the magnetic field shows an opposite behavior. The rate of heat transfer and mass transfer are observed to increase for increasing values of Casson and magnetic field parameters. The effect of increasing values of stretching ratio has a decreasing influence on the x component of surface shear stress while an opposite effect on its corresponding y- component. The Nusselt number and Sherwood number are observed to enhance with increasing c. The effect of porous parameter on the skin friction coefficient in both the directions, Nusselt number and Sherwood number is qualitatively similar to that of the magnetic field. Table 2. Skin friction coefficient, Nusselt number and Sherwood number for various values of pertinent parameters where Pr = 1.8; Nr = ; Q = ; Sc =.8; γ = ; Bi 1 = and Bi 2 = β M c k p (1 + 1 β ) f () (1 + 1 β ) g () θ () φ ()

16 228 K. Kalyani, K. Sreelakshmi and G. Sarojamma Table 3 shows that increasing Prandtl number and heat transfer Biot number increase the local Nusselt number. It is observed that the heat source and thermal radiation are found to enhance rate of heat transfer. Table 3. Nusselt number for various values of pertinent parameters β = ; M = ; c = ; Sc =.8; γ = and Bi 2 = Pr Nr Q Bi 1 θ ()

17 The three-dimensional flow of a non-newtonian fluid over a stretching flat 229 Table 4. Sherwood number for various values of pertinent parameters where β = ; M = ; c = ; Pr = 1.8; Nr = ; Q = and Bi 1 = Sc γ Bi 2 φ () Table 4 illustrates that Schmidt number, chemical reaction parameter and mass transfer Biot number are found to have increasing influence on the Sherwood number. 5 CONCLUSIONS The steady three dimensional MHD Casson fluid saturated porous medium over a stretching surface is investigated to analyze the effects of magnetic field and thermal radiation with convective boundary conditions on temperature and concentration. Casson parameter has a decreasing influence on velocities Increasing values of stretching parameter enhance the y-component of the velocity while an opposite effect on the x-component is observed. Lorentz force and porous parameter decelerated flow. Rate of heat transfer is enhanced with heat source parameter.

18 221 K. Kalyani, K. Sreelakshmi and G. Sarojamma Schmidt number and chemical reaction parameters show a decreasing influence on species concentration while solutal Biot number and Schmidt number have reverse effect. REFERENCES [1]. Eldahab, E. M. A., Aziz, M. A. E., Salem, A. M., ad Jaber, K. K., 27, Hall current on MHD mixed convection flow from an inclined continuously stretching surface with blowing/suction and internal heat generation/absorption, Applied Mathematical Modelling, 31(9), pp [2]. Jat, R. N., and Gopi Chand, 213, MHD flow and heat transfer over an exponentially stretching sheet with viscous dissipation and radiation effects, Applied Mathematical Sciences, 7(4), pp [3]. Nadeem, S., Zaheer, S., and Fang, T., 211, Effects of thermal radiation on the boundary layer flow of a Jeffrey fluid over an exponentially stretching surface, Numerical Algorithms, 57(2), pp [4]. Sajid, M., and Hayat, T., 28, Influence of thermal radiation on the boundary layer flow due to an exponentially stretching sheet, Int. J. Heat Mass Transfer, 35(3), pp [5]. Sreelakshmi, K., Nagendramma, V., Sarojamma, G., 215, Unsteady boundary layer flow induced by a stretching sheet in a rotating fluid with thermal radiation, Procedia Engineering, 127, pp [6]. Makinde, O. D., and Olanrewaju, P. O., 21, Buoyancy effects on thermal layer over a vertical plate with a convective surface boundary condition, J. Fluid Engineering, 132(4), pp [7]. Murthy, P. V. S. N., Ram Reddy, Ch., Chamkha, A. J., and Rashad, A. M., 213, Magnetic effect on thermally stratified nano fluid saturated non-darcy porous medium under convective boundary condition, International Communications in Heat and Mass Transfer, 47(1), pp [8]. Ahmad, R., Khan, W. A., 214, Numerical study of heat and mass transfer MHD viscous flow over a moving wedge in the presence of viscous dissipation and heat source/sink with convective boundary condition, Heat Transfer Asian Research, 43(1), pp [9]. Wang, C. Y., 1984, The three-dimensional flow due to a stretching flat surface, The Physics of fluids, 27(8), pp

19 The three-dimensional flow of a non-newtonian fluid over a stretching flat 2211 [1]. Ashraf, M. B., Hayat, T., and Alhuthali, M. S., 216, Three-dimensional flow of Maxwell fluid with Soret and Dufour effects, Journal of Aerospace Engineering, 29(3), [11]. Ashraf, M. B., Hayat, T., and Alsaedi, A., 215, Three-dimensional flow of Eyring-Powell nanofluid by convectively heated exponentially stretching sheet, The European Physical Journal Plus, 13(5), [12]. Ramzan, M., Inan, S., and Shehzad, S. A., 215, Three dimensional boundary layer flow of a viscoelastic nanofluid with Soret and Dufour effects, Alexandria Engineering Journal, 55(1), pp [13]. Nadeem, S., Rizwan UI Haq, Akbar, N. S., and Khan, Z. H., 213, MHD three-dimensional Casson fluid flow past a porous linearly stretching sheet, Alexandria Engineering Journal, 52(1), pp [14]. Nadeem, S., Rizwan UI Haq and Akbar, N. S., 214, MHD three dimensional boundary layer flow of Casson nanofluid past a linearly stretching sheet with convective boundary condition, IEEE Tramsactions on nanotechnology, 13(1), pp [15]. Shehzad, S. A., Hayat, T., and Alsaedi, A., 216, Three-Dimensional MHD Flow of Casson Fluid in Porous Medium with Heat Generation, Journal of Applied Fluid Mechanics, 9(1), pp [16]. Sharada, K, and Shankar, B., 216, Three-dimensional MHD Mixed Convection Casson Fluid Flow over an Exponential Stretching Sheet with the Effect of Heat Generation, British Journal of Mathematics & Computer Science, 19(6), pp.1 8. [17]. Butt, A. S., Tufail, M. N., and Ali, A., 216, Three-dimensional flow of a magnetohydrodynamic Casson fluid over an unsteady stretching sheet embedded into a porous medium, Journal of Applied Mechanics and Technical Physics, 57(2), pp [18]. Eldabe, N. T. M., and Salwa, M. G. E., 1995, Heat transfer of MHD non- Newtonian Casson fluid flow between two rotating cylinders, J. Phys. Soc. Japan, 64, pp [19]. Lakshmisha K. N., Venkateswaran, S., and Nath, G., 1988, Three dimensional unsteady flow with heat and mass transfer over a continuous stretching surface, J. Heat Trans., 11(3), pp [2]. Rajeswari, V., Kumari, M. and Nath, G., 1993, Unsteady three dimensional boundary layer flow due to a stretching surface, Acta Mechanica, 98(1), pp

20 2212 K. Kalyani, K. Sreelakshmi and G. Sarojamma