Quasi-linearization Approach to MHD Effects on Boundary Layer Flow of Power-Law Fluids Past A Semi Infinite Flat Plate with Thermal Dispersion

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1 ISSN (print), (online) International Journal of Nonlinear Science Vol.11(2011) No.3,pp Quasi-linearization Approach to MHD Effects on Boundary Layer Flow of Power-Law Fluids Past A Semi Infinite Flat Plate with Thermal Dispersion Kishan Naikoti 1, Shashidar Reddy Borra 2 1 Department of Mathematics, University College of Science, Osmania University, Hyderabad -7, A.P., INDIA. 2 Department of Sciences and Humanities, Sreenidhi Institute of Science and Technology, Ghatkesar, Ranga Reddy District , A.P., INDIA. (Received 27 February 2010, accepted 14 December 2010) Abstract: The momentum and heat transfer in laminar boundary layer flow of non-newtonian fluids past a semi-infinite flat plate with the thermal dispersion in the presence of a uniform magnetic field are analyzed. Both the cases of static plate and continuous moving plate were considered. The resulting governing equations are transformed into non-linear ordinary differential equations using appropriate transformations. The system of coupled non-linear ordinary equations was linearized using Quasi-linearization technique and then solved numerically based on finite difference scheme. The effects of various physical parameters such as power-law fluid index n, heat transfer index s, Prandtl number P r and magnetic field parameter M, which determine the velocity profiles and temperature profiles are shown graphically. The generalized local Nusselt number is tabulated for various parameters such as power-law fluid index n, heat transfer index s and stream wise coordinate x. Keywords: magnetic field effects; non-newtonian power-law index; heat transfer index; thermal dispersion and Local Nusselt number 1 Introduction Non-Newtonian fluids have many applications in modern technology. The study of flow and heat transfer for an electrically conducting fluid past a heated surface has attracted the interest of many investigations, in view of its applications in many physical, geophysical and industrial fields. To be more specific, it may be pointed out that many manufacturing processes involve the cooling of continuous sheets or filaments by drawing them through a quiescent fluid. These sheets or filaments are usually stretched during the drawing process. It is known that the properties of the final product depend to a great extent on the rate of cooling. By drawing such sheets in an electrically conducting fluid subject to a magnetic field, the rate of cooling can be controlled and the final product with required characteristics can be obtained. The subject of the boundary- layer flow on a continuously moving surface traveling through a quiet ambient fluid is currently one of important field in view of its relevance to a number of engineering processes. Flows due to a continuously moving surface is encountered in several processes for thermal and moisture treatment of materials, particularly in processes involving continuous pulling of a sheet through a reaction zone, as in metallurgy, in textile and paper industries, in the manufacture of polymeric sheets, sheet glass and crystalline materials. The flow and heat transfer problems of visco-elastic fluids over a stretching sheet subject to a transverse magnetic field have been considered by several investigators. Most of the investigations on boundary layer flow in these fluids are concerned with a simplified model neglecting the dispersion effect from the energy equation. However, the non- Newtonian fluids have viscosity and hence dispersion of energy may significantly effect the heat transfer parameters. Without dispersion term in the energy equation, the problems have been considered by Acrivos et al. [1], Schowalter [2], Lee and Ames [3], Fox et al. [4], Lin and Shih [5]. It is known from these studies that similarity solutions do not exist for the above mentioned problems except for the limiting case of infinite Prandtl numbers. Thus Fox et al. [4] applied the Corresponding author. address: kishan n@rediffmail.com Copyright c World Academic Press, World Academic Union IJNS /477

2 302 International Journal of Nonlinear Science, Vol.11(2011), No.3, pp Karman Pohlhausen method to obtain an approximate solution while Lin and Shih [5] used the local similarity method by omitting the derivatives with respect to the coordinate along the plate. Vasantha et al. [6] studied a non-newtonian flow past a wedge with non-isothermal surfaces using numerical method. For the non-newtonian power-law fluids, the hydrodynamic problem of the MHD boundary layer flow over a continuously moving surface has been dealt with by several authors such as Andersson et al. [7], Cortell [8], and Mahmoud et al. [9]. However, relatively less attention has been paid to the accompanying heat transfer problem of power-law fluids past a stretching surface in the presence of a magnetic field. The MHD free-convection flow of a non-newtonian power-law fluid at a stretching surface with a uniform free-stream was studied by E. M. Abo-Eldahab and A. M. Salem [10]. It is worth mentioning to this end that the effect of dispersion on heat transfer in non-newtonian fluids has been previously studied, primarily in pipe and duct flows under various boundary conditions by Skelland [11], Froyshteter [12]. It was shown in these papers that the dispersion plays a great role in non-newtonian than Newtonian fluids and in some cases dispersion may be a significant factor. Recently Kumari et al. [13] investigated the thermal dispersion effect for the boundary layer flow of a non-newtonian fluid past a semi-infinite flat plate. Both the cases of static and continuous moving plate were considered. In the present work, we investigated the MHD effect on laminar boundary layer flow of a non-newtonian fluid past a semi infinite flat plate by considering the thermal dispersion. The flow is subjected to external magnetic field. The governing non-linear partial differential equations under the assumption of small magnetic Reynolds number transformed into a system of ordinary differential equations which are solved numerically using the implicit finite difference scheme. 2 Mathematical formulation Consider the problem of laminar flow of a non-newtonian fluid with constant thermo physical properties over a static or due to a continuous moving semi-infinite flat plate at the following transport properties.[14] τ ij = pδ ij + K 1 2 I 2 1 q = K 1 2 I 2 s/2 (n 1) 2 e ij (1) grad T (2) Here τ ij and e ij are the tensors of stress and strain-rate, δ ij is the unit tensor, I 2 is the second invariant of the strainrate tensor, K and K 1 are the consistency index and the conductivity parameter of the fluid, n and s are superscripts identifying non-newtonian behavior in the flow and heat transfer respectively, p is the pressure, q is the heat flux and T is the temperature. We assume that the plate is maintained at the constant temperature T w which is the greater than the ambient temperature T. We also designate by U the flow velocity of the ambient fluid along the static plate and the velocity of the moving plate. The analysis will be restricted to the case in which the usual boundary layer approximation can be used. With this assumption and using equations (1), (2) the boundary layer equations can be written in nondimensional form as u x + v y = 0 (3) u u u + v x y = y [ u y n 1 u y u θ θ + v x y = 1 [ u s ] θ P r y y y The boundary conditions to be satisfied by these equations are (I) Static plate u = v = 0, θ = 1 on y = 0 u 1, θ 0 as y (II) Moving plate u = 1,v = 0, θ = 1 on y = 0 u 0, θ 0 as y ] σb2 0 (u 1) (4) ρ } } (5) (6) (7) IJNS for contribution: editor@nonlinearscience.org.uk

3 K. Naikoti, S. Borra: Quasi-linearization Approach to MHD Effects on Boundary Layer Flow of Power-Law Fluids 303 Here x and y are coordinates measuring distance along the plate and normal to it, u is the velocity component along x-direction and v is the velocity component along y-direction and P r is the generalized Prandtl number. The non- dimensional quantities x, y, u, v and θ are related to their dimensional counterpart by x, y, u, v and T as x = Lx, y = LyRe 1 1 n+1, u = Uu, v = UvRe n+1, θ = T T T w T, where Re = U 2 n L n K/ρ is the generalized Reynolds number, M = σb2 0 x ρ is the magnetic field parameter, P r = ( U 1 s L 1+s K 1 length of the plate and ρ is the density of the fluid. 3 Method of solution )Re (s+1) (n+1) is the modified Prandtl number with L being a characteristic We shall further transform equations (4) and (5) into a set of ordinary differential equations amenable to a numerical solution. For this purpose we introduce the variables ψ = x 1 n+1 f(η), θ = g(x, η), η = yx 1 n+1. (8) where η is the pseudo-similarity variable and the stream function ψ is defined in the usual way as u = ψ y and v = ψ x. It can be shown that the velocity components in terms of the new variables are u = f (η), v = 1 n x n+1 (f ηf ). (9) n + 1 using these in equations (4) and (5), we obtain the following coupled non-linear differential equations. d dη ( f n 1 f ) + 1 n + 1 ff + M(1 f ) = 0. (10) x (n 1 s)/(n+1) 1 P r in which primes denote differentiation with respect to η. The boundary conditions (6) and (7) will be reduced as (I) Static plate f(0) = f (0) = 0, g(x, 0) = 1 f ( ) = 1, g(x, 0) = 0 (II) Moving plate η [ f s g ] + 1 n + 1 fg = xf g x. (11) f(0) = 0, f (0) = 1, g(x, 0) = 1 f ( ) = 0, g(x, 0) = 0 To solve the system of transformed governing equations (10) and (11) with the boundary conditions (12) for static plate and (13) for moving plate, first we applied the Quasi linearization technique [15], to equation (10) to obtain where } A 0 f + A 2 f + A 3 f + A 4 f = A 5 A 1 [f ] n 1 M (14) A 0 = n[f ] n 1, A 1 = nf, A 2 = 1 n + 1 F, A 3 = M, A 4 = 1 n + 1 F, A 5 = n[f ] n 1 F + 1 F F n + 1 here F is assumed to be a known function. Now using the finite difference scheme, the equations (14) and (11) are transformed to B 0 [i]f[i + 1] + B 1 [i]f[i] + B 2 [i]f[i 1] B 3 [i]f[i 2] = B 4 [i] (15) } (12) (13) IJNS homepage:

4 304 International Journal of Nonlinear Science, Vol.11(2011), No.3, pp D 1 [i]g[i + 1] D 2 [i]g[i] + D 3 [i]g[i 1] = 0 (16) where B 0 [i] = 2A 0 [i] + 2hA 2 [i] + h 2 A 3 [i], B 1 [i] = 2h 3 A 4 [i] 6A 0 [i] 4hA 2 [i], B 2 [i] = 6A 0 [i] + 2hA 2 [i] h 2 A 3 [i], B 3 [i] = 2A 0 [i], B 4 [i] = 2h 3 {A 5 [i] A 1 [i](f 2 [i]) n 1 M}, D 1 [i] = C 1 [i] + hc 2 [i], D 2 [i] = 2C 1 [i] + hc 2 [i], D 3 [i] = C 1 [i], C 1 [i] = (x) n 1 s n+1 (n + 1)[f ] s, C 2 [i] = (x) n 1 s n+1 (n + 1)s[f ] s 1 f + P r (ihf + f) here h represents the mesh size in η direction. The system of equations (15) & (16) are solved under the boundary conditions (12) & (13) by Gauss-Seidel iteration method and computations were carried out by using C programming. The numerical solutions of f are considered as (n+1) th order iterative solutions and F are the n th order iterative solutions. After each cycle of iteration the convergence check is performed, and the process is terminated when F f < Heat transfer The rate of heat transfer in terms of the generalized local Nusselt number Nu x can be computed from (s+1) n+1 Nu x Rex = f (0) s g (x, 0) (17) where Re x is the generalized local Reynolds number and Nu x is given by Nu x = Ls+1 U s. q w x s+1 K 1 (T w T ) with heat flux at the plate obtained from equation (2). 5 Results and discussion The parametric study is performed to explore the effects of various governing parameters on the fluid flow and heat transfer characteristics. These major parameters include the Magnetic field parameter M, Prandtl number P r, Power law fluid index n and Power law index for heat transfer s. The Computations are carried on the assumption that the liquids obey the Fourier and non-fourier thermal conductivity equations with s = 0 and s = n 1 for pseudo-plastic fluids (n = 0.6), Newtonian fluids (n = 1.0) and dilatant fluids (n = 1.8). IJNS for contribution: editor@nonlinearscience.org.uk

5 K. Naikoti, S. Borra: Quasi-linearization Approach to MHD Effects on Boundary Layer Flow of Power-Law Fluids 305 Figure 1: Velocity Profiles f for different values of n and M for static plate with x = 0.1 ) n = 1.0 and ( d ) n = 1.8 ( a ) M = 0, ( b ) n = 0.6, ( c Figure 2: Velocity Profiles f for different values of n and M for static plate with x = 0.1 ( a ) M = 0, ( b ) n = 0.6, ( c ) n = 1.0 and ( d ) n = 1.8 IJNS homepage:

6 306 International Journal of Nonlinear Science, Vol.11(2011), No.3, pp Figure 3: Temperature Profiles g for various values of P r for static plate with x = 0.1, M = 0 and s = 0. ( a ) n = 0.6, ( b ) n = 1.0 and ( c ) n = 1.8 Figure 4: Temperature Profiles g for various values of P r for static plate with x = 0.1, M = 0 and s = n 1. ( a ) n = 0.6, ( b ) n = 1.8 We discuss the velocity profiles f, f and temperature profiles g for the static plate and moving plate by taking the stream wise co-ordinate x as 0.1. Fig.1 show the velocity profiles f for static plate, where we can see from them that at any η, the velocity increases with the increase of power law fluid index n and the magnetic field parameter M. Similar behavior of the velocity profiles f for static plate can be seen from Fig.2. Figs. 3 & 4 elucidates the influence of the generalized Prandtl number on the temperature profiles for static plate. It is evident from the figures that increase in Prandtl number results in decrease of temperature profiles when n = 0.6, n = 1.0 and n = 1.8 for s = 0 and s = n 1. Figs. 5 & 6 show how the temperature profiles for static plate change with the magnetic field parameter M for different n values when s = 0 and s = n 1. We can observe that as M increases the temperature profiles decrease when n = 0.6, n = 1.0 and n = 1.8. We notice that the influence of magnetic field parameter on the temperature profiles for static plate is less when n = 1.8. The effect of power law fluid index n on temperature profiles for static plate is shown by the fig. 7, in which the temperature profiles decreases with the increase of n for s = 0 and s = n 1. From the Figs. 8 & 9, we can observe that the effect of power law fluid index n and the effect of magnetic field parameter M on velocity profiles f and f for moving plate are almost same as in case of static plate. Similarly Figs.10 & 11 show that the effect of the generalized Prandtl number P r and the effect of power law fluid index n on temperature profiles for moving plate behaves as almost same as in case of static plate, while the temperature profiles for moving plate were found to be insensible to change in magnetic field parameter M. IJNS for contribution: editor@nonlinearscience.org.uk

7 K. Naikoti, S. Borra: Quasi-linearization Approach to MHD Effects on Boundary Layer Flow of Power-Law Fluids 307 Figure 5: Temperature Profiles g for different values of M for static plate with x = 0.1, P r = 100 and s = 0. ( a ) n = 0.6, ( b ) n = 1.0 and ( c ) n = 1.8 Figure 6: Temperature Profiles g for different values of M for static plate with x = 0.1, P r = 100 and s = n 1. ( a ) n = 0.6 and ( b ) n = 1.8 Figure 7: Temperature Profiles g for different values of n for static plate with x = 0.1, P r = 100 and M = 0, s = 0, s = n 1. IJNS homepage:

8 308 International Journal of Nonlinear Science, Vol.11(2011), No.3, pp Figure 8: Velocity Profiles f for different values of n and M for moving plate with x = 0.1 ( a )M = 0, ( b ) n = 0.6, ( c ) n = 1.0 and ( d ) n = 1.8 n Table 1: Nu x Re (s+1)/(n+1) x for P r = 100 X Static plate Moving plate s=0 s=n-1 s=0 s=n IJNS for contribution: editor@nonlinearscience.org.uk

9 K. Naikoti, S. Borra: Quasi-linearization Approach to MHD Effects on Boundary Layer Flow of Power-Law Fluids 309 Figure 9: Velocity Profiles f for different values of n and M for moving plate with x = 0.1 ( a )M = 0, ( b ) n = 0.6, ( c ) n = 1.0 and ( d ) n = 1.8 Figure 10: Temperature Profiles g for various values of P r for moving plate with x = 0.1, M = 0 and s = 0. ( a ) n = 0.6, ( b ) n = 1.0 and ( c ) n = 1.8 IJNS homepage:

10 310 International Journal of Nonlinear Science, Vol.11(2011), No.3, pp Figure 11: Temperature Profiles g for different values of n for moving plate with x = 0.1, P r = 100, M = 0 and s = 0. Figure 12: Variation of Local Nusslet number for different values of x with P r = 100, M = 0 and s = 0 (a) Static Plate and ( b ) Moving Plate. The resulting solutions for the local Nusselt number are given in the Table and are also displayed in Fig.12 for both the cases which show that the local Nusselt number increases for Pseudo plastic fluids, remains constant for Newtonian fluids and decreases for dilatant fluids with the increase of the stream wise coordinate x when s = 0. The local Nusselt number is independent of stream wise co-ordinate x when s = n 1. On the other hand, the effect of heat transfer index s is to increase the local Nusselt number for non-newtonian fluids for both static and moving plate. The local Nusselt number is higher for moving plate than the static plate except for pseudo platic fluids when s = n 1. 6 Conclusions 1. The effect of magnetic field parameter is to increase the velocity profiles in both the cases of static and moving plate. 2. The effect of magnetic field is to decrease the temperature profiles in the both the cases of static and moving plate. 3. The effect of Prandtl number is to decrease the temperature profiles in both the cases of static and moving plate. 4. The temperature profiles for moving plate were found to be insensible to change in magnetic field parameter M. 5. The effect of heat transfer index s is to decrease the temperature profiles for non-newtonian fluids. References [1] A. Acrivos, M. J. Shah, E. E. Peterson. Momentum and heat transfer in laminar boundary-layer flows of non- Newtonian fluids past external surfaces. AIChE JI, 6(1960): [2] W. R. Schowalter. The application of boundary-layer theory to power-law pseudo-plastic fluids: similar solution. AIChE JI, 6(1960): [3] S. Y. Lee, W.F. Ames. Similarity solutions for non-newtonian fluids. AIChE JI, 12(1966): [4] V. G. Fox, L. E. Erickson, L. T. Fan. The laminar boundary layer on a moving continuous flat plate immersed in a non-newtonian fluid. AIChE JI, 15(1966): IJNS for contribution: editor@nonlinearscience.org.uk

11 K. Naikoti, S. Borra: Quasi-linearization Approach to MHD Effects on Boundary Layer Flow of Power-Law Fluids 311 [5] H. T. Lin, Y. P. Shih. Laminar boundary layer heat transfer to power-law fluids. Chem. Engineering comm., 4(1980): [6] R. Vasantha, I. Pop, G. Nath. A numerical solution for the heat transfer in non-newtonian flow past a wedge with non-isothermal surface. Engineering Transactions, 36(1988): [7] H. I. Anderrson, K. H. Bech, B. S. Dandapat. Magneto hydrodynamic flow of a power-law fluid over a stretching sheet. Int.J. Non-linear Mech., 27(1992): [8] R. Cortell. A note on magneto hydrodynamic flow of a power law fluid over a stretching sheet. Appl. Math. Comput., 168(2005): [9] M. A. A. Mahmoud, M. A. E. Mahmoud. Analytical solutions of hydro magnetic boundary-layer flow of a non- Newtonian power-law fluid past a continuously moving surface. Acta Mech., 181(2006): [10] E. M. Abo-Eldahab, A. M. Salem. MHD Free-convection flow of a non-newtonian power-law fluid at a stretching surface with a uniform free-stream. Applied Mathematics and Computation, 169(2005): [11] A. H. P. Skelland. Non-Newtonian flow and heat transfer. Oxford, Univ. Press, [12] G. B. Froyshteter. Generalized theory of non-isothermal laminar pipe flow and heat transfer with non-newtonian fluids. Case of internal heat sources. Heat Transfer- Soviet Research, 9(1977): [13] M. Kumari, I. Pop, G. Nath. Analysis of thermal dispersion effect on laminar boundary layer in flow of non- Newtonian fluids. ZAMM., 72(1992)(8): [14] Yu. I. Shvets, V. K. Vishnevskiy. Effect of dissipation on convective heat transfer in flow on non-newtonian fluids. Heat Transfer-Soviet Research, 19(1987): [15] R. E. Bellman, R. E. Kalaba. Quasi-Linearization and Non-linear boundary value problems. Elsevier, NewYork, IJNS homepage:

MHD Non-Newtonian Power Law Fluid Flow and Heat Transfer Past a Non-Linear Stretching Surface with Thermal Radiation and Viscous Dissipation

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