The Multiple Solutions of Laminar Flow in a. Uniformly Porous Channel with Suction/Injection

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1 Adv. Studies Theor. Phys., Vol. 2, 28, no. 1, The Multiple Solutions of Laminar Flow in a Uniformly Porous Channel with Suction/Injection Botong Li 1, Liancun Zheng 1, Xinxin Zhang 2, Lianxi Ma 3 1 Department of Mathematics and Mechanics University of Science and Technology Beijing, Beijing 183, China leedonlion48@163.com, liancunzheng@sina.com 2 Mechanical Engineering School University of Science and Technology Beijing, Beijing 183, China 3 Department of Physics, Blinn College, Bryan, TX 7785, USA malianxi@gmail.com Abstract: This paper presents a numerical investigation for laminar flow in a uniformly porous channel with suction/injection at both moving walls. The characteristics for the existence of multiple solutions to the problem are numerically established for values of Reynolds number and the velocity coefficient. Mathematics Subject Classification: Numerical hydromechanics Keywords: Porous channel, moving walls, shooting method, multiple solutions 1. Introduction Following the pioneering work of Berman [1], the problems of the steady, incompressible, laminar flow in channels or circular pipes with uniformly porous walls with suction/injection at both walls have attracted considerable attention during the last few decades. The main reason for it is probably that fluid flow is produced

2 474 Botong Li, Liancun Zheng, Xinxin Zhang, Lianxi Ma industrially in increasing quantities and is therefore just likely to be pumped in a plant. The great majority of theoretical investigations in this field described the fluid flow in the vicinity of the surface with the aid of similarity solutions [2-6]. The purpose of this paper is to present a numerical investigation for this problem. A special emphasis is given to the formulation of boundary layer equations, which may provide the multiple similarity solutions. 2. Formulation of the problem Consider the steady, incompressible, laminar flow along a two-dimensional channel with porous walls through which fluid is injected/extracted with uniform speedv w. Let the channel width be 2h and introducing the dimensionless variable: y η = (1) h the Navier-Stokes equations and the continuity equation are written as: 2 2 u v u 1 P u 1 u u + = + ν + x h x x 2 h 2 2 η ρ η (2) 2 2 v v v 1 P v 1 v u + = + ν + x h h x 2 h 2 2 η ρ η η (3) The boundary conditions are: u ux± (, 1) =, vx (, ± 1) =± vw, ( x,) = η, vx (,) = (4) The problem is reduced to the following equation: 2 f + R[ f ff ] = k where k a constant and the suction Reynolds number of the flow is taken as: vh R ( w ) (6) ν The boundary conditions (4) become: f () =, f () =, f (1) = 1, f (1) = (7) (5)

3 Multiple solutions of laminar flow 475 Watson investigated analytically a similar flow of fluid which is driven by uniform steady suction through the porous and accelerating walls of the channel. The problem is also reduced to equation (5) with the following boundary conditions: f () =, f () =, f (1) = 1+ k f (1) = k, Where k is the velocity coefficient of accelerating walls ( ( 1,] ). (8) 3. Numerical Investigation In order to obtain the numerical solution, we transfer the problems (5) and (8) to a system of four first-order equations as follows f = u, v = w,v = w,w = Rfw Ruv (9) The corresponding boundary conditions are: f() =, v() =, f (1) = 1 + k, u(1) = k (1) We introduce the parameters of t and s such that u() = t, w() = s (11) The problem now is to find the parameters ts., We denote the solutions of (9)-(1) as f η, t 1, t ), u η, t, t ), v η, t, t ) and w η, t, t ) Thus the following equations ( 2 ( 2 ( 2 are solved by using the Newtonian technique. ( 2 φ t, t ) = f (1, t, t ) 1, φ t, t ) = u(1, t, t ) (12) ( = ( = 4. Numerical Results The Watson problems is discussed by dividing the value of the suction Reynolds number into three sections for different velocity coefficient and some interesting numerical results are presented in figures 1-4. Figures 1-2 show the distribution characteristic of values of f (1) R ( R > or R < ).The results will be divided into three sections. with

4 476 Botong Li, Liancun Zheng, Xinxin Zhang, Lianxi Ma For k =, Only a single solution is observed for each value of R [,12.165) ; Triple solutions are found for each value of R (12.165, + ) ; Only a single solution is observed for R <. For k =.5, only a single solution for each value of R [, 25) ;Triple solutions for each value of R (25, + ) ; Only a single solution for R <. For k =.75, only a single solution for each R [, 48) ;Triple solutions for each value of R (48, + ) ; Only a single solution for R <. Figures 3-4 show the unique velocity profiles and the unique shear profiles for R = 1 with different values of velocity parameters. 5. Conclusions The multiple solutions are investigated for laminar uniform flow in a channel with special suction or injection at both porous walls for certain parameters and the transfer characteristics are discussed in detail. Acknowledgement: The work is supported by the National Natural Science Foundations of China (No ). References [1] A.S.Berman, Laminar flow in channels with porous walls, J.Appl. Phys, 24(1953), [2] A. MG, C Lu and S P H, Asymptotic behavior of solutions of a similarity equation for laminar flows in channels with porous walls, SIAM. J. Applied Mathematics, 49(1992), [3] C.L, On existence of multiple solutions of a boundary value problem from pipe flow, Q. A. M., 2(1994), 361.

5 Multiple solutions of laminar flow 477 [4] C.L, On the asymptotic solution of laminar channel flow with large suction, J. M. A, 28 (1997), [5] E.B.B.Watson, W.H.H.Banks, M.B.Zaturska, et. al., On transition to chaos in two-dimensional channel flow symmetrically driven by accelerating walls, J.Fluid Mech., 212(199), [6] S. P. Hastings, C. Lu and A. D. Macgillivray, A boundary value problem with multiple solutions from the theory of laminar flow, SIAM. J. Math. Anal, 23(1992), Received: November 1, 27

6 478 Botong Li, Liancun Zheng, Xinxin Zhang, Lianxi Ma -f''(1) = = f''(1) = = = = R R Fig.1 Multiple values of f (1) with R ( R > ) Fig.2 Unique value of f (1) with R ( < R ) f ' = -6 = = -. 5 = f ' ' -3 = -. 5 = η η Fig.3 Unique velocity profiles for R = 1 Fig.4 Unique shear stress profile for R = 1