Ali J. Chamkha Assistant Professor, Mechanical Engineering Department, Kuwait University, Safat, 13060 Kuwait Mem. ASME Analytical Solutions for Flow of a Dusty Fluid Between Two Porous Flat Plates Equations governing flow of a dusty fluid between two porous flat plates with suction and injection are developed and closed-form solutions for the velocity pro- files, displacement thicknesses, and skin friction coefficients for both phases are obtained. Graphical results of the exact solutions are presented and discussed. Introduction This paper deals with the two-dimensional, steady, laminar, fully developed flow of a dusty fluid between two parallel porous flat plates. The plates are infinitely long and separated by a fixed distance of h\ {h multiplied by 1, a constant) with the lower plate being coincident with the plane y - 0. The flow takes place due to the action of a constant pressure gradient applied in the ^-direction. Uniform fluid-phase suction and injection are imposed at the lower and upper plates, respec- tively. The fluid and particulate phases are both assumed in- compressible. In the present work both phases (the fluid phase and the particle cloud) are treated as continua. The basic assumption in the theoretical analysis of such a suspension is that the average properties of the particles are described in terms of continuous variables. Extensive work based on the continuum modeling of particulate (particle-fluid) suspensions has been reported (see, for instance, Marble, 1970; Di Giovanni and Lee, 1974; Ishii, 1975; and Drew, 1979, 1983). The mathematical model employed in the present work rep- resents a generalization of the original dusty-gas model (a model restricted for particulate suspensions having small vol- ume fraction. See, for instance, Marble, 1970) by allowing for finite particulate volume fraction. In this case the particle- phase viscous effects are important. In the absence of particle-phase viscous effects (small par- ticulate volume fraction), it was reported by Chamkha (1992) that a difficulty exists as to the appropriate particle-phase boundary conditions that need to be used for this problem. The purpose of this paper is to obtain a closed-form solution for the problem described above for uniform and finite par- ticle-phase volume fraction by applying slip boundary con- ditions familiar from rarefied gas dynamics on the particle phase. This allows one to explore the qualitative behavior brought about by changes in boundary conditions and various parameters of the system. Governing Equations Consider steady laminar flow of a suspension of solid spher- Contributed by the Fluids Engineering Division for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received by the Fluids Engineering Division June 29, 1992; revised manuscript received May 24, 1993. Associate Technical Editor: M. W. Reeks. ical particles uniformly distributed in a continuous carrier fluid between two infinite parallel porous flat plates due to a con- stant applied pressure gradient. The governing equations for this investigation are based on the balance laws of mass and linear momentum for both phases. These are given by V((l-tf)V) = 0, (\a) (16) y(«V„) = 0 P (i -40V- vv= - v((i -<t>)p) + v .(/i(i -</>)( vv+ vv 7 )) + PP 4>{\ P -V)/T, (2a) P^V P . W p = V.(M P *(W I ,+ VV l))-f>,My p -\)/T {lb) where V is the gradient operator, <j> is the particle volume fraction, V is the fluid-phase velocity vector, V p is the particle- phase velocity vector, p is the fluid-phase density, p p is the particle-phase density, p is the fluid pressure, n is the fluid- phase dynamic viscosity, r is the momentum relaxation time (time needed for the relative velocity between the two phase to decrease e" 1 of its original value), and \x p is the particle- phase dynamic viscosity, and a superposed Tdenotes the trans- pose of a second order tensor. It can be seen from Eq. (2b) that the partial pressure contributed by the particle phase and gravity are neglected. This situation arises when inertia and drag dominate over gravity forces. This obtains when the ve- locity in the x-direction and the suction velocity are large com- pared to the particles settling velocity. The last term in Equation (2a) accounts for the interaction between the two phases and is based on Stoke's linear drag theory. In the present work, (j>, p, p p , ix, n P , and T will all be treated as constants. It is convenient to nondimensionalize the governing equa- tions given earlier by using the following equations: y = U, V = e x V c F( v ) y p = e x V c F p ( n ) Cy V W -e y V w , dP/dx=-iiV c G/\ 2 (3) where e* and e y are unit vectors in the x and y directions, respectively, K c is a characteristic velocity, and V w is the suction (or injection) velocity and is a constant and positive. It can be noticed from Eqs. (2) that for steady-state and constant par- ticulate volume fraction conditions the cross stream velocities for both phases have to be equal. The resulting nondimensional equations can be shown to be 354/Vol. 116, JUNE 1994 Transactions of the ASME Copyright © 1994 by ASME Downloaded 01 Oct 2012 to 109.171.137.210. 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