A. J. Chamkha Assistant Professor, Department of Mechanical Engineering, Kuwait University, Safat 13060 Kuwait Mem. ASME J. Peddieson, Jr. Professor, Department of Mechanical Engineering, Tennessee Technological University, Cookeville, TN 38505 Boundary Layer Theory for a Particulate Suspension Order of magnitude considerations are employed to develop boundary layer equa- tions for two phase particle/fluid suspension flows. It is demonstrated that a variety of possibilities exist and three of these are examined in detail. Two are applied to the problem of flow past a semi-infinite flat plate. Introduction This paper is concerned with boundary layer theory for a particulate suspension. Given the importance of boundary lay- ers in applications, this topic has received surprisingly little attention. There is a history of work on the problem of the steady laminar boundary layer on a semi-infinite flat plate with recent contributions by Osiptsov (1980), Prabha and Jain (1980), and Wang and Glass (1988). References to earlier work can be found in these papers. In contrast to investigations of the flat plate and a few other specific geometries, there appears to have been little effort devoted to determining the general form of boundary layer equations. The present paper deals with this topic. A boundary layer order of magnitude analysis is carried out using a typical set of two fluid equations representative of the current literature. It is found that a variety of outcomes are possible depending on the order of magnitude assumptions selected. Three of the most interesting cases are singled out for explicit presentation. Some specific numerical results are then given for the problem of steady laminar boundary layer flow past a semi-infinite flat plate. It is shown that the bound- ary layer model employed greatly influences predictions. Governing Equations The boundary layer analysis to be described in the present work is based on the following typical set of two phase flow equations. W-V-((l-0)v c ) = O, d,4.+ V-W>v d ) = 0 (1) represent respective balances of mass for the fluid and partic- ulate phases (with the true densities of both phases assumed constant). In Eqs. (1) V is the gradient operator, t is time, <j> is the particulate volume fraction, v c is the fluid phase velocity vector, and \ d is the particulate phase velocity vector. /o c (l-</>)(d,v c + v c -W c )=V-g c -f p rf <M«,v d + v d -W d )=V-£ d +f (2) Contributed by the Fluids Engineering Division for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received by the Fluids Engineering Division May 7, 1992; revised manuscript received March 18, 1993. Associate Technical Editor: M. W. Reeks. represent respective balances of linear momentum for the fluid and particulate phases (with external body forces neglected). In Eqs. (2) p c is the fluid true density, p d is the particulate true density, g c is the fluid phase stress tensor, g d is the particulate phase stress tensor, and f is the interphase force per unit volume acting on the particulate phase. The balance laws discussed above will be supplemented by the constitutive equations g c =-(\-M)pl +2 A t c (l-<«D c ; D c =(vv c +W c r )/2 g d = -{\<f>p + q)l +2 Mrf </.D d ; D d= ( V v d + VvJ)/2 f = p d 4>y r /r + \pV<t>;\ r = \ d -v c (3) wherep is the indeterminate pressure, q is the particulate phase dynamic pressure, X is a coefficient which determines the ap- portionment of the indeterminate pressure gradient between the phases (see below), ^ c and fi d are dynamic viscosity coef- ficients, T is the interphase relaxation time, 4 is the unit tensor, and a superposed T indicates the transpose of a second order tensor. In general q, X, LI C , \i. d , and T are functions of such quantities as 4>, v r , and the invariants of D c and D rf . The following brief comments about Eqs. (3) are in order. The forms of Eqs. (3b,c) should be such that Eq. (2b) will reduce to the Eulerian form of the equation of motion for a single particle when the volume fraction is sufficiently small. Angular momentum considerations do not require that g c and g d be individually symmetric but do require that the combina- tion a c + g d be symmetric (in the absence of body moments). Several particle phase stress mechanisms have been proposed. These include direct contact between particles, wall effects, local particle deformations, and the consequences of the av- eraging required to model a system containing discrete particles as a continuum. The precise forms of X, /n c , \s, d , and q are model dependent. In the present work it is not necessary to restrict attention to a specific model and this has not been done. The focus of the present paper is on laminar flow. The inclusion of D c and D rf in the list of arguments given above formally allows Eqs. (3) to include algebraic turbulence models. It is highly doubtful, however, that such models are sufficiently inclusive to simulate two phase turbulence under any but the simplest conditions. Journal of Fluids Engineering MARCH 1994, Vol. 116/147 Copyright © 1994 by ASME Downloaded 01 Oct 2012 to 109.171.137.210. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm