Abstract
Hele-Shaw cells are used to model creeping flow through porous media (where Darcy's law is valid). The effects of inertia on flow about obstructions in a Hele-Shaw cell can be calculated by a perturbation method if one can determine a solution to Laplace's equation.
Results of a computer solution for flow about circular, square and elliptical obstructions are presented These results show that for a modified presented These results show that for a modified Reynolds number of less than 1, the inertia terms are small; and for values of less than 3, the average streamline predicts the ideal flow. Therefore, the analogy might be used for studying flow in porous media up to a modified Reynolds number of at least 3.
Introduction
The nature of fluid flow in porous media is of interest in the fields of soil mechanics, ground water flow, petroleum production, filtration and flow, in packed beds. Because it is very difficult to study the phenomenological behavior of flow in porous media, homologs and analogs are used to study flow characteristics. A Hele-Shaw model, made of two closely spaced plates - usually glass - is often used as an analogy to two-dimensional flow in porous media. Hele-Shaw showed experimentally that the streamline configuration for creeping flow around an obstacle located between two closely spaced parallel plates is the same as for two-dimensional parallel plates is the same as for two-dimensional ideal flow about the same obstacle. Stokes verified these observations mathematically. The usual equation of motion for flow in porous media is Darcy's law. The form of the mathematical statement of Darcy's law is identical, within a multiplicative constant, to the expression for the average velocity over the place gap in the plane of a Hele-Shaw model. These models may be used to describe flow in both homogeneous and heterogeneous porous media.
In the mathematical proof of the Hele-Shaw analogy it is assumed that the convective terms in the Navier-Stokes equations are negligible and that the equations of motion degenerate to Laplace's equation, with pressure the dependent variable. Whenever a Hele-Shaw model is used as an analogy to flow in porous media, the validity of this assumption is in question. Riegels showed that if convection is not neglected, the velocity distribution around a cylindrical obstruction in the flow field depends on a Reynolds number, the plate spacing, and a dimension characteristic of the obstacle. Riegels solution, a perturbation solution, uses the boundary condition that the flow rate into the obstacle averaged over the plate gap at any point on the obstacle is zero. The method requires that a solution to Poisson's equation for the perturbation pressure be found. Riegels evaluated this solution pressure be found. Riegels evaluated this solution for the case of the cylindrical obstruction. His method may be simplified by eliminating the need for solving Poisson's equation for the perturbation pressure. Instead, an analytic expression for the pressure. Instead, an analytic expression for the perturbation pressure gradient is obtained (valid perturbation pressure gradient is obtained (valid for arbitrary shapes) and used to eliminate pressure from the equations for the perturbation velocities. The results show, for symmetrical shapes, that if N'Re less than 1, the convective acceleration terms are small, and that the average velocities represent ideal flow up m at least N'Re 3, where:
..........................................(1)
L is a characteristic dimension of the obstacle perpendicular to flow, b is the plate spacing, mu is perpendicular to flow, b is the plate spacing, mu is viscosity, va is velocity of approach and p is density.
SPEJ
P. 434
Hydromagnetic Blood Flow through a Uniform Channel with Permeable Walls Covered by Porous Media of Finite Thickness