Local volume-averaging of the equations of continuity and of notion over a porous medium, is discussed. For steady-state flow such that inertial effects can be neglected, a resistance transformation is introduced which in part transforms the local average velocity vector into the local force per unit volume which the fluid exerts on the pore walls. It is suggested that for a randomly deposited, though perhaps layered, porous structure this resistance transformation is invertible, symmetric and positive-definite. Finally, for an isotropic porous structure [the proper values of the resistance transformation are all equal and are termed the resistance coefficient] and an incompressible fluid, the functional dependence of the resistance coefficient is discussed using the Buckingham-Pi theorem for an Ellis model fluid, a power-model fluid, a Newtonian fluid and a Noll simple fluid. Based on the discussion of the Noll simple fluid, a suggestion is made for the correlation and extrapolation of experimental data for a single viscoelastic fluid in a set of geometrically similar porous structures.
Darcy's law, involving a parameter k termed the permeability, was originally proposed as a correlation of experimental data for the flow of an incompressible Newtonian fluid of viscosity mu moving axially with a volume flow rate Q through a cylindrical packed bed of cross-section A and length under the influence of a pressure difference [Ref. 1, p. 634),
Eq. 1 has suggested for isotropic porous media a vector form of Darcy's law,
A major difficulty of this equation has been that, since it was not derived, the average pressure P and average velocity z were undefined. Whitaker has recently derived a generalization of Eq. 2 appropriate to anisotropic porous media by taking a local average of the equation f motion. In his result, P and V are local surface averages of pressure and velocity, respectively.
The object of this paper is to develop by a method considerably different from Whitaker's an extension of Darcy's law which is appropriate to viscoelastic fluids. [Viscoelastic is used here in the sense that the materials obey neither of the classical linear relations: Newton's law of viscosity and Hooke's law of elasticity. We being by discussing in the first section the problem, of local volume averaging of the equation of motion as opposed to the local surface averaging explained by Whitaker. In the second section a resistance transformation [the words transformation and tensor are used interchangeably here] is introduced to describe in part the force per unit volume which the fluid exerts on the pore walls; we discuss this transformation for randomly deposited, though perhaps layered, porous media. In the third and fourth sections we specialize to isotropic media and consider the functional dependence of the resistance parameter by means of the Buckingham-Pi theorem. In the third section we take up two simple empirical models which do not account for normal stress effects or the possible memory of the fluid. In the fourth section we consider the problem for the incompressible Noll simple fluid, currently believed to be a general description of a wide variety of memory fluids.