Abstract 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. Introduction 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), ........................................(1) Eq. 1 has suggested for isotropic porous media a vector form of Darcy's law, ........................................(2) 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.

Abstract This work was undertaken to define the conditions under which the Deborah number characteristic of the flow process may become large enough to cause significant deviations from the usual drag coefficient-Reynolds number relationships for purely viscous fluids flowing through porous media under non-inertial conditions. The analysis suggests that major [order-of-magnitude] effects may be expected to occur at Deborah number levels in the range 0.1 to 1.0. Experimental studies using a porous medium, having a permeability of 46.4 × 10-0 sq cm. support this analysis and define the critical value of the Deborah number at which viscoelastic effects are first found to be measurable. Prior studies, in which no effects attributable to fluid elasticity were found, are seen to have been confined to lower Deborah number levels. The influences of a high Deborah number level upon the uniformity of the flow and upon the distribution of the fluid residence times in the porous medium are considered briefly. Introduction While the pragmatic significance of studies of flows through porous media requires no discussion, there is additionally a very strong motivation for such studies from a strictly theoretical point of view: flows in this geometry provide an excellent opportunity for a study of the behavior of viscoelastic 25 fluids at high levels of the Deborah number. This dimensionless group, representing a ratio of time scales of the material and the flow process, may be defined as: ........................................(1) in which f1 denotes the relation time of the fluid under the conditions of interest in the problem under consideration and IId represents the second invariant of the deformation rate tensor. This latter term depicts the intensity or the magnitude of the deformation rate process, and the dimensionless group defined by Eq. 1 may be considered to represent the ratio of the size interval required for that fluid to respond to a change in imposed conditions of deformation rate as compared to the time interval between such changes. It is thus an index of the extent to which the velocity field is unsteady from a Lagrangian viewpoint [i.e., from, the viewpoint of an observer moving with a given fluid element as it proceeds its course or trajectory in a process], using the relaxation time of the fluid as a wait of time. For perfectly steady flows (e.g., under laminar flow conditions in a very long tube the Deborah number is identically zero; for highly unsteady processes it may be large. It has been shown elsewhere that quantitative mathematical descriptions of the properties of viscoelastic fluids rather generally predict a fluid-like response to be exhibited whenever the Deborah number is sufficiently low, and that the same materials will exhibit an essentially solid-like response whenever the Deborah number becomes large. In the case of dilute polymeric solutions in steady laminar shearing flows [NDeb = 01], the fluid-like response is, of course, well known and requires no further discussion. That the same materials may behave as elastic solids when deformed suddenly enough [NDeb large] may be demonstrated dramatically by impacting a blunt object suddenly upon a pool of such a "fluid": in this case the material may deform appreciably [sheets 6 to 20 in. in diameter are readily formed], but it retracts elastically to its initial configuration, rather than flowing or splashing as a Newtonian fluid does.