Viscous properties of flows of an incompressible liquid in an anisotropic heterogeneous granular medium

Abstract

The flow of a liquid or a gas in an anisotropic heterogeneous granular medium is studied. It is shown that the behavior of a medium in which there is a nonuniform velocity field is described by a tensor of “effective” viscous stresses related to the transfer of liquid particles in the transverse direction as the liquid or gas flows past granules of the medium in the longitudinal direction. The structure of the effective viscosities is determined, and hypotheses allowing one to consider them independent of the flow velocity are formulated. Comparison with experimental data shows that the effective viscosities exceed the viscosity of the flowing liquid by a factor of several hundred. A set of differential equations is proposed in which the equations have the form of equations of flow of a viscous and the bulk forces are proportional to the velocity. Darcy’s law turns out to be a particular case of this set. It is demonstrated that the equations obtained allow a solution of the boundary-layer type. To determine the structure of the anisotropic effective viscosities, the concepts of tensors of scales of transverse transfer velocities and “mixing” lengths are introduced. Dimensionless numbers of the type of Reynolds, Euler, Prandtl, and Strouhal numbers are determined. A new number, so, is defined, which characterizes the ratio of Darcy’s pore resistance forces to the effective viscosity forces. A number of practically important problems are solved.