Formulation of a statistical equation of motion of a viscous fluid in an anisotropic non-rigid porous solid

Abstract

The motion of a viscous fluid through an anisotropic non-rigid porous solid is studied. The porous solid considered in this study is a solid medium involving a very dense and fine network of tubules, whose diameter is much smaller than the characteristic length of the flow system. Kinematically, such a medium should behave like a continuum, i.e. the state of deformation can be well described macroscopically. Dynamically, this type of medium has to be distinguished from the true solid continuum, because one additional body force in addition to its weight (the surface traction exerted by the viscous fluid moving through it) has to be considered in the deformation of the porous solid. Their own weight is the only body force in a true solid continuum. The equations governing the macroscopic motion of the viscous fluid flow through the porous medium of this type are derived by averaging the motion of the fluid through individual elements of pores over a small volume of the porous medium. The physics of the derived equations and the possible applications to the blood flow in the network of capillary blood vessels and to the motion of extracellular fluid through the network of interstitial space are discussed.