Fluid Flow in Restricted Geometries

This paper addresses two topics concerning fluid flow in restricted geometries. The first topic is on the relationship between the dynamic permeability κ(ω), where ω denotes frequency, and the microstructure of a porous medium. It is shown that when κ is normalized by its static value κ₀, and the frequency ω by a characteristic ω₀ particular to the sample, the resulting dimensionless function (Formula Presented.) , with (Formula Presented.) and \documentclass{article}\pagestyle{empty}\begin{document}$ {\rm \tilde \omega = }\omega {\rm /}\omega _{\rm 0} $\end{document}, is dominated by the geometry of the throat regions in a porous medium. If the pore cross‐sectional area S varies slowly near the throats, i.e., dS/dz ≅ 0 where z is the distance normal to the cross section, then \documentclass{article}\pagestyle{empty}\begin{document}$ \tilde \kappa ({\rm \tilde \omega)} $\end{document} is an approximate universal function independent of other features of the microstructure. Both theoretical and experimental evidences are presented for such scaling behavior. When scaling holds, the dynamic permeability κ(ω) is found to contain only two pieces of geometric information, and the knowledge of either the low‐frequency or the high‐frequency asymptotic constants of κ(ω) would enable one to deduce the other missing parameters. In particular, since the high‐frequency asymptotic constants of κ(ω) can be related to the electrical formation factor and a weighted volume‐to‐surface ratio, the static permeability value κ₀ may be directly deduced from the knowledge of these parameters. Whereas the first topic concerns single‐phase flow, the second topic of this paper involves the displacement of a fluid by another immiscible fluid in a capillary tube. Here the basic physics lies in the dynamics of the moving contact line, defined as the intersection of the moving fluid‐fluid interface with the tube wall. Through first‐principle hydrodynamic calculations and comparison with experiments, the macroscopic behavior of fluid displacement is linked to the microscopic parameters governing the region around the contact line. It is shown that there are two frictional forces associated with the motion of the contact line. One is the viscous stress, which is primarily responsible for the deformation of the fluid‐fluid interface as the velocity of the moving interface increases. The other frictional force, found experimentally to vary as the square root of velocity, is explained on the basis of capillary‐wave excitation at the fluid‐fluid interface due to contact line motion over wall roughness. Excellent agreement between theory and experiment is obtained.