Flow of spatially non-uniform suspensions. Part III: Closure relations for porous media and spinning particles

The methods developed in the earlier papers of this series are applied to the systematic derivation of averaged equations for two situations: slow viscous flow past a system of rigid spheres fixed in space (which may be considered as approximating a porous medium), and the flow induced by a system of fixed spheres all spinning with the same angular velocity. When the same closure relations used in the earlier papers are applied, it is found that the closure coefficients are different. This finding implies that broadly applicable closure relations expressed solely in terms of volume fraction, velocities, and pressure (as usually found in models of the `two-fluidʹ type) are insufficient: it must be that one or more additional variables need to be specified to achieve some degree of universality independent of the particular flow considered. It is also shown that the difficulties in the prescription of the viscosity parameter for use in the Brinkman equation derive from the fact that the correct parameter is actually the combination of two different viscosities that accidentally end up combined into a single term when the particles are fixed.