Viscous flow and collapse of macroscopic cavities in a granular material

Abstract

The viscous flow in a granular material containing macroscopic cavities (or clear fluid regions) is determined on the basis of the generalized Darcy equation (outside the cavity region) and the Stokes equation (inside the cavity region). A three-dimensional singularity termed a “Darcylet”, which is a fundamental solution in the granular material, is distributed over the surface of a spherical cavity of unit size that surrounds the isolated singularity. The latter is calibrated with the exact solution and applied to describe the interaction between two cavities, which successfully reproduces previous analytic results. The proposed method is applicable to two cavities until they come into contact, thus providing a theoretical explanation of the fundamental collapse-and-merger process of two cavities. Furthermore, we extend this method to the interaction among multiple cavities, which reveals the effect on the flow field of an arbitrary number of cavities in an arbitrary configuration. To demonstrate the potential of the proposed method, we first apply it to a one-dimensional array of cavities of arbitrary orientation and curvature and estimate the convective diffusion of fluid volume, before considering a two-dimensional slightly disordered array of cavities. The results indicate the development and creation of new fluid passages, which may provide clues for predicting the anomalously fast spread of contaminants in inhomogeneous granular material; the formation of waterways, collapse of river banks, and landslides in civil engineering; and fluidization in chemical engineering. Our results may also provide insights into medical engineering techniques for creating new blood veins for micro-circulation (angiogenesis).