Variational bound finite element methods for three-dimensional creeping porous media and sedimentation flows

Abstract

SUMMARY We present an analytico-computational methodology for the prediction of the effective properties of two types of three-dimensional particulate Stokes flows: porous media and sedimentation flows. In particular, we determine the permeability and average settling rate of media that consist of non-colloidal monodisperse solid spherical particles immersed in a highly viscous Newtonian fluid. Our methodology recasts the original problem into three scale-decoupled subproblems: the macro-, meso- and microscale subproblems. In the macroscale analysis the appropriate effective property is used to calculate the bulk quantity of interest. The mesoscale problem provides this effective property through the finite element solution of the transport equations in a periodic cell containing many particles distributed according to a prescribed joint probability density function. Finally, the microscale analysis allows us to accommodate mesoscale realizations in which two or more inclusions are in very close proximity; this geometrical stiffness is alleviated by introducing simple domain modifications that relax the mesh generation requirements while simultaneously yielding rigorous bounds for the effective property. Our methodology can treat random particle distributions as well as regular arrays; in the current paper we analyse only the latter. # 1998 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids, 26: 145‐175 (1998)