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Variational-bound finite-element methods for three-dimensional low-Reynolds-number porous media and sedimentation flows

Two phase media are of great importance in science and engineering. In general, two phase systems can be divided into particulate and fibrous materials. This thesis focuses on the former type of material, in which many inclusions are dispersed in a continuous matrix. In particular, we analyze three-dimensional porous media and sedimentation flows at negligible Reynolds numbers, when the particulate phase consists of monodisperse spheres. Because of the complex microstructure of such systems, it is convenient to focus on their macroscopic behavior rather than on their detailed microscopic workings: it is common practice to replace the inhomogeneous materials with a homogenized medium having the appropriate effective property. For this reason, we focus on procedures that yield the permeability and the average settling speed that characterize porous media and suspensions, respectively. Our methodology consists of the analysis of three scale-decoupled subproblems: the macro-, meso-, and micro-scale subproblems. In the macro-scale analysis the bulk quantity of interest is determined by using the effective property. The mesoscale subproblem yields the effective macroscopic property of interest, provided the statistics related to the spatial distribution of the inclusions are known. In fact, the solution is achieved by solving the transport equations over a statistically significant periodic cell extracted from the original medium. The solution of the governing equations is approximated by the finite-element solution over the appropriate spaces. Lastly, the micro-scale treatment is useful when two or more meso-scale inclusions are very close to each other. This geometrically stiff problem is alleviated by introducing geometry changes that also have bounding properties on the effective property of interest. Although our methodology can treat statistically random distributions of spherical inclusions, we focus on very simple periodic sphere distributions to validate our porous media and sedimentation formulations. Thesis Supervisor: Anthony T. Patera Title: Professor of Mechanical Engineering