Micromechanical Modeling of Heterogeneous Dispersions

Abstract

In this thesis, we present a theoretical study of the micromechanics of heterogeneous dispersions. Heterogeneous dispersions are distinguished by their statistical non-uniformity and the presence of one (or more) internal length scale over which the statistical properties of the dispersion vary. At the macroscale, this statistical non-uniformity manifests as flow instabilities, such as shear banding or structure formation during sedimentation, or through experimental measurements of, e.g. viscosity or pressure drop through a pipe, that depend on the size of the experimental apparatus. At the microscale, heterogeneity is accompanied by a breakdown of the theoretical methods developed to describe statistically homogeneous dispersions. Here, we present four studies investigating the computation of the properties of heterogeneous dispersions by developing mathematical methods to overcome the breakdown associated with the loss of homogeneity. We begin with a study on the non-equilibrium forces that arise on probe particles driven through a dispersion, and demonstrate that these forces tend to cause particle pairs to align in the flow direction. We continue with a mathematical proof of a conjectured relation between Fax\'en's laws and Stokes flow singularity solutions, and demonstrate how it may be used to compute particle motion in the presence of heterogeneous velocity or phoretic (concentration, temperature, electric, etc.) fields. Following this, we derive a framework for the computation of the average material stress arising from statistical heterogeneity at the microscale, which we use to derive constitutive equations for the heterogeneous flow of dilute suspensions of spheres of arbitrary material composition. We conclude with the development of an integral theorem from which material properties in semi-dilute heterogeneous dispersions of hydrodynamically interacting spheres may be computed. A connection between microscopic suspension statistics and macroscopic observable flows is emphasized throughout.