Homogenization of Coupled Problems of a Fluid Flow with Magnetic Particles

Abstract

This thesis is devoted to the asymptotic analysis of systems of partial differential equations arising from the physical models of complex fluids. We aim to derive the effective behavior of such mixtures from the properties of their constituents via rigorous mathematical methods, in particular, the homogenization method. The latter is a set of techniques and toolboxes that are suitable to deal with partial differential equations with rapidly oscillating coefficients.

The mixture of interest is a colloid consisting of a large number of fairly small magnetizable particles suspended in a carrier fluid, which is usually called a magnetorheological fluid. We consider two mathematical models describing the fluid, corresponding to a weakly coupled and a strongly coupled system of partial differential equations. Both systems depend on a small parameter that represents the size of heterogeneities, and thus, they are called fine-scale systems. Our main aim is to find equivalent systems that govern the behaviors of the quantities of interest as the size of heterogeneities approaches zero. Such equivalent systems are called effective systems.

For the weakly coupled model, we start with formal asymptotic analysis, then move on to rigorous homogenization with the aid of the two-scale convergence theory. The effective system is derived, and connections to the properties of each constituent are explicitly demonstrated by the effective tensors’ formulas and cell problems. Corrector results, which provide the strong convergences of fine-scale solutions, are also established. With a different boundary condition, we are able to relax a strict regularity assumption by utilizing the compactness method. For the strongly coupled model, the saddle point method and a corrector result of two-scale convergence allow us to obtain the effective systems as well as the explicit formulas for effective quantities.

Our results confirm mathematically that a suspension of magnetizable particles in a nonconducting carrier fluid behaves like liquid metal.