Homogenization of Non-dilute Suspension of Viscous Fluid with Magnetic Particles

TL;DR

This paper rigorously derives the effective macroscopic behavior of a non-dilute suspension of viscous fluid with magnetizable particles using homogenization and two-scale convergence, providing mathematical justification for previous asymptotic analyses.

Contribution

It presents the first rigorous homogenization of a non-dilute magnetizable particle suspension in Stokes flow, extending prior formal asymptotic methods.

Findings

Derived the effective response of magnetizable particle suspensions.

Provided mathematical justification for formal asymptotic analysis.

Applied two-scale convergence to complex particulate flows.

Abstract

This paper seeks to carry out the rigorous homogenization of a particulate flow consisting of a non-dilute suspension of a viscous Newtonian fluid with magnetizable particles. The fluid is assumed to be described by the Stokes flow, while the particles are either paramagnetic or diamagnetic, for which the magnetization field is a linear function of the magnetic field. The coefficients of the corresponding partial differential equations are locally periodic. A one-way coupling between the fluid domain and the particles is also assumed. The homogenized or effective response of such a suspension is derived, and the mathematical justification of the obtained asymptotics is carried out. The two-scale convergence method is adopted for the latter. As a consequence, the presented result provides a justification for the formal asymptotic analysis of L\'{e}vy and Sanchez-Palencia for particulate steady-state Stokes flows.