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 Lévy and Sanchez-Palencia [18] for particulate steady-state Stokes flows.

This paper generalizes the results obtained by the authors in [DGB21] concerning the homogenization of a non-dilute suspension of magnetic particles in a viscous flow. More specifically, in this paper, a restrictive assumption on the coefficients of the coupled equation, made in [DGB21], that significantly narrowed the applicability of the homogenization results obtained, is relaxed and a new regularity of the solution of the fine-scale problem is proven. In particular, we obtain a global L ∞ -bound for the gradient of the solution of the scalar equation − div [a (x/ε) ∇ϕ ε (x)] = f (x), uniform with respect to microstructure scale parameter ε 1 in a small interval (0, ε 0 ), where the coefficient a is only piecewise Hölder continuous. Thenceforth, this regularity is used in the derivation of the effective response of the given suspension discussed in [DGB21].

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