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This paper generalizes the results obtained by the authors in\cite{dangHomogenizationNondiluteSuspension2021} concerning the homogenizationof a non-dilute suspension of magnetic particles in a viscous flow. Morespecifically, in this paper, a restrictive assumption on the coefficients ofthe coupled equation, made in \cite{dangHomogenizationNondiluteSuspension2021},that significantly narrowed the applicability of the homogenization resultsobtained, is relaxed and a new regularity of the solution of the fine-scaleproblem is proven. In particular, we obtain a global $L^{\infty}$-bound for thegradient of the solution of the scalar equation $-\mathrm{div} \left[\mathbf{a} \left( x/\varepsilon \right)\nabla \varphi^{\varepsilon}(x) \right]= f(x)$, uniform with respect to microstructure scale parameter $\varepsilon\ll1$ in a small interval $(0,\varepsilon_0)$, where the coefficient $\mathbf{a}$is only \emph{piecewise} H\"{o}lder continuous. Thenceforth, this regularity isused in the derivation of the effective response of the given suspensiondiscussed in \cite{dangHomogenizationNondiluteSuspension2021}.

Global gradient estimate for a divergence problem and its application to the homogenization of a magnetic suspension