  In this paper, we study Brinkman's equations with microscale properties that
are highly heterogeneous in space and time. The time variations are controlled
by a stochastic particle dynamics described by an SDE. The particle dynamics
can be thought as particle deposition that often occurs in filter problems. Our
main results include the derivation of macroscale equations and showing that
the macroscale equations are deterministic. The latter is important for our
(also many other) applications as it greatly simplifies the macroscale
equations. We use the asymptotic properties of the SDE and the periodicity of
the Brinkman's coefficient in the space variable to prove the convergence
result. The SDE has a unique invariant measure that is ergodic and strongly
mixing. The macro scale equations are derived through an averaging principle of
the slow motion (fluid velocity) with respect to the fast motion (particle
dynamics) and also by averaging the Brinkman's coefficient with respect to the
space variable. Our results can be extended to more general nonlinear diffusion
equations with heterogeneous coefficients.
