Asymptotic Analysis of Stokes Flow Through a Filter

This paper investigates the Stokes flow through a thin porous layer composed of a rigid, thin, periodic, and closely packed array of parallel long rods with a noncircular, anisotropic cross‐section in the shape of a slot. This shape allows for the construction of a thicker permeable filter than what is known as the Brinkman critical size. The analysis is restricted to regions far from the top and bottom boundaries, where the flow becomes effectively two‐dimensional due to the invariance of the structure in the third dimension. We explore the critical scaling of the filter permeability and various regimes based on the relationships between the obstacle distance, length, and thickness. In addition, the paper presents three new tools: (1) estimates for small periodic obstacles with normal nonpenetration and tangential slip on all rigid boundaries, (2) a rescaled version of the Nečas inequality for thin domains, analogous to Korn's inequalities, and (3) a demonstration of the invariance of estimates under local diffeomorphisms.