Asymptotic Analysis of Stokes Flow Through a Filter
Résumé
This paper investigates Stokes flow through a filter composed of a rigid, thin, periodic, and closely packed array of parallel long rods with a non-circular, anisotropic cross-section, having the shape of a slot. 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. In 2D formulation, we will refer to the cross-section of the rods as obstacles. The obstacles are much thinner than the distance between them, so the filter has wide channels. The form of the obstacles is parametrized to enable detailed analysis. 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 unilateral constraints; (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 (changes in obstacle shape), including the derivation of a sharp constant.
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