Similarity Solutions for the Two-Dimensional Diffusion–Advection Equation in Sharp-Corner Geometries

Abstract

Problems involving corner geometries arise frequently in science and engineering, including in fluid mechanics. Low-Reynolds-number flow patterns near sharp corners, including Moffatt eddies, have been studied extensively in the past. Here, we focus on the effect of Moffatt eddies on concentration profiles near sharp corners. The Lie symmetry method is applied to the 2D diffusion-advection equation to obtain a similarity ansatz, reducing the equation to a time-independent second-order PDE. By modifying the ansatz, the problem can be recast as an eigenvalue problem. Exploiting elliptic regularity and the strong maximum principle, we invoke the Krein-Rutman theorem to establish the existence and domain-size monotonicity of the principal eigenvalue. We then investigate how varying domain length and corner angle affect the magnitude of the principal eigenvalue, observing an inverse relationship in both cases. The behavior of the eigenfunction is sensitive to small changes in the domain shape and size, so we truncate the domain along separating streamlines. This study adds to our understanding of the low-Reynolds-number transport dynamics near sharp corners from a pattern formation perspective, and can be extended to other systems.