Finiteness of corner vortices

Abstract

Till date, the sequence of vortices present in the solid corners of steady internal viscous incompressible flows, widely known as Moffatt vortices was thought to be infinite. However, the already existing and most recent geometric theories on incompressible viscous flows that express vortical structures in terms of critical points in bounded domains, indicate a strong opposition to this notion of infiniteness. In this study, we endeavor to bridge the gap between the two opposing stream of thoughts by addressing what might have gone wrong and pinpoint the shortcomings on the assumptions of the existing theorems on Moffatt vortices. We provide our own set of proofs for establishing the finiteness of the sequence of Moffatt vortices by making use of the continuum hypothesis and Kolmogorov scale, which guarantee a non-zero scale for the smallest vortex structure possible in incompressible viscous flows. We point out that the notion of infiniteness resulting from discrete self-similarity of the vortex structures is not physically feasible. The centers of these vortices have been quantified by us as fixed points through Brouwer fixed-point theorem and boundary of a vortex as circle cell. With the aid of these new developments and making use of some existing theorems in topology along with some elementary concept of mathematical analysis, we provide several approaches to delve into this issue. All these approaches converge to the same conclusion that the sequence of Moffatt vortices cannot be infinite; in fact, it is at most finite.