The role of Cheeger sets in the steady flows of viscoplastic fluids in pipes: A survey

Abstract

Given a convex or a Jordan domain Ω, let Ω′ be a subset of this domain, with P(Ω′) denoting its perimeter and A(Ω′) its area. If a subset Ωc exists such that h=P(Ωc)/A(Ωc) is a minimum, the subset Ωc is called the Cheeger set of Ω and h, the Cheeger constant of the given domain. If one considers the reciprocal of this minimum or the maximum ratio of the area of the subset to its perimeter, t∗=1/h. It follows from the work of Mosolov and Miasnikov that the minimum pressure gradient G to sustain the steady flow of a viscoplastic fluid in a pipe, with a cross section defined by Ω, is given by G>τy/t∗, where τy is the constant yield stress of the fluid. In this survey, we summarize several results to determine the constant h when the given domain is self-Cheeger or a Cheeger-regular set that touches each boundary of a convex polygon and when the Cheeger-irregular set does not do so. The determination of the constant h for an arbitrary ellipse, a strip, and a region with no necks is also mentioned.