Critical yield numbers of rigid particles settling in Bingham fluids and Cheeger sets

TL;DR

This paper investigates the critical yield number preventing particle motion in Bingham fluids, linking it to Cheeger set problems, and develops explicit solutions and computational methods for this eigenvalue problem.

Contribution

It introduces a novel eigenvalue problem for critical yield numbers in Bingham fluids, connecting it to Cheeger set optimization and providing explicit solutions and computational approaches.

Findings

Critical yield numbers relate to Cheeger set eigenvalues.

Explicit solutions are found for specific geometries.

A computational method effectively approximates solutions.

Abstract

We consider the fluid mechanical problem of identifying the critical yield number $Y_c$ of a dense solid inclusion (particle) settling under gravity within a bounded domain of Bingham fluid, i.e. the critical ratio of yield stress to buoyancy stress that is sufficient to prevent motion. We restrict ourselves to a two-dimensional planar configuration with a single anti-plane component of velocity. Thus, both particle and fluid domains are infinite cylinders of fixed cross-section. We show that such yield numbers arise from an eigenvalue problem for a constrained total variation. We construct particular solutions to this problem by consecutively solving two Cheeger-type set optimization problems. We present a number of example geometries in which these geometric solutions can be found explicitly and discuss general features of the solutions. Finally, we consider a computational method for the eigenvalue problem, which is seen in numerical experiments to produce these geometric solutions.