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

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.

30 pages, 8 figures

References in corpus (1)

• TV denoising of two balls in the plane

Cited by in corpus (4)

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• Critical yield numbers and limiting yield surfaces of particle arrays settling in a Bingham fluid
• On extremal points for some vectorial total variation seminorms