Critical yield numbers and limiting yield surfaces of particle arrays settling in a Bingham fluid

TL;DR

This paper analyzes the flow cessation of particle arrays in a Bingham fluid by formulating it as a variational problem, identifying critical yield numbers, and demonstrating the existence of limiting yield surfaces through theoretical and numerical methods.

Contribution

It introduces a variational framework for determining critical yield numbers and limiting yield surfaces in particle-laden Bingham fluids, with proofs of existence and a numerical solution approach.

Findings

Existence of limiting yield surfaces as rescaled limits of physical velocities.

Development of a numerical method based on nonlinear finite differences.

Numerical examples illustrating geometric properties of solutions.

Abstract

We consider the flow of multiple particles in a Bingham fluid in an anti-plane shear flow configuration. The limiting situation in which the internal and applied forces balance and the fluid and particles stop flowing, that is, when the flow settles, is formulated as finding the optimal ratio between the total variation functional and a linear functional. The minimal value for this quotient is referred to as the critical yield number or, in analogy to Rayleigh quotients, generalized eigenvalue. This minimum value can in general only be attained by discontinuous, hence not physical, velocities. However, we prove that these generalized eigenfunctions, whose jumps we refer to as limiting yield surfaces, appear as rescaled limits of the physical velocities. Then, we show the existence of geometrically simple minimizers. Furthermore, a numerical method for the minimization is then considered. It is based on a nonlinear finite difference discretization, whose consistency is proven, and a standard primal-dual descent scheme. Finally, numerical examples show a variety of geometric solutions exhibiting the properties discussed in the theoretical sections.