Computing the Yield Limit in Three-dimensional Flows of a Yield Stress Fluid About a Settling Particle

Calculating the yield limit $Y_c$ (the critical ratio of the yield stress to the driving stress), of a viscoplastic fluid flow is a challenging problem, often needing iteration in the rheological parameters to approach this limit, as well as accurate computations that account properly for the yield stress and potentially adaptive meshing. For particle settling flows, in recent years calculating $Y_c$ has been accomplished analytically for many antiplane shear flow configurations and also computationally for many geometries, under either two dimensional (2D) or axisymmetric flow restrictions. Here we approach the problem of 3D particle settling and how to compute the yield limit directly, i.e. without iteratively changing the rheology to approach the yield limit. The presented approach develops tools from optimization theory, taking advantage of the fact that $Y_c$ is defined via a minimization problem. We recast this minimization in terms of primal and dual variational problems, develop the necessary theory and finally implement a basic but workable algorithm. We benchmark results against accurate axisymmetric flow computations for cylinders and ellipsoids, computed using adaptive meshing. We also make comparisons of accuracy in calculating $Y_c$ on comparable fixed meshes. This demonstrates the feasibility and benefits of directly computing $Y_c$ in multiple dimensions. Lastly, we present some sample computations for complex 3D particle shapes.