Critical Yield Numbers and Limiting Yield Surfaces of Particle Arrays Settling in a Bingham Fluid

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.

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

“…It is important to use a discretization that takes into account derivatives in all directions as equally as possible, since we aim to resolve sharp geometric interfaces of the discontinuous flows. Illustrations of the directional behaviour of different finite difference schemes when finding interfaces in the anti-plane case can be found in [17], where the discretization of [41] is found to be particularly symmetric, as expected by its construction. We remark in any case that when only forward differences are used, The geometry of the interfaces is distorted according to their orientation.…”

“…With regard to perspectives, we postpone proposing a more suitable/accurate discretization of the primal dual problem (17) to a future study, e.g. using adaptive finite element schemes.…”

“…This can be seen as the price to pay for a direct method in which only one limiting field is computed, instead of a fixed approximation by computing Stokes flows. Given this situation, we use the non-accelerated version of the primal-dual proximal splitting algorithm of [36] applied to a discretization of the primal-dual formulation (17). For it we have chosen (in their notation) the dual step size as τ = 0.1, the primal step size as σ = 1/τ /L 2 , where L 2 is an estimation of the norm of the discrete operator (∇, div), and the relaxation parameter as θ = 1.…”

“…We now define the discrete domain Ω n and particle X n , and formulate a convergence result to demonstrate that the chosen discretization and penalization scheme is consistent and correctly accounts for the boundary conditions in the limit. The definitions and analysis follow the same lines as for the anti-plane case as presented in [17]. In this section, for simplicity, we assume that n t = n z = n and Ω (0, 1) 3 .…”

“…More general anti-plane flows, including non-convex and multiple particle arrays were considered in [17], again for anti-plane shear flows.…”