Critical Yield Numbers and Limiting Yield Surfaces of Particle Arrays Settling in a Bingham Fluid
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Critical Yield Numbers and Limiting Yield Surfaces of Particle Arrays Settling in a Bingham Fluid José A. Iglesias1 · Gwenael Mercier1 · Otmar Scherzer1,2
© The Author(s) 2018
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. Keywords Bingham fluid · Exchange flow · Settling · Critical yield number · Total variation · Piecewise constant solutions Mathematics Subject Classification 49Q20 · 76A05 · 76T20 · 49M25
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José A. Iglesias [email protected] Gwenael Mercier [email protected] Otmar Scherzer [email protected]
1
Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Linz, Austria
2
Computational Science Center, University of Vienna, Vienna, Austria
123
Applied Mathematics & Optimization
1 Introduction In this article, we investigate the stationary flow of particles in a Bingham fluid. Such fluids are important examples of non-Newtonian fluids, describing for instance cement, toothpaste, and crude oil [31]. They are characterized by two numerical quantities: a yield stress τY that must be exceeded for strain to appear, and a fluid viscosity μ f that describes its linear behaviour once it starts to flow (see Fig. 1). An important property of Bingham fluid flows is the occurrence of plugs, which are regions where the fluid moves like a rigid body. Such rigid movements occur at positions where the stress does not exceed the yield stress. In this paper we consider anti-plane shear flow in an infinite cylinder, where an ensemble of inclusions move under their own weight inside a Bingham fluid of lower density, and in which the gravity and viscous forces are in equilibrium [cf (6)], therefore inducing a flow which is steady or stationary, that is, in which the velocity does not depend on time. For such a configuration, we are interested in determining the ratio between applied forces and the yield stress such that the Bi
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