Yield limit analysis of particle motion in a yield-stress fluid

Abstract

A theoretical and numerical study of yield-stress fluid creeping flow about a particle is presented. Yield-stress fluids can hold rigid particles statically buoyant if the yield stress is large enough. In addressing sedimentation of rigid particles in viscoplastic fluids, we should know this critical ‘yield number’ beyond which there is no motion. As we get close to this limit, the role of viscosity becomes negligible in comparison to the plastic contribution in the leading order, since we are approaching the zero-shear-rate limit. Admissible stress fields in this limit can be found by using the characteristics of the governing equations of perfect plasticity (i.e. the sliplines). This approach yields a lower bound of the critical plastic drag force or equivalently the critical yield number. Admissible velocity fields also can be postulated to calculate the upper bound. This analysis methodology is examined for three families of particle shapes (ellipse, rectangle and diamond) over a wide range of aspect ratios. Numerical experiments of either resistance or mobility problems in a viscoplastic fluid validate the predictions of slipline theory and reveal interesting aspects of the flow in the yield limit (e.g. viscoplastic boundary layers). We also investigate in detail the cases of high and low aspect ratio of the particles.

Viscoplastic squeeze flow between two identical infinite circular cylinders

“…We then investigate the validity of the small gap approximation used to develop a leading order viscoplastic lubrication solution in section III B. Following this we compare pressure profiles along the the axis of symmetry, and the resultant normal force exerted on the cylinders, to solutions from viscoplastic lubrication theory in section III C. Finally, we investigate viscous dissipation in the system in section III D. points, unyielded plugs in the equatorial planes of the cylinders, and a yield envelope fully surrounding the two cylinder system [19,27,[33][34][35][36]. As the Bingham number increases the unyielded stagnation caps and the equatorial plugs grow while the yield envelope shrinks.…”

“…Instead, we use an iterative method based on the variational form of the Bingham problem, established by Duvaut and Lions [25], which forms the basis for the widely used augmented Lagrangian (AL) first proposed by Glowinski [26]. This formulation is commonly known as ALG2 and is used extensively in the literature, see Yu and Wachs [8], Muravleva [11], Chaparian and Frigaard [27] and references therein, so we do not give details here. For its solution we use the Uzawa type algorithm of Olshanskii [28] and Muravleva and Olshanskii [29].…”

Computational modeling of multiphase viscoelastic and elastoviscoplastic flows

“…47,48 In the case of a pure visco-plastic (VP) suspending fluid, there is an abundance of computational studies of single and multiple particles. [49][50][51][52][53] Full 3D suspension flows for VP fluids are time consuming, and thus limited to a few benchmark calculations and lower mesh resolutions. 54,55 However, 2D suspension flows are feasible.…”

Stability of particles inside yield-stress fluid Poiseuille flows

“…where Y * c is the critical value of the yield number which decides the stability of the particle (see § 2.3). Chaparian & Frigaard (2017b) followed the same framework and also revealed the relevance of perfect-plasticity theories in the study of yield limit or particle stability. Exploiting the cloaking phenomenon (Chaparian & Frigaard 2017a), they systematically presented a model for calculating Y * c for a single symmetric particle by finding the unyielded envelope enclosing the particle and postulating an admissible stress/velocity field about the particle.…”

“…This is sometimes referred to as static stability. The mathematical definition of Y * c could be extracted from the energy equation (Putz & Frigaard 2010;Chaparian & Frigaard 2017b) as…”