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.
In Stokes flow of a particle settling within a bath of viscoplastic fluid, a critical resistive force must be overcome in order for the particle to move. This leads to a critical ratio of the buoyancy stress to the yield stress: the critical yield number. This translates geometrically to an envelope around the particle in the limit of zero flow that contains both the particle and encapsulated unyielded fluid. Such unyielded envelopes and critical yield numbers are becoming well understood in our previous studies for single (2D) particles as well as the means of calculating. Here we address the case of having multiple particles, which introduces interesting new phenomena. First, plug regions can appear between the particles and connect them together, depending on the proximity and yield number. This can change the yielding behaviour since the combination forms a larger (and heavier) “particle.” Moreover, small particles (that cannot move alone) can be pulled/pushed by larger particles or assembly of particles. Increasing the number of particles leads to interesting chain dynamics, including breaking and reforming.
A numerical and theoretical study of yieldstress fluid flows in two types of model porous media is presented. We focus on viscoplastic and elastoviscoplastic flows to reveal some differences and similarities between these two classes of flows. Small elastic effects increase the pressure drop and also the size of unyielded regions in the flow which is the consequence of different stress solutions compare to viscoplastic flows. Yet, the velocity fields in the viscoplastic and elastoviscoplastic flows are comparable for small elastic effects. By increasing the yield stress, the difference in the pressure drops between the two classes of flows becomes smaller and smaller for both considered geometries. When the elastic effects increase, the elastoviscoplastic flow becomes timedependent and some oscillations in the flow can be observed. Focusing on the regime of very large yield stress effects in the viscoplastic flow, we address in detail the interesting limit of 'flow/no flow': yieldstress fluids can resist small imposed pressure gradients and remain quiescent. The critical pressure gradient which should be exceeded to guarantee a continuous flow in the porous media will be reported. Finally, we propose a theoretical framework for studying the 'yield limit' in the porous media.
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