Particle settling in yield stress fluids: Limiting time, distance and applications

We examine the problem of a single heavier solid particle settling in a yield stress fluid that behaves as a classical Bingham plastic. The flow configuration we are interested in is the transient dynamics from a particle settling in a Newtonian fluid to a Bingham plastic. Depending on the magnitude of the yield stress (or dimensionlessly the Bingham number), the particle and the surrounding fluid may return to rest in a finite time or reach another steady but lower settling velocity. At the analytical level, we write the total kinetic energy decay of the system. We evidence the existence of a critical Bingham number beyond which motion is suppressed and derive upper bounds for the finite stopping time as well as the maximum path length. These estimates can be obtained in 2D only while the extension to 3D remains an open question. At the numerical level, we design a robust and efficient Lagrange multiplier based algorithm that enables us to compute actual finite time decay. The algorithm combines an Augmented Lagrangian outer loop to treat the exact Bingham law to a Distributed Lagrange Multiplier/Fictitious Domain inner loop to account for freely-moving particles. We show that the ability to compute the balance between net weight (weight plus buoyancy) and yield stress resistance is the key point. The algorithm is implemented together with a Finite Volume/Staggered Grid algorithm in the numerical platform PeliGRIFF. We investigate 2D configurations with the following particle shape: (i) a circular disc and (ii) a 2:1 rectangle.