A fictitious domain method for dynamic simulation of particle sedimentation in Bingham fluids

Abstract

In this study, we present a fictitious domain method for the dynamic simulation of particle motion in a Bingham viscoplastic fluid at moderate Bingham numbers. Our method is built on the framework established by Glowinski and his coworkers, in the sense that we use their formulation and their operator-splitting idea to simplify the computation, but differs from their method in both spatial and temporal discretizations of the governing equations. Concerning the space scheme, we use a finite-difference method to discretize equations and a collocation-element method to enforce the rigid-body motion constraint inside the particle boundaries. Concerning the time scheme, the combined system is decoupled into three sub-systems: a Navier–Stokes problem, a plasticity problem and a rigid-body-motion problem, and we solve the Navier–Stokes problem with the classic projection method. The plasticity and rigid-body-motion multipliers at the previous time-level are retained in the Navier–Stokes problem to reduce the splitting error. In addition, the present study shows that retaining a viscous diffusion term in the plasticity problem is not necessary for the convergence of the iteration. The method is verified by comparing our results on the lid-driven cavity flow, the drag coefficient for a sphere settling in a tube and the hydrodynamic interactions between two spheres translating along the tube axis to the data available in the literature. Our results confirm that the drag coefficient for a sphere settling in a Bingham fluid at non-zero Reynolds numbers can be well correlated with an effective Reynolds number. For two approaching spheres, there exists a critical separation distance above which a drag-reduction is observed and below which a drag-enhancement takes place, compared to the drag for a single sphere. The drag-reduction however does not happen to a sphere falling towards a solid wall at the moderate Bingham numbers we studied. These observations are explained by a consideration of the competition between a shear-thinning plastic force and a repulsive lubrication force on the sphere occurring in the squeezing flow.