Dynamics of sedimentation of particle in a viscous fluid in the presence of two flat walls

Abstract

The model problem of sedimentation of a solid spherical particle in a viscous fluid bordering two solid planar surfaces is considered. To find the solution of the hydrodynamic equations in the approximation of small Reynolds numbers with boundary conditions on a particle and on two planes, a procedure developed for numerical simulation of the dynamics of a large number of particles in a viscous fluid with one plane wall is used. The procedure involves usage of fictive particles located symmetrically to real ones with respect to the plane. To solve the problem of the real particle’s sedimentation in the presence of two planes, a system of fictive particles is introduced. An approximate solution was found using four fictive particles. Basing on this solution, numerical results are obtained on dynamics of particle deposition for the cases of planes oriented parallel and perpendicular to each other. In particular, the values of linear and angular velocities of a particle are found, depending on the distance to each plane and on the direction of gravity. In the limiting case, when one of the planes is infinitely far from the particle, we obtain known results on the dynamics of particle sedimentation along and perpendicular to one plane.