Interactions between two touching spherical particles in sedimentation

A contact of a falling spherical particle with another fixed one in an unbounded viscous fluid is theoretically investigated based on a model of adding the contact interaction to the gravitational and hydrodynamic forces. The hydrodynamic interaction between the two particles is dealt with using an extended successive reflection method, with which the complete solution to the exterior velocity field around the two-particle system is constructed on the basis of the general expression given by Lamb, and then the hydrodynamic forces and torques on the two particles are obtained by integrating the fluid stress over each particle surface. The mechanical contact force is characterized by the standard friction theory with a criterion responsible for the transition from pure rolling to rolling with slip. Resorting to the dynamical equations of motion including the gravitational, hydrodynamic, and contact forces, the settling motion of a spherical particle in the vicinity of another fixed one is depicted using the fourth-order Runge-Kutta-Fehlberg method. Compared with the experimental results available in the literature, the theoretical prediction confirms two moving patterns at contact: pure rolling and rolling with slip, analyzes the dependence of the transition from one to another on the static friction coefficient and the contact separation distance between the particle surfaces, and accounts for a limitation of the quasisteady description of two interacting noncolloidal particles.