A New method for computing particle collisions in Navier-Stokes flows

Particle collisions in fluids are ubiquitous, but to compute the collision dynamics in a Navier-Stokes flow remains challenging. In addition to capturing the two-way coupling between the fluid and the particles, a key difficulty is to resolve the collision dynamics mediated by the flow. The gap between particles during collision is minuscule. This introduces a small length scale which needs to be resolved simultaneously with the flow at the large scale. Our goal is to develop a numerical scheme that is accurate and efficient in computing the Navier-Stokes flow around moving particles while taking into account the effect of the lubrication forces on the collisions. Our method integrates the immersed interface method with the lubrication theory in a way that directly couples all three parts, the bulk flow, the flow in the gap, and the dynamics of the freely moving particles. We present a general algorithm for computing the collision. To test the method, we study four fundamental cases involving normal and tangential collisions, so that we can compare numerics against analytic solutions in the lubrication layer. In addition, we provide the lubrication solution needed for computing collisions between surfaces of any shapes in arbitrary relative motions, so that the method can be applied to other cases.