Boundary conditions in PDE model of collisions of swimmers

The goal of the paper is to determine boundary conditions in PDE models of collisions of microswimmers in a viscous fluid. We consider two self-propelled spheres (microswimmers) moving towards each other in viscous fluid. We first show that under commonly used no-slip boundary conditions on the fluid-solid interface the microswimmers do not collide which is a generalization of the well-known no-collision paradox for solid bodies (with no self-propulsion) in a viscous fluid. Secondly, we show that the microswimmers do collide when the no-slip boundary conditions are replaced by the Navier boundary conditions which therefore provides an adequate model of microswimmers such as swimming bacteria. The self-propulsion mechanism generates a drag force pulling a bacterium backwards and the collision problem is reduced to the analysis of competition between the drag and self-propulsion. For no-slip this is done by utilizing the Lorentz Reciprocal Theorem and the analytical solution for two solid spheres in the fluid. The analysis for the Navier boundary conditions is based on the variational formulation of Stokes problem. A Poincare type inequality for symmetrized gradient is introduced in this work.