  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.
