The Motion of a Fluid-Rigid Ball System at the Zero Limit of the Rigid Ball Radius

We study the limiting motion of a system of rigid ball moving in a Navier–Stokes fluid in \({\mathbb{R}^3}\) as the radius of the ball goes to zero. Recently, Dashti and Robinson solved this problem in the two-dimensional case, in the absence of rotation of the ball (Dashti and Robinson in Arch Rational Mech Anal 200:285–312, 2011). This restriction was caused by the difficulty in obtaining appropriate uniform bounds on the second order derivatives of the fluid velocity when the rigid body can rotate. In this paper, we show how to obtain the required uniform bounds on the velocity fields in the three- dimensional case. These estimates then allow us to pass to the zero limit of the ball radius and show that the solution of the coupled system converges to the solution of the Navier–Stokes equations describing the motion of only fluid in the whole space. The trajectory of the centre of the ball converges to a fluid particle trajectory, which justifies the use of rigid tracers for finding Lagrangian paths of fluid flow.