The motion of a fluid-rigid ball system at the zero limit of the rigid ball radius

Ana Leonor Silvestre (1) , Takéo Takahashi (2, 3)

1 CEMAT - Center for Computacional and Stochastic Mathematics
2 CORIDA - Robust control of infinite dimensional systems and applications
3 IECL - Institut Élie Cartan de Lorraine

We study the limiting motion of a system of a rigid ball moving in a Navier-Stokes fluid flow in ${\mathbb R}^3$ as the radius of the ball goes to zero. Recently, Dashti and Robinson solved this problem in the 2D case, in the absence of rotation of the ball. 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 3D case. These estimates then allow 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.

Domaines: Equations aux dérivées partielles [math.AP]

Ana Leonor Silvestre, Takéo Takahashi. The motion of a fluid-rigid ball system at the zero limit of the rigid ball radius. Archive for Rational Mechanics and Analysis, 2014, 211 (3), pp.991-1012. ⟨10.1007/s00205-013-0696-7⟩. ⟨hal-00914936⟩