The vanishing limit of a rigid body in three-dimensional viscous incompressible fluid

TL;DR

This paper proves that as a small rigid body in a viscous incompressible fluid shrinks to a point, the coupled system's solution converges to the Navier-Stokes solution in the whole space, using semigroup estimates and fixed point methods.

Contribution

It establishes the vanishing limit of a rigid body in 3D viscous fluid, showing convergence to Navier-Stokes equations with new uniform velocity estimates.

Findings

Solution converges to Navier-Stokes equations as body shrinks

Uniform velocity estimates for the rigid body

Construction of test functions for limit passage

Abstract

We consider the evolution of a small rigid body in an incompressible viscous fluid filling the whole space $\rline^3$. When the small rigid body shrinks to a "massless" point in the sense that its density is constant, we prove that the solution of the fluid-rigid body system converges to a solution of the Navier-Stokes equations in the full space. Based on some $L^p-L^q$ estimates of the fluid-structure semigroup and a fixed point argument, we obtain a uniform estimate of velocity of the rigid body. This allows us to construct admissible test functions which plays a key role in the procedure of passing to the limit.