On the motion of rigid bodies in a viscous incompressible fluid

Abstract

The motion of one or several rigid bodies in a viscous incompressible fluid has been a topic of numerous theoretical studies. The time evolution of the fluid density ϱ f = ϱ f (t, x) and the velocity u f = u f (t, x) is governed by the Navier-Stokes system of equations $$ {\partial _{t}}{\varrho ^{f}} + {\text{div(}}{\varrho ^{f}}{u^{f}}{\text{)}} {\text{ = }} {\text{0,}} $$ (1.1) $$ {\partial _{t}}({\varrho ^{f}}{u^{f}}) + {\text{div(}}{\varrho ^{f}}{u^{f}} \oplus {u^{f}}{\text{)}} + \nabla p = {\text{div}} \mathbb{T} + {\varrho ^{f}}{g^{f}} $$ (1.2) satisfied in a region Q f of the space-time occupied by the fluid. We focus on linearly viscous (Newtonian) incompressible fluids where the stress tensor \( \mathbb{T} \) is determined through the constitutive relation $$ \mathbb{T} = \mathbb{T}(u) \equiv 2\mu \mathbb{D}(u), \mathbb{D}(u) \equiv \frac{1}{2}(\nabla u{\text{ + }}\nabla {u^{t}}),\mu > 0, $$ (1.3) and the velocity satisfies the incompressibility condition $$ {\text{div}} {u^{f}} = 0. $$ (1.4) .