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))

.