Dynamics of a small rigid body in a perfect incompressible fluid

Presentation of the model: a rigid body immersed in an incompressible perfect fluid We consider the motion of a rigid body immersed in an incompressible perfect fluid in a regular domain Ω ⊂ R 2. S(t) F (t) Ω Ω = R 2 or is a bounded domain. The solid occupies at each instant t ≥ 0 a closed subset S(t) ⊂ Ω, and the fluid occupies F(t) := Ω \ S(t). In F(t), the fluid satisfies the incompressible Euler equation: ∂u ∂t At the boundaries, the fluid satisfies the no-penetration/slip condition: u · n = 0 for x ∈ ∂Ω, u · n = V S · n = [h (t) + ϑ (t)(x − h(t)) ⊥ ] · n for x ∈ ∂S(t). Here: u = u(t, x) : F(t) → R 2 is the fluid velocity, p = p(t, x) : F(t) → R the pressure, n is the outer normal to the boundaries ∂Ω and ∂S(t), h(t) is the position of its center of mass (say h(0) = 0), ϑ is the angle with respect to the initial position (so ϑ(0) = 0). The dynamics of the solid is driven by the action of the pressure on its surface: m h (t) = ∂S(t) p n ds, J ϑ (t) = ∂S(t) p (x − h(t)) ⊥ · n ds, where m > 0 is the mass of the body, J > 0 denotes the moment of inertia. Remark. D'Alembert's paradox does not apply here, because it concerns fluids which are potential in R 2 , stationary and constant at infinity. In that case (only), D'Alembert's paradox states that the fluid does not influence the dynamics of the solid. Other formulations Vorticity formulation. In 2-D, the fluid part of the system can also be written ∂ t ω + (u · ∇)ω = 0 in F(t), and curl u = ω in F(t), div u = 0 in F(t), ∂S(t) u · τ ds = ∂S0 u 0 · τ ds = γ (Kelvin's law), + boundary conditions on u · n. As for the Euler equation alone, the complete system can be viewed as an equation of geodesics on an infinite dimensional Riemannian manifold, in the spirit of Arnold's work, see also Ebin-Marsden. References for the Cauchy problem Classical solutions (say at least C 1) solutions with finite energy: Ortega-Rosier-Takahashi in the …