Small moving rigid body into a viscous incompressible fluid

Christophe Lacave (1) , Takéo Takahashi (2, 3)

1 IMJ - Institut de Mathématiques de Jussieu
2 EDP - Equations aux dérivées partielles
3 SPHINX - Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization

We consider rigid bodies moving under the influence of a viscous fluid and we study the asymptotic as the size of the solids tends to zero. In a bounded domain, if the solids shrink to " massive " pointwise particles, we obtain a convergence to the solution of the Navier-Stokes equations independently to any possible collision of the bodies with the exterior boundary. In the case of " massless " pointwise particles, we obtain a result for a single disk moving in the full plane. In this situation, the energy equality is not sufficient, and we obtain a uniform estimate for the solid velocity thanks to the optimal L p − L q decay estimates of the Stokes semigroup.

Domaines: Equations aux dérivées partielles [math.AP]; Recherche d'information [cs.IR]; Mathématiques générales [math.GM]

Christophe Lacave, Takéo Takahashi. Small moving rigid body into a viscous incompressible fluid. 2015. ⟨hal-01169436v1⟩