On the motion of a nearly incompressible viscous fluid containing a small rigid body

Abstract

We consider the motion of a compressible viscous fluid containing a moving rigid body confined to a planar domain Ω ⊂ R 2 . The main result states that the influence of the body on the fluid is negligible if (i) the diameter of the body is small and (ii) the fluid is nearly incompressible (the low Mach number regime). The specific shape of the body as well as the boundary conditions on the fluid-body interface are irrelevant and collisions with the boundary ∂Ω are allowed. The rigid body motion may be enforced externally or governed solely by its interaction with the fluid.

On the Velocity of a Small Rigid Body in a Viscous Incompressible Fluid in Dimension Two and Three

“…) are uniformly bounded in ε from (11) and Korn inequality, 1/m ε and ũin ε L 2 (F ε (0)) are uniformly bounded by hypothesis, 1/ log(α ε ) −→ 0 from the choice α ε −→ +∞ and | in ε − in | −→ 0 by hypothesis. It only remains to show a uniform bound for ũε L 2 (0,T ;L 4 (R 2 )) .…”

“…Putting estimates ( 14)-( 15)-( 16)-( 17)-( 18) together and using the uniform estimates (11), the convergence of the initial data and the hypothesis that m ε /ε 1/2 −→ +∞, we deduce from (12) that…”

“…where in the third inequality we used points 1 and 3 of Lemma 1 for q = 2. Moreover notice that u ε L ∞ (0,T ;L 2 (F ε (t))) is uniformly bounded in ε from (11).…”

“…where in the last step we use the Korn inequality. Notice that the right hand side of the above inequality is uniformly bounded due to (11).…”

“…Notice that there are many results in this direction. For example in [28] the authors studied the vanishing object problem in dimension two under the assumption m ε = ε 2 m. For viscous compressible fluid the three dimensional case was tackled in [4] and improved in [12] and more recently in [11] was studied the two dimensional case for a weakly compressible fluid. Notice that for viscous compressible fluid the vanishing object problem in two dimensions is still an open problem due to the lack of integrability of the pressure.…”