On the motion of a small rigid body in a viscous compressible fluid

Abstract

We consider the motion of a small rigid object immersed in a viscous compressible fluid in the 3-dimensional Eucleidean space. Assuming the object is a ball of a small radius $\epsilon$ we show that the behavior of the fluid is not influenced by the object in the asymptotic limit $\epsilon \to 0$. The result holds for the isentropic pressure law $p(\varrho )=a{\varrho }^{\gamma }$ for any $\gamma >\frac{3}{2}$ under mild assumptions concerning the rigid body density. In particular, the latter may be bounded as soon as $\gamma >3$. The proof uses a new method of construction of the test functions in the weak formulation of the problem, and, in particular, a new form of the so-called Bogovskii operator.