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

TL;DR

This paper proves that a small rigid body's influence on a viscous compressible fluid vanishes as its size approaches zero, using novel test function constructions and a new Bogovskii operator form.

Contribution

It introduces a new method for constructing test functions and a novel form of the Bogovskii operator to analyze the asymptotic behavior of small rigid bodies in compressible fluids.

Findings

Fluid behavior is unaffected by a small rigid body as its radius tends to zero.

The result applies for isentropic pressure laws with gamma > 3/2.

New analytical techniques are developed for weak formulation analysis.

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 $\varepsilon$ we show that the behavior of the fluid is not influenced by the object in the asymptotic limit $\varepsilon \to 0$. 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.