# Wall Effect on the Motion of a Rigid Body Immersed in a Free Molecular Flow

## TL;DR

This paper analyzes how a wall influences the motion of a rigid body in a free molecular flow, revealing that the wall causes the body to approach its terminal velocity more slowly than in unbounded space.

## Contribution

It provides a detailed analysis of the wall's effect on the velocity decay rate of a body in free molecular flow, deriving a new power law with a smaller exponent.

## Key findings

- Velocity approaches terminal velocity as a power law with exponent d-1.
- Presence of the wall slows down the convergence rate compared to unbounded flow.
- The decay rate differs from classical results without walls.

## Abstract

Motion of a rigid body immersed in a semi-infinite expanse of gas in a $d$-dimensional region bounded by an infinite plane wall is studied for free molecular flow on the basis of the free Vlasov equation under the specular boundary condition. We show that the velocity $V(t)$ of the body approaches its terminal velocity $V_{\infty}$ according to a power law $V_{\infty}-V(t)\approx Ct^{-(d-1)}$ by carefully analyzing the pre-collisions due to the presence of the wall. The exponent $d-1$ is smaller than $d+2$ for the case without the wall found in the classical work by Caprino, Marchioro and Pulvirenti~[Comm. Math. Phys., \textbf{264} (2006), pp. 167--189] and thus slower convergence rate results from the presence of the wall.