Wall effect on the motion of a rigid body immersed in a free molecular flow

Kai Koike

doi:10.3934/krm.2018020

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2018, Volume 11, Issue 3: 441-467. Doi: 10.3934/krm.2018020

This issue Previous Article Next Article Incompressible Limit of isentropic Navier-Stokes equations with Navier-slip boundary

Wall effect on the motion of a rigid body immersed in a free molecular flow

Kai Koike

School of Fundamental Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan

Received: February 28, 2017

Revised: May 31, 2017

Published: May 2018

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Abstract

Motion of a rigid body immersed in a semi-infinite expanse of free molecular gas in a $ d$-dimensional region bounded by an infinite plane wall is studied. The free molecular flow is described by the free Vlasov equation with the specular boundary condition. We show that the velocity $ V(t)$ of the body approaches its terminal velocity $ V_{∞}$ according to a power law $ V_{∞}-V(t)≈ t^{-(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., 264 (2006), 167-189] and thus slower convergence results from the presence of the wall.

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Mathematics Subject Classification: Primary: 35Q83, 35R37; Secondary: 70F40.

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Figure 1. A two dimensional picture of a cylinder immersed in a semi-infinite expanse of gas in a region bounded by an infinite plane wall is shown. The radius of the cylinder is $R$ and the height is $h$. The distance between the cylinder and the wall is denoted by $X(t)$ and the velocity by $V(t) = dX(t)/dt$. A constant force $E$ is applied in the direction of the axis of the cylinder and a drag force $D_V(t)$ is exerted to the cylinder from the surrounding gas.

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Figure 2. A two dimensional picture of a pre-collision at $C_{W}^{+}(\tilde{\tau}_1)$ is shown. The horizontal distance traversed by the cylinder and the characteristic curve $x(s)$ from $\tilde{\tau}_1$ to $t$ coincide.

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Figure 3. Two dimensional picture of a pre-collision at $C_{W}^{-}(\tau_2)$ via pre-collision at the plane wall. The sum of the horizontal distance traversed by the cylinder and the characteristic curve $x(s)$ from $\tau_2$ to $t$ equals $2X(t)$.

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