Effects of Concavity on the Motion of a Body Immersed in a Vlasov Gas

Abstract

We consider a body immersed in a perfect gas, moving under the action of a constant force E along the x axis. We assume the gas to be described by the mean-field approximation and interacting elastically with the body. Such a dynamic was studied in [S. Caprino, C. Marchioro,. In these studies the asymptotic trend showed no sensitivity whatsoever to the shape of the object moving through the gas. In this work we investigate how a simple concavity in the shape of the body can affect its asymptotic behavior; we thus consider the case of a hollow cylinder in three dimensions or a box-like body in two dimensions. We study the approach of the body velocity V (t) to the limiting velocity V∞ and prove that, under suitable smallness assumptions, the approach to equilibrium is |V∞ − V (t)| ≈ Ct −3 both in two or three dimensions, with C a positive constant. This approach is not exponential, as is typical in friction problems, and even slower than for the simple disk and the convex body in R 2 or R 3 .

Uniqueness of Solutions to a Gas-Disk Interaction System

“…In this scenario the velocity of the disk can be found by solving a linear ODE. Here we consider a more realistic model as in [1][2][3][4][5][6][7][8][10][11][12][13], where the evolution of the gas and the disk satisfies a coupled system of integro-differential equations. The coupling is through collisions of gas particles with the disk: these collisions produce a drag force on the disk through momentum exchange and provide a boundary condition for the gas.…”

“…Regarding the long-time asymptotics, it is now fairly wellunderstood that due to the effect of re-collisions, the relaxation of the disks velocity toward its equilibrium state may not be exponential as in the simplified model where re-collisions are ignored. In fact, one may obtain algebraic decay rates [1][2][3][4][5][6][7][8][10][11][12][13]. Moreover, depending on the shape of the body, such rates may or may not depend on the spatial dimension [4,10].…”

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In this paper we give an elementary proof of uniqueness of solutions to a gas-disk interaction system with diffusive boundary condition. Existence of near-equilibrium solutions for this type of systems with various boundary conditions has been extensively studied in [1][2][3][4][5][6][7][8]10]. However, the uniqueness has been an open problem, even for solutions near equilibrium.

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Convergence to Equilibrium of a Body Moving in a Kinetic Sea

“…Some numerical investigations, mostly dealing with a diffuse reflection which ejects Maxwellians, have appeared in [3,16,17,18]. Some other related investigations that deal with the speed of approach to equilibrium in the specular reflection case are [2,7,8,13]. See also the survey in [4].…”

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

“…Note that due to the presence of the wall, even if the body is convex the asymptotic convergence rate may be different from t −(d+2) . In the presence of the wall, general convex bodies or a U-shaped body treated in [12] may be included but we shall restrict ourselves to the case of a cylinder for simplicity. We note that for the model considered in this paper, the drag force D V (t) is time dependent even when the velocity V (t) is constant.…”