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

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∞ according to a power law V∞ − V (t) ≈ 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., 264 (2006), pp. 167-189] and thus slower convergence rate results from the presence of the wall.

Motion of a Rigid Body in a Special Lorentz Gas: Loss of Memory Effect

“…Note that |W (t)| ≤ γ by ineqs. (19) and (20) (take γ sufficiently small so that 2 3−d γ 2 A 1 ≤ 1); therefore, C 0 ≤ K(W (t)) ≤ C γ by Lemma 1.…”

“…Next, I show that V W satisfies ineq. (20) with W (t) replaced by V W (strictly for t ≥ 0). First, by Propositions 1 and 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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