On a Microscopic Model of Viscous Friction

Abstract

We consider a body moving along the x-axis under the action of an external force E and immersed in an infinitely extended perfect gas. We assume the gas to be described by the mean-field approximation and interacting elastically with the body. In this setup, we discuss the following statement: "Let V0 be the initial velocity of the body and V∞ its asymptotic velocity, then for |V0 - V∞| small enough it results |V(t) - V∞| ≈ C t-d-2 for t large, where V(t) is the velocity of the body at time t, d the dimension of the space and C is a positive constant depending on the medium and on the shape of the body". The reason for the power law approach to the stationary state instead of the exponential one (usually assumed in viscous friction problems), is due to the long memory of the dynamical system. In a recent paper by Caprino, Marchioro and Pulvirenti,3 the case of E constant and positive, with 0 < V0 < V∞, for a disk orthogonal to the x-axis has been discussed. Here we complete the analysis in the cases E > 0 with V0 > V∞ and E = 0. We also approach the problem of an x-dependent external force, by choosing E of harmonic type. In this case we obtain the power-like asymptotic time behavior for the body position X(t). The investigation is done in detail for a disk orthogonal to the x-axis and then, by a sketched proof, extended to a body with a general convex shape.

On the motion of a body in thermal equilibrium immersed in a perfect gas

“…An investigation on the approach to equilibrium in such a model has been done in [3,4]. A comparison between the results obtained in these references and the results given in the present paper will be done just after the statement of Theorem 2.1.…”

“…By using the results of Section 3 it is easy to prove the existence of the pair (V, f ), by studying the fixed point of the map W → V W by means of Schauder's theorem (see [4] for details).…”

Moving boundary problems in kinetic theory of gases: Spatially one-dimensional problems

“…More precisely, the decay rate depends on the dimension d of the plate [in [13], d = 1 indicates the present one-dimensional problem with an infinitely wide plate, d = 2 the two-dimensional problem with an infinitely long plate with a finite width, and d = 3 the three-dimensional (axisymmetric) problem with a circular disk] and is given as |x w (t)| ≈ const ×t −d−1 . Incidentally, under the specular reflection boundary condition on the plate, the decay rate is rigorously proved to be |x w (t)| ≈ const ×t −d−2 in the case of a monotonic decay without oscillation [14]. This slow decay in the free-molecular gas is attributed to the long-memory effect caused by the molecules that are reflected by the plate at early times and hit the plate again, with keeping their velocity, at later times (direct recollisions of the gas molecules) [14].…”

“…Incidentally, under the specular reflection boundary condition on the plate, the decay rate is rigorously proved to be |x w (t)| ≈ const ×t −d−2 in the case of a monotonic decay without oscillation [14]. This slow decay in the free-molecular gas is attributed to the long-memory effect caused by the molecules that are reflected by the plate at early times and hit the plate again, with keeping their velocity, at later times (direct recollisions of the gas molecules) [14]. In fact, if we devise an artificial gas (a special Lorentz gas) in which the possibility of the direct recollisions is negligibly small, the decay is almost exponential in time and is much faster than the inverse power of time [13].…”

Approach to equilibrium of a rotating sphere in a Stokes flow

“…Finally we mention the results obtained in case of a body moving in a gas (see [1,[7][8][9] for analytical results, and [2,24] for numerical results), for which the approach of the body velocity towards its asymptotic value follows an inverse power of time, due to the memory effect created by the recollisions between the gas particles and the body. In case that the medium is a fluid, we can think that the counterpart of recollisions is the Basset memory term.…”