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On the motion of a body in thermal equilibriumimmersed in a perfect gas

Published online by Cambridge University Press:  27 March 2008

Kazuo Aoki ,
Guido Cavallaro ,
Carlo Marchioro  and
Mario Pulvirenti
Kazuo Aoki
Affiliation:
Department of Mechanical Engineering and Science and Advanced Research Institute of Fluid Science and Engineering, Graduate School of Engineering, Kyoto University, Kyoto 606-8501, Japan. aoki@aero.mbox.media.kyoto-u.ac.jp
Guido Cavallaro
Affiliation:
Dipartimento di Matematica, Università di Roma “La Sapienza", Piazzale A. Moro 2, 00185, Roma, Italy. cavallar@mat.uniroma1.it; marchior@mat.uniroma1.it; pulvirenti@mat.uniroma1.it
Carlo Marchioro
Affiliation:
Dipartimento di Matematica, Università di Roma “La Sapienza", Piazzale A. Moro 2, 00185, Roma, Italy. cavallar@mat.uniroma1.it; marchior@mat.uniroma1.it; pulvirenti@mat.uniroma1.it
Mario Pulvirenti
Affiliation:
Dipartimento di Matematica, Università di Roma “La Sapienza", Piazzale A. Moro 2, 00185, Roma, Italy. cavallar@mat.uniroma1.it; marchior@mat.uniroma1.it; pulvirenti@mat.uniroma1.it
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Abstract

We consider a body immersed in a perfect gas and moving under the action of a constant force. Body and gas are in thermal equilibrium. We assume a stochastic interaction body/medium: when a particle of the medium hits the body, it is absorbed and immediately re-emitted with a Maxwellian distribution. This system gives rise to a microscopic model of friction. We study the approach of the body velocity V(t) to the limiting velocity $V_\infty$ and prove that, under suitable smallness assumptions, the approach to equilibrium is $$ |V(t)-V_\infty|\approx \frac{C}{t^{d+1}}, $$ where d is the dimension of the space, and C is a positive constant. This approach is not exponential, as typical in friction problems, and even slower than for the same problem with elastic collisions.

Keywords

Kinetic theory of gasesBoltzmann equationfree molecular gasfriction problemapproach to equilibrium.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 42 , Issue 2 , March 2008 , pp. 263 - 275
Copyright
© EDP Sciences, SMAI, 2008

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