Ghost effect by curvature in planar Couette flow

Leif Arkeryd; Raffaele Esposito; Rossana Marra; Anne Nouri

doi:10.3934/krm.2011.4.109

Article Contents

2011, Volume 4, Issue 1: 109-138. Doi: 10.3934/krm.2011.4.109

This issue Previous Article On the speed of approach to equilibrium for a collisionless gas Next Article A numerical model of the Boltzmann equation related to the discontinuous Galerkin method

Ghost effect by curvature in planar Couette flow

1.
Chalmers, 41296 Gothenburg
2.
Dipartimento di Matematica pura ed Applicata, Università dell’Aquila, Via Vetoio - Coppito, L’Aquila, 67100
3.
Dipartimento di Fisica and Unità INFN, Università di Roma Tor Vergata, 00133 Roma
4.
LATP, Université d’Aix-Marseille I, Marseille

Received: September 2010

Revised: November 2010

Published: March 2011

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Abstract

We study a rarefied gas, described by the Boltzmann equation, between two coaxial rotating cylinders in the small Knudsen number regime. When the radius of the inner cylinder is suitably sent to infinity, the limiting evolution is expected to converge to a modified Couette flow which keeps memory of the vanishing curvature of the cylinders ( ghost effect [18]). In the $1$-d stationary case we prove the existence of a positive isolated $L_2$-solution to the Boltzmann equation and its convergence. This is obtained by means of a truncated bulk-boundary layer expansion which requires the study of a new Milne problem, and an estimate of the remainder based on a generalized spectral inequality.

Keywords:

Mathematics Subject Classification: Primary: 82B40, 82C26; Secondary 76P05.

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References

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