Ghost effect by curvature in planar Couette flow

TL;DR

This paper investigates the ghost effect in planar Couette flow of a rarefied gas, demonstrating the convergence of solutions to a modified flow influenced by the curvature of the cylinders, using advanced mathematical analysis.

Contribution

It introduces a new analysis of the Boltzmann equation in the context of curved boundary effects, including a novel Milne problem and spectral estimates for convergence.

Findings

Existence of a positive isolated L_2-solution to the Boltzmann equation.

Convergence of the solution to a modified Couette flow with curvature effects.

Development of a truncated bulk-boundary layer expansion and spectral inequality estimates.

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). 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.