Heat transfer problem for the Boltzmann equation in a channel with diffusive boundary condition

TL;DR

This paper analyzes the steady Boltzmann equation in a channel with diffusive boundaries, incorporating slip boundary conditions, and rigorously derives the associated compressible Navier-Stokes equations with temperature jumps.

Contribution

It provides a formal asymptotic analysis and rigorous derivation of slip boundary conditions for the Boltzmann equation in a channel, extending previous work by including slip phenomena.

Findings

Flow approximated by compressible Navier-Stokes with temperature jump

Established uniform $L^ abla$ estimate on the remainder

Derived slip boundary condition rigorously

Abstract

In this paper, we study the 1D steady Boltzmann flow in a channel. The walls of the channel are assumed to have vanishing velocity and given temperatures $\theta_0$ and $\theta_1$. This problem was studied by Esposito et al [13,14] where they showed that the solution tends to a local Maxwellian with parameters satisfying the compressible Navier-Stokes equation with no-slip boundary condition. However, a lot of numerical experiments reveal that the fluid layer does not entirely stick to the boundary. In the regime where the Knudsen number is reasonably small, the slip phenomenon is significant near the boundary. Thus, we revisit this problem by taking into account the slip boundary conditions. Following the lines of [9], we will first give a formal asymptotic analysis to see that the flow governed by the Boltzmann equation is accurately approximated by a superposition of a steady CNS equation with a temperature jump condition and two Knudsen layers located at end points. Then we will establish a uniform $L^\infty$ estimate on the remainder and derive the slip boundary condition for compressible Navier-Stokes equations rigorously.