A Fokker-Planck model of the Boltzmann equation with correct Prandtl number

TL;DR

This paper extends the Fokker-Planck model of the Boltzmann equation to accurately reproduce the Prandtl number in fluid dynamics by introducing a temperature tensor, supported by theoretical and numerical validation.

Contribution

It introduces a novel Fokker-Planck model with a temperature tensor to achieve the correct Prandtl number, improving upon previous models.

Findings

The model satisfies conservation laws and an H-theorem.

Numerical tests confirm the Prandtl number of 2/3.

Chapman-Enskog analysis supports the model's validity.

Abstract

We propose an extension of the Fokker-Planck model of the Boltzmann equation to get a correct Prandtl number in the Compressible Navier-Stokes asymptotics. This is obtained by replacing the diffusion coefficient (which is the equilibrium temperature) by a non diagonal temperature tensor, like the Ellipsoidal-Statistical model (ES) is obtained from the Bathnagar-Gross-Krook model (BGK) of the Boltzmann equation. Our model is proved to satisfy the properties of conservation and a H-theorem. A Chapman-Enskog analysis and two numerical tests show that a correct Prandtl number of 2/3 can be obtained.