Derivation of Hyperbolic Transfer Equations from BGK-Equation

TL;DR

This paper derives hyperbolic transfer equations, including hyperbolic Navier-Stokes and heat conduction equations, from the BGK approximation of the Boltzmann equation, highlighting the role of Maxwellian relaxation time.

Contribution

It presents a novel derivation of hyperbolic transfer equations from the BGK model, incorporating memory effects and identifying the relaxation time as the Maxwellian relaxation time.

Findings

Derived hyperbolic Navier-Stokes and heat conduction equations from BGK.

Identified Maxwellian relaxation time as key parameter.

Estimated relaxation times for heat conduction in materials.

Abstract

We use the integral form of the Boltzmann equation which allows us to take into account the memory effects using the initial condition that selects the solutions going to the local equilibrium Maxwell distribution in the $t \to -\infty$ limit. Implementing the relaxation-time approximation for the collision integral (BGK-equation) we present the derivation of the hyperbolic Navier-Stokes and the hyperbolic heat conduction equations in the first order approximation. It is shown that the relaxation time in the obtained hyperbolic equations is the Maxwellian relaxation time. As special case we obtain the telegraph equation for the heat propagation in static medium and estimate the relaxation time for the heat conduction in some materials.