[Paper Review] Derivation of Hyperbolic Transfer Equations from BGK-Equation

Andrew Terentyev, Skryl, Yu.|arXiv (Cornell University)|Jul 14, 2005

Gas Dynamics and Kinetic Theory7 references3 citations

TL;DR

This paper derives hyperbolic Navier-Stokes and heat conduction equations from the Boltzmann equation using the BGK relaxation-time approximation in integral form, incorporating memory effects via an initial condition that enforces convergence to Maxwellian equilibrium as t → -∞. The key result is that the relaxation time in the resulting hyperbolic equations is the Maxwellian relaxation time, validated by agreement with experimental values for nitrogen and aluminum.

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 $t o -\infty$ 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.

Motivation & Objective

• To derive hyperbolic transport equations from the Boltzmann equation with physically consistent memory effects.
• To establish a theoretical link between the relaxation time in hyperbolic heat conduction and the Maxwellian relaxation time.
• To validate the derived equations by estimating relaxation times for nitrogen and aluminum and comparing them with experimental values.
• To demonstrate that the hyperbolic form emerges naturally from the BGK approximation when memory effects are properly accounted for in the integral formulation.

Proposed method

• Uses the integral form of the Boltzmann equation with an initial condition that selects solutions approaching the local Maxwellian distribution as t → -∞, incorporating memory effects.
• Applies the BGK relaxation-time approximation to the collision integral, replacing the full collision term with a single relaxation time τ₀.
• Expands the distribution function in first-order gradients of hydrodynamic variables (density, velocity, temperature) to derive fluxes and balance equations.
• Derives the hyperbolic heat flux equation (Cattaneo-type) and the telegraph equation for heat conduction in static media.
• Computes the pressure tensor and heat flux in terms of hydrodynamic variables and the relaxation time τ₀.
• Estimates the Maxwellian relaxation time τ_r = μ/p for gases and τ_r from phonon gas model for solids, using material-specific parameters.

Experimental results

Research questions

• RQ1 Can hyperbolic transport equations for heat and momentum be derived from the Boltzmann equation using a memory-inclusive initial condition?
• RQ2 What is the physical origin of the relaxation time τ_g in the Cattaneo-type hyperbolic heat flux equation?
• RQ3 Does the relaxation time in the derived hyperbolic equations correspond to the Maxwellian relaxation time?
• RQ4 How do the theoretical estimates of relaxation time for nitrogen and aluminum compare with experimental values?

Key findings

• The relaxation time τ_r in the derived hyperbolic equations is identified as the Maxwellian relaxation time, τ_r = μ/p for gases.
• For nitrogen at 1000 K and atmospheric pressure, the theoretical τ_r is 4.096 × 10⁻¹⁰ s, matching experimental estimates.
• For aluminum, the derived τ_r is 1.024 × 10⁻¹¹ s, consistent with experimental values from Lykov (1967).
• The telegraph equation for heat conduction in static media is derived as a special case, confirming the hyperbolic nature of heat propagation.
• The derived equations are valid for time scales of order the mean free time τ₀, where non-equilibrium dynamics are significant.
• The method yields consistent expressions for the heat flux and pressure tensor in the first-order approximation, with the relaxation time emerging naturally from the kinetic theory framework.