Extensions to the Navier-Stokes-Fourier Equations for Rarefied Transport: Variational Multiscale Moment Methods for the Boltzmann Equation

TL;DR

This work addresses modeling gases in the transition regime by deriving a fourth-order, entropy-stable extension to the Navier–Stokes–Fourier equations via a variational multiscale closure of the Boltzmann equation. By projecting the Boltzmann dynamics onto collision invariants and constructing entropy-stable fine-scale closures (both linear and nonlinear), the authors avoid Burnett- and Super-Burnett-type instabilities while preserving key thermodynamic structure. The linearized theory is benchmarked against classical Boltzmann results for stationary heat transfer and Poiseuille flow, achieving remarkable agreement across Knudsen numbers and even into the transition regime; they also fit boundary-condition parameters to optimize this match. The work highlights the importance of unconditional entropy stability in higher-order closures and lays out a path toward robust nonlinear extensions and boundary-condition derivations with potential broad impact on rarefied gas dynamics and microflow simulations.

Abstract

We derive a fourth order entropy stable extension of the Navier-Stokes-Fourier equations into the transition regime of rarefied gases. We do this through a novel reformulation of the closure of conservation equations derived from the Boltzmann equation that subsumes existing methods such as the Chapman-Enskog expansion. We apply the linearized version of this extension to the stationary heat problem and the Poiseuille channel and compare our analytical solutions to asymptotic and numerical solutions of the linearized Boltzmann equation. In both model problems, our solutions compare remarkably well in the transition regime. For some macroscopic variables, this agreement even extends far beyond the transition regime.

Figures (8)

• Figure 1: The setup consists of two parallel plates, assumed to be infinite in two dimensions, that are a distance $l$ apart. A constant force $G$ parallel to the surface of the plate is applied to the gas within the channel.
• Figure 2: Heat flux as a function of the Knudsen number in the stationary heat problem. All the methods compared agree in the small $\epsilon$ regime but the Navier-Stokes-Fourier solution without a jump condition (green dotted line) deviates first as $\epsilon$ increases. The Navier-Stokes-Fourier solution with a jump condition (maroon dotted line) remains relatively close to the linearized Boltzmann solution and even converges to the correct collisionless limit heat flux. The Grad-Hilbert solution (orange dot-dashed line) does better than the Navier-Stokes-Fourier solution with a jump condition for $\epsilon\le1$ but converges to the wrong value in the collisionless limit. Finally, the solution due to our entropy stable extension agrees with the linearized Boltzmann solution remarkably well over the entire range of Knudsen numbers observed.
• Figure 3: Comparison of the non-dimensional temperature distribution obtained via the entropy stable extension (blue line), the Grad-Hilbert expansion (orange dashed) and the Navier-Stokes-Fourier equations with the jump boundary conditions described in the text (red dotted). The three solutions agree strongly in the interior of the channel for smaller Knudsen numbers with a discrepancy at the walls that extends into the domain as the Knudsen number increases ($\epsilon = 0.125,\,0.25,\,\text{and } 1$). The solution due to the entropy stable extension hews more closely to the Grad-Hilbert temperature distribution than the Navier-Stokes-Fourier solution does. Because all three solutions converge to the correct collisionless limit temperature distribution, they again agree strongly for very large Knudsen numbers ($\epsilon=100$).
• Figure 4: We use $C(\epsilon) = - \epsilon\,(2.1246\ln(1 + \epsilon ) + 2.3066)$ in \ref{['eq:altBurnMassFlux']} and plot $\frac{2}{B\gamma_1\sqrt{\pi}}Q(\frac{2}{\sqrt{\pi}}k)$ against $k$ (blue line). Also plotted are the linearized Boltzmann data points due to Ohwada, Sone and Aoki Ohwada1989 (red dots), the Navier-Stokes-Fourier solution ($C(\epsilon)=0$) (black dash line) and the R13 moments solution from Struchtrup and Torrilhon Struchtrup2007 (magenta dot-dash). We observe a reasonable agreement between our function and the data points from Ohwada et al., quantified by $R^2= 0.978$. The Knudsen minimum for the extension occurs around attained at $k = 0.48$. The Stokes solution for the mass flux (black dashed) does not exhibit a Knudsen minimum and decreases monotonically whilst R13 solution is really good for smaller values of $k$ but diverges from the Boltzmann solution as $k$ increases.
• Figure 5: We use $C(\epsilon) = - \epsilon\,(2.1246\ln(1 + \epsilon ) + 2.3066)$ in \ref{['eq:AltBPoiss']} to compare $\frac{2}{\gamma_1\sqrt{\pi}}\,u(x;\, \frac{2}{\sqrt{\pi}}k)$ (blue line) to the data points for the velocity profile provided in Ohwada1989 (red dots), the reduced R13 moments solution in Struchtrup2008 (magenta dot-dash) and the Navier-Stokes-Fourier solution (black dash). w Notice that the Navier-Stokes-Fourier solution and the entropy stable extension always agree at $x=1$. The best results occur for low intermediate Knudsen numbers (represented by $k=0.4$). We omit the R13 solution at $k=2$ and $k=8$ because these are outside its range of validity.

Theorems & Definitions (9)

• Remark 2.1
• Remark 2.2
• Remark 3.1
• Remark 3.2
• Remark 3.3
• Theorem 4.1: Theorem 10.1 of Baidoo2025
• Remark 4.1
• Corollary 4.2
• Definition F.1: The Kronecker Phi tensor