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Time-dependent Regularized 13-Moment Equations with Onsager Boundary Conditions in the Linear Regime

Bo Lin, Haoxuan Wang, Siyao Yang, Zhenning Cai

TL;DR

The paper advances time-dependent, regularized 13-moment (R13) equations for general elastic collision models in the linear regime, achieving symmetric, thermodynamically consistent closures with super-Burnett accuracy. It derives the equations via a Chapman–Enskog-like projection that yields a 37-moment system whose reduction to 13 primary moments preserves the higher-order Knudsen effects while maintaining $L^2$ stability. A key contribution is the Onsager-consistent boundary conditions, adapted to general collision models, and a systematic method to remove unphysical boundary layers by modifying the boundary operator to enforce a null-space constraint. One-dimensional simulations demonstrate good agreement with DSMC results and confirm the effective elimination of spurious boundary layers, highlighting the model’s potential for microflow applications with moderate Knudsen numbers. The framework lays groundwork for robust multidimensional implementations and further extension to nonlinear regimes.

Abstract

We develop the time-dependent regularized 13-moment equations for general elastic collision models under the linear regime. Detailed derivation shows the proposed equations have super-Burnett order for small Knudsen numbers, and the moment equations enjoy a symmetric structure. A new modification of Onsager boundary conditions is proposed to ensure stability as well as the removal of undesired boundary layers. Numerical examples of one-dimensional channel flows is conducted to verified our model.

Time-dependent Regularized 13-Moment Equations with Onsager Boundary Conditions in the Linear Regime

TL;DR

The paper advances time-dependent, regularized 13-moment (R13) equations for general elastic collision models in the linear regime, achieving symmetric, thermodynamically consistent closures with super-Burnett accuracy. It derives the equations via a Chapman–Enskog-like projection that yields a 37-moment system whose reduction to 13 primary moments preserves the higher-order Knudsen effects while maintaining stability. A key contribution is the Onsager-consistent boundary conditions, adapted to general collision models, and a systematic method to remove unphysical boundary layers by modifying the boundary operator to enforce a null-space constraint. One-dimensional simulations demonstrate good agreement with DSMC results and confirm the effective elimination of spurious boundary layers, highlighting the model’s potential for microflow applications with moderate Knudsen numbers. The framework lays groundwork for robust multidimensional implementations and further extension to nonlinear regimes.

Abstract

We develop the time-dependent regularized 13-moment equations for general elastic collision models under the linear regime. Detailed derivation shows the proposed equations have super-Burnett order for small Knudsen numbers, and the moment equations enjoy a symmetric structure. A new modification of Onsager boundary conditions is proposed to ensure stability as well as the removal of undesired boundary layers. Numerical examples of one-dimensional channel flows is conducted to verified our model.
Paper Structure (33 sections, 1 theorem, 166 equations, 6 figures, 4 tables)

This paper contains 33 sections, 1 theorem, 166 equations, 6 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Linearization and nondimensionalization
  3. Linearization and nondimensionalization of the Boltzmann equation
  4. Linearization and nondimensionalization of the conservation laws
  5. Linear R13 equations
  6. Choice of variables
  7. Moment equations and boundary conditions
  8. Second law of thermodynamics
  9. Derivation of R13 equations
  10. Choice of variables
  11. Second-order variables
  12. Derivation of R13 equations
  13. Verification of the super-Burnett order
  14. First Maxwellian iteration
  15. Second Maxwellian iteration
  16. ...and 18 more sections

Key Result

Lemma 1

If $\{ \lambda, (\boldsymbol{r}_{\mathrm{o}}^{\intercal},\boldsymbol{r}_{\mathrm{e}}^{\intercal})^{\intercal} \}$ is an eigen-pair of the matrix in eq:vrhsmat, then $\{ -\lambda, (-\boldsymbol{r}_{\mathrm{o}}^{\intercal},\boldsymbol{r}_{\mathrm{e}}^{\intercal})^{\intercal} \}$ is also an eigen-pair

Figures (6)

  • Figure 1: $\bar{q}$ and $\theta$ of the one-dimensional problem \ref{['eq:1deqrho']}--\ref{['eq:1dbcsigma']}. $\theta^W=0$ in the left boundary and $\theta^{W}=0.2$ in the right one.
  • Figure 2: $\bar{q}$ and $\theta$ of the one-dimensional problem \ref{['eq:1deqrho']}--\ref{['eq:1dbcsigma']} with modified coefficient $\hat{m}_{ij} \to {m}_{ij}$. $\theta^W=0$ in the left boundary and $\theta^{W}=0.2$ in the right one.
  • Figure 3: Results of steady-state example when $\overline{\mathrm{Kn}}=0.1$. Top subfigures use coefficients $\hat{m}_{ij}$ and bottom subfigures use modified ones $m_{ij}$. DSMC solutions are given by dotted lines of the same colors.
  • Figure 4: Results of steady-state example when $\overline{\mathrm{Kn}}=0.05$. Top subfigures use coefficients $\hat{m}_{ij}$ and bottom subfigures use modified ones $m_{ij}$. DSMC solutions are given by dotted lines of the same colors.
  • Figure 5: Results of the Couette flow. Left subfigures use previous Onsager boundary conditions and right subfigures use our modified boundary conditions.
  • ...and 1 more figures

Theorems & Definitions (4)

  • Remark 1
  • Remark 2
  • Lemma 1
  • proof