# Accurate and efficient computation of the Boltzmann equation for Couette flow: influence of intermolecular potentials on Knudsen layer function and viscous slip coefficient

Wei Su, Peng Wang, Haihu Liu, Lei Wu

Research output: Contribution to journalArticle

### Abstract

The Couette flow is one of the fundamental problems of rarefied gas dynamics, which has been investigated extensively based on the linearized Boltzmann equation (LBE) of hard-sphere molecules and simplified kinetic model equations. However, how the different intermolecular potentials affect the viscous slip coefficient and the structure of Knudsen layer remains unclear. Here, a novel synthetic iteration scheme (SIS) is developed for the LBE to find solutions of Couette flow accurately and efficiently: the velocity distribution function is first solved by the conventional iterative scheme, then it is modified such that in each iteration i) the flow velocity is guided by an ordinary differential equation that is asymptotic-preserving at the Navier-Stokes limit and ii) the shear stress is equal to the average shear stress. Based on the Bhatnagar-Gross-Krook model, the SIS is assessed to be efficient and accurate. Then we investigate the Knudsen layer function for gases interacting through the inverse power-law, shielded Coulomb, and Lennard-Jones potentials, subject to diffuse-specular and Cercignani-Lampis gas-surface boundary conditions. When the tangential momentum accommodation coefficient (TMAC) is not larger than one, the Knudsen layer function is strongly affected by the potential, where its value and width increase with the effective viscosity index of gas molecules. Moreover, the Knudsen layer function exhibits similarities among different values of TMAC when the intermolecular potential is fixed. For Cercignani-Lampis boundary condition with TMAC larger than one, both the viscous slip coefficient and Knudsen layer function are affected by the intermolecular potential, especially when the backward'' scattering limit is approached. With the asymptotic theory by Jiang and Luo (\textit{J. Comput. Phys}., vol. 316, 2016, pp. 416--434) for the singular behavior of the velocity gradient in the vicinity of solid surfaces, we find that the whole Knudsen layer function can be well fitted by the power series $\sum_{n=0}^{2}\sum_{m=0}^{2}c_{n,m}x^n(x\ln x)^m$, where $x$ is the distance to the solid surface. Finally, the experimental data of the Knudsen layer profile are explained by the LBE solution with proper values of the viscosity index and TMAC.
Language English 573-590 18 Journal of Computational Physics 378 13 Nov 2018 10.1016/j.jcp.2018.11.015 E-pub ahead of print - 13 Nov 2018

### Fingerprint

Boltzmann equation
Couette flow
slip
accommodation coefficient
Momentum
coefficients
iteration
momentum
Shear stress
Gases
Boundary conditions
Viscosity
solid surfaces
Lennard-Jones potential
shear stress
Molecules
Gas dynamics
rarefied gas dynamics
gases
Velocity distribution

### Keywords

• Kramer's problem
• Boltzmann equation
• gas-kinetic theory
• Knudsen layer

### Cite this

@article{283ce77b3c804232bf821c1993b342ff,
title = "Accurate and efficient computation of the Boltzmann equation for Couette flow: influence of intermolecular potentials on Knudsen layer function and viscous slip coefficient",
abstract = "The Couette flow is one of the fundamental problems of rarefied gas dynamics, which has been investigated extensively based on the linearized Boltzmann equation (LBE) of hard-sphere molecules and simplified kinetic model equations. However, how the different intermolecular potentials affect the viscous slip coefficient and the structure of Knudsen layer remains unclear. Here, a novel synthetic iteration scheme (SIS) is developed for the LBE to find solutions of Couette flow accurately and efficiently: the velocity distribution function is first solved by the conventional iterative scheme, then it is modified such that in each iteration i) the flow velocity is guided by an ordinary differential equation that is asymptotic-preserving at the Navier-Stokes limit and ii) the shear stress is equal to the average shear stress. Based on the Bhatnagar-Gross-Krook model, the SIS is assessed to be efficient and accurate. Then we investigate the Knudsen layer function for gases interacting through the inverse power-law, shielded Coulomb, and Lennard-Jones potentials, subject to diffuse-specular and Cercignani-Lampis gas-surface boundary conditions. When the tangential momentum accommodation coefficient (TMAC) is not larger than one, the Knudsen layer function is strongly affected by the potential, where its value and width increase with the effective viscosity index of gas molecules. Moreover, the Knudsen layer function exhibits similarities among different values of TMAC when the intermolecular potential is fixed. For Cercignani-Lampis boundary condition with TMAC larger than one, both the viscous slip coefficient and Knudsen layer function are affected by the intermolecular potential, especially when the backward'' scattering limit is approached. With the asymptotic theory by Jiang and Luo (\textit{J. Comput. Phys}., vol. 316, 2016, pp. 416--434) for the singular behavior of the velocity gradient in the vicinity of solid surfaces, we find that the whole Knudsen layer function can be well fitted by the power series $\sum_{n=0}^{2}\sum_{m=0}^{2}c_{n,m}x^n(x\ln x)^m$, where $x$ is the distance to the solid surface. Finally, the experimental data of the Knudsen layer profile are explained by the LBE solution with proper values of the viscosity index and TMAC.",
keywords = "Kramer's problem, Boltzmann equation, gas-kinetic theory, Knudsen layer",
author = "Wei Su and Peng Wang and Haihu Liu and Lei Wu",
year = "2018",
month = "11",
day = "13",
doi = "10.1016/j.jcp.2018.11.015",
language = "English",
volume = "378",
pages = "573--590",
journal = "Journal of Computational Physics",
issn = "0021-9991",

}

In: Journal of Computational Physics, Vol. 378, 01.02.2019, p. 573-590.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Accurate and efficient computation of the Boltzmann equation for Couette flow

T2 - Journal of Computational Physics

AU - Su,Wei

AU - Wang,Peng

AU - Liu,Haihu

AU - Wu,Lei

PY - 2018/11/13

Y1 - 2018/11/13

N2 - The Couette flow is one of the fundamental problems of rarefied gas dynamics, which has been investigated extensively based on the linearized Boltzmann equation (LBE) of hard-sphere molecules and simplified kinetic model equations. However, how the different intermolecular potentials affect the viscous slip coefficient and the structure of Knudsen layer remains unclear. Here, a novel synthetic iteration scheme (SIS) is developed for the LBE to find solutions of Couette flow accurately and efficiently: the velocity distribution function is first solved by the conventional iterative scheme, then it is modified such that in each iteration i) the flow velocity is guided by an ordinary differential equation that is asymptotic-preserving at the Navier-Stokes limit and ii) the shear stress is equal to the average shear stress. Based on the Bhatnagar-Gross-Krook model, the SIS is assessed to be efficient and accurate. Then we investigate the Knudsen layer function for gases interacting through the inverse power-law, shielded Coulomb, and Lennard-Jones potentials, subject to diffuse-specular and Cercignani-Lampis gas-surface boundary conditions. When the tangential momentum accommodation coefficient (TMAC) is not larger than one, the Knudsen layer function is strongly affected by the potential, where its value and width increase with the effective viscosity index of gas molecules. Moreover, the Knudsen layer function exhibits similarities among different values of TMAC when the intermolecular potential is fixed. For Cercignani-Lampis boundary condition with TMAC larger than one, both the viscous slip coefficient and Knudsen layer function are affected by the intermolecular potential, especially when the backward'' scattering limit is approached. With the asymptotic theory by Jiang and Luo (\textit{J. Comput. Phys}., vol. 316, 2016, pp. 416--434) for the singular behavior of the velocity gradient in the vicinity of solid surfaces, we find that the whole Knudsen layer function can be well fitted by the power series $\sum_{n=0}^{2}\sum_{m=0}^{2}c_{n,m}x^n(x\ln x)^m$, where $x$ is the distance to the solid surface. Finally, the experimental data of the Knudsen layer profile are explained by the LBE solution with proper values of the viscosity index and TMAC.

AB - The Couette flow is one of the fundamental problems of rarefied gas dynamics, which has been investigated extensively based on the linearized Boltzmann equation (LBE) of hard-sphere molecules and simplified kinetic model equations. However, how the different intermolecular potentials affect the viscous slip coefficient and the structure of Knudsen layer remains unclear. Here, a novel synthetic iteration scheme (SIS) is developed for the LBE to find solutions of Couette flow accurately and efficiently: the velocity distribution function is first solved by the conventional iterative scheme, then it is modified such that in each iteration i) the flow velocity is guided by an ordinary differential equation that is asymptotic-preserving at the Navier-Stokes limit and ii) the shear stress is equal to the average shear stress. Based on the Bhatnagar-Gross-Krook model, the SIS is assessed to be efficient and accurate. Then we investigate the Knudsen layer function for gases interacting through the inverse power-law, shielded Coulomb, and Lennard-Jones potentials, subject to diffuse-specular and Cercignani-Lampis gas-surface boundary conditions. When the tangential momentum accommodation coefficient (TMAC) is not larger than one, the Knudsen layer function is strongly affected by the potential, where its value and width increase with the effective viscosity index of gas molecules. Moreover, the Knudsen layer function exhibits similarities among different values of TMAC when the intermolecular potential is fixed. For Cercignani-Lampis boundary condition with TMAC larger than one, both the viscous slip coefficient and Knudsen layer function are affected by the intermolecular potential, especially when the backward'' scattering limit is approached. With the asymptotic theory by Jiang and Luo (\textit{J. Comput. Phys}., vol. 316, 2016, pp. 416--434) for the singular behavior of the velocity gradient in the vicinity of solid surfaces, we find that the whole Knudsen layer function can be well fitted by the power series $\sum_{n=0}^{2}\sum_{m=0}^{2}c_{n,m}x^n(x\ln x)^m$, where $x$ is the distance to the solid surface. Finally, the experimental data of the Knudsen layer profile are explained by the LBE solution with proper values of the viscosity index and TMAC.

KW - Kramer's problem

KW - Boltzmann equation

KW - gas-kinetic theory

KW - Knudsen layer

UR - https://www.sciencedirect.com/journal/journal-of-computational-physics

U2 - 10.1016/j.jcp.2018.11.015

DO - 10.1016/j.jcp.2018.11.015

M3 - Article

VL - 378

SP - 573

EP - 590

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

ER -