TY - JOUR

T1 - Analysis and accurate numerical solutions of the integral equation derived from the linearized BGKW equation for the steady Couette flow

AU - Jiang, Shidong

AU - Luo, Li Shi

N1 - Funding Information:
The authors would like to thank the anonymous referees for their useful comments and suggestions, which helped improve the presentation of the paper. S. Jiang was supported by the National Science Foundation under grant DMS-1418918 and would like to thank Dr. Zydrunas Gimbutas at NIST for helpful discussions. L.-S. Luo would like to acknowledge the support from the Richard F. Barry Jr. Endowment from Old Dominion University . Part of this work was done in the Summer of 2014 when S. Jiang was visiting the Computational Science Research Center in Beijing, China.
Publisher Copyright:
© 2016 Elsevier Inc.

PY - 2016/7/1

Y1 - 2016/7/1

N2 - The integral equation for the flow velocity u(x;k) in the steady Couette flow derived from the linearized Bhatnagar-Gross-Krook-Welander kinetic equation is studied in detail both theoretically and numerically in a wide range of the Knudsen number k between 0.003 and 100.0. First, it is shown that the integral equation is a Fredholm equation of the second kind in which the norm of the compact integral operator is less than 1 on Lp for any 1≤p≤∞ and thus there exists a unique solution to the integral equation via the Neumann series. Second, it is shown that the solution is logarithmically singular at the endpoints. More precisely, if x=0 is an endpoint, then the solution can be expanded as a double power series of the form ∑n=0∞∑m=0∞cn,mxn(xln x)m about x=0 on a small interval x∈(0, a) for some a>0. And third, a high-order adaptive numerical algorithm is designed to compute the solution numerically to high precision. The solutions for the flow velocity u(x;k), the stress Pxy(k), and the half-channel mass flow rate Q(k) are obtained in a wide range of the Knudsen number 0.003≤k≤100.0; and these solutions are accurate for at least twelve significant digits or better, thus they can be used as benchmark solutions.

AB - The integral equation for the flow velocity u(x;k) in the steady Couette flow derived from the linearized Bhatnagar-Gross-Krook-Welander kinetic equation is studied in detail both theoretically and numerically in a wide range of the Knudsen number k between 0.003 and 100.0. First, it is shown that the integral equation is a Fredholm equation of the second kind in which the norm of the compact integral operator is less than 1 on Lp for any 1≤p≤∞ and thus there exists a unique solution to the integral equation via the Neumann series. Second, it is shown that the solution is logarithmically singular at the endpoints. More precisely, if x=0 is an endpoint, then the solution can be expanded as a double power series of the form ∑n=0∞∑m=0∞cn,mxn(xln x)m about x=0 on a small interval x∈(0, a) for some a>0. And third, a high-order adaptive numerical algorithm is designed to compute the solution numerically to high precision. The solutions for the flow velocity u(x;k), the stress Pxy(k), and the half-channel mass flow rate Q(k) are obtained in a wide range of the Knudsen number 0.003≤k≤100.0; and these solutions are accurate for at least twelve significant digits or better, thus they can be used as benchmark solutions.

KW - Boltzmann equation

KW - Couette flow

KW - Integral equation with end-point singularities

KW - Knudsen layer

KW - Linearized BGKW equation

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U2 - 10.1016/j.jcp.2016.04.011

DO - 10.1016/j.jcp.2016.04.011

M3 - Article

AN - SCOPUS:84963815717

SN - 0021-9991

VL - 316

SP - 416

EP - 434

JO - Journal of Computational Physics

JF - Journal of Computational Physics

ER -