Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6930074 | Journal of Computational Physics | 2016 | 19 Pages |
Abstract
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 1â¤pâ¤â 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 ân=0ââm=0âcn,mxn(xlnâ¡x)m about x=0 on a small interval xâ(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 0.003â¤kâ¤100.0; and these solutions are accurate for at least twelve significant digits or better, thus they can be used as benchmark solutions.
Related Topics
Physical Sciences and Engineering
Computer Science
Computer Science Applications
Authors
Shidong Jiang, Li-Shi Luo,