Article ID Journal Published Year Pages File Type
6930074 Journal of Computational Physics 2016 19 Pages PDF
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 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.
Related Topics
Physical Sciences and Engineering Computer Science Computer Science Applications
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