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

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 L-p for any 1 <= p <= infinity 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 Sigma(infinity)(n=0) Sigma(infinity)(n=0) c(n,m)x(n)(x ln x)(m) about x = 0 on a small interval x is an element of (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 P-xy(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. (C) 2016 Elsevier Inc. All rights reserved.