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

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 L p 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 ∞ c n , m x n ( x ln ⁡ 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 P x y ( 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.