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 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.

Application of quantile functions for the analysis and comparison of gas pressure balance uncertainties

“…Later numerical simulations undertaken by Jiang and Luo [21] also in the context of Couette flow similar to that studied by Li et al for the flow distribution again reported results of a slightly modified integral equation for a Knudsen number range (using their symbols) of 0.003 k 100. Their final results accurate to twelve decimal places were reported for the half-channel velocity and halfchannel flow rate.…”

Kinetic multiscale scheme based on the discrete-velocity and lattice-Boltzmann methods

“…A highly nonequlibrium gas in the Knudsen layer is described using the BGK equation, while the LBGK model is employed for the internal zone. For the BGK model, this problem can be reduced to a one-dimensional integral equation, which has been solved with high accuracy in [57,56]. Due to the lack of data on longitudinal heat flux in the mentioned works, we have reimplemented (in Python) the adaptive collocation method based on the generalized Gauss quadratures [56] for computing the benchmark solutions.…”

“…The black lines are the high-accuracy benchmark solution. The black boxes correspond to the tabulated values from[56].…”

Evaluation of Abramowitz functions in the right half of the complex plane

“…are frequently encountered in kinetic theory (cf., e.g., [7,14]), where the integral equations resulting from linearization of the Boltzmann equation have these functions (cf., e.g., [7,14,19,16]) as the kernels. The n-th order Abramowitz function J n satisfies the third order ODE [1,2] zJ ′′′ n − (n − 1)J ′′ n + 2J n = 0 (2) and the recurrence relations…”