Accurate solution and approximations of the linearized BGK equation for steady Couette flow

• Benchmark solution for Couette flow with Knudsen number between 0.003 and 10.
• High-order method for integral equation with singular kernel.
• Chebyshev based collocation method.
• Models of Knudsen layer.

We present an accurate and efficient high-order collocation method to solve the integral equation with a singular kernel derived from the linearized BGK equation in kinetic theory. In particular, we use a Chebyshev based collocation method to solve the integral equation with singular kernel for the steady Couette flow in a wide range of Knudsen numbers, i.e.  , 0.003⩽k⩽10.00.003⩽k⩽10.0. We compute the flow velocity u(y,k)u(y,k), the stress Pxy(k)Pxy(k), and the half-channel total mass flow rate Q(k)Q(k). Our results are uniformly accurate to 11 significant digits or better, thus they can serve as benchmark data. We also construct approximations of the velocity, the slip velocity, and the total mass flow rate as functions of the Knudsen number k.