The Response of a Gas in a Micro-channel to Periodic Boundary Heating

We study the flow-field generated in a one-dimensional wall-bounded gas layer due to periodic small-amplitude time variation in the temperature of its boundaries. We focus on the Navier-Stokes limit, where the layer width is large compared to the mean free path and the characteristic time-scale of temperature variations is long compared with the mean free time between collisions. The viscous-compressible Navier-Stokes equations with slip-flow boundary conditions are solved analytically for the case of sinusoidal heating. The analysis is then extended to study the system response to arbitrary periodic heating. Results are presented for both triangle- and square-wave heating profiles. These solutions are found to be in good agreement with low-variance Monte-Carlo simulations of the Boltzmann equation, validating the present analysis as an accurate and simple alternative to expensive molecular computations. In addition, the analysis is applied for quantitative examination of the conditions for breakdown of the slip-flow description in non-isothermal flows.