Gas motion induced by unsteady boundary heating in a small-scale slab

We study the response of a gas confined in a small-scale gap to a small time-dependent change in the temperature of the gap boundaries. Using the collisionless Boltzmann equation, a general scheme for the calculation of the probability density function and the respective hydrodynamic fields in response to any heating history is developed. Asymptotic analytical results are obtained for the cases of “ramp” (linearly varying with a cutoff value) and oscillatory boundary heating. The ramp solution can be used to approximate the system response to arbitrarily slow and fast process time scales (compared to the mean free time) and thus complement previous analyses focusing on relatively slow or instantaneous heating. For oscillatory heating at frequencies significantly higher than the collision frequency, we find that, at late times, the hydrodynamic fields decay proportionally to exp[−(ωδw)2/3], where ω is the oscillation frequency and δw is the distance from the wall. As a result, for sufficiently high frequencies, the steady-state gas motion is confined to narrow “bounded” layers in the vicinity of each wall. The results are compared with low-variance particle simulations of the linearized Boltzmann equation using the recently developed low-variance deviational simulation Monte Carlo method. Good agreement is found between the analysis and simulations, suggesting the former to be an accurate and simple means for calculating the hydrodynamic response of systems of arbitrary size within one mean free path from the heated boundary.