Acoustic flows in a slightly rarefied gas

Abstract

The Boltzmann equation provides a rigorous description of gas flows at all degrees of gas rarefaction. Asymptotic analyses of this equation yield valuable insight into the physical mechanisms underlying gas flows. In this article, we report an asymptotic analysis of the Boltzmann-BGK equation for a slightly rarefied gas when the acoustic wavelength is comparable to the macroscopic characteristic length scale of the flow. This is performed using a three-way matched asymptotic expansion, which accounts for the Knudsen layer, the viscous layer, and the outer Hilbert region - these are separated by asymptotically disparate length scales. Transport equations and boundary conditions for these regions are derived. The utility of this theory is demonstrated by application to three problems: (1) flow generated by uniformly heating two plates, (2) oscillatory thermal creep induced between two plates, and (3) the flow generated by an oscillating sphere. Comparisons to numerical simulations of the Boltzmann-BGK equation and previous asymptotic theories (for long wavelength) are performed. The present theory is distinct from previous asymptotic analyses that implicitly assume long or short acoustic wavelength. This theory is expected to find application in the design and characterization of nanoelectromechanical devices, which often generate acoustic oscillatory flows of a rarefied nature.