Analytical results for sound wave propagation in small-scale two-dimensional channels

When the characteristic scale of a channel decreases, wave propagation is increasingly dominated by viscous effects. This was first realized by Lamb who predicted that when the channel size is small compared to the diffusion length based on the oscillation frequency, the gas inertia becomes negligible (the fluid motion is effectively quasi-steady) and the flow is isothermal. This parameter regime is referred to as the narrow channel regime. We take advantage of this observation to derive analytical results for wave propagation in small scale channels for arbitrary Knudsen numbers, since due to their small transverse dimensions micro and nanochannels satisfy the narrow channel requirement except at very high frequencies. In the slip-flow regime in particular where the equations of motion can be integrated analytically, we show that thermal effects are always negligible and that the long wavelength approximation is always satisfied for narrow channels. We also discuss how this theory can be extended beyond the narrow channel approximation, that is, to include the effects of inertia and heat conduction to first order. Our results are verified by direct Monte Carlo simulations of the Boltzmann equation.