The transition from Rayleigh to baroclinic acoustic streaming

Rayleigh [Phil. Trans. R. Soc. Lond. 175, 1(1884)] computed the time-mean Eulerian flow of a homogeneous fluid driven by the viscous attenuation of standing acoustic waves in oscillatory boundary layers, commonly referred to as Rayleigh streaming. Recently, Chini et.al. [J. Fluid Mech., Vol. 744 (2014), pp. 329-351] demonstrated that, owing to fluctuating baroclinicity, streaming flows in strongly inhomogeneous (thermally-stratified) gases are orders of magnitude faster than those realized in Rayleigh streaming. While these two streaming regimes have been studied separately, an analytical solution capturing the transition between them has not yet been discussed. To address this omission, we perform a systematic asymptotic analysis of an acoustically-driven weakly inhomogeneous ideal gas confined in a horizontal channel. We obtain a strictly analytical description of the resulting steady streaming flow, and validate the analytical solution via comparisons with existing experimental and numerical results. The analysis elucidates the relative contributions of baroclinic and viscous torques to the resulting streaming flow, and reveals the distinguished scaling of the temperature gradient required to trigger a transition from Rayleigh to baroclinic acoustic streaming.