The study of acoustic streaming, in which the nonlinear interaction of time-periodic sound waves of small amplitude $\epsilon$ drives $\mathit{O}(\epsilon^2)$ time-mean flows, dates back to the work of Lord Rayleigh. Owing to their ability to effect a \emph{net} transport of heat, mass, and momentum, acoustic streaming flows are employed in a range of modern technologies. Surprisingly, when an acoustic wave interacts with a stratified fluid, a distinct and much stronger type of acoustic streaming can occur. This \emph{baroclinic} acoustic streaming is investigated here via the derivation of a set of wave/mean-flow interaction equations for a system comprising a standing acoustic wave in a channel with walls maintained at differing temperatures. Unlike classical Rayleigh streaming, the resulting mean flow arises at $\mathit{O}(\epsilon)$ rather than at $\mathit{O}(\epsilon^2)$. Thus, fully two-way coupling between the waves and the mean flow is possible: the streaming is sufficiently strong to induce $\mathit{O}(1)$ rearrangements of the imposed background temperature and density fields, which modifies the spatial structure and frequency of the acoustic mode on the streaming time scale. A novel WKBJ analysis is developed to average over the fast wave dynamics, enabling the coupled system to be integrated strictly on the slow time scale of the streaming flow. Numerical simulations of the asymptotically-reduced system are shown to reproduce results from prior simulations of the instantaneous compressible Navier--Stokes and heat equations with remarkable accuracy for a fraction of the computational expense. The reduced simulations shed light on the potential for baroclinic acoustic streaming to be used as an effective means to enhance heat transfer, particularly in micro-gravity environments, where weight is a primary concern and natural convective flows are absent.
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