1. A path from Faraday’s observations of acoustic streaming

Three steps forwards, two back, this is the way the progress of science is sometimes presented: one step after the other, regularly coming back to fundamental works to go forward in a strict and confident way. This can also be a first picture of acoustic streaming, in which a progressive motion originates from an oscillating one. But acoustic streaming is not only kinematics and has been the subject of numerous studies since the observations of Faraday (Reference Faraday1831) reported by Rayleigh (Reference Rayleigh1884): ‘In these problems the fluid may be treated as incompressible… It was discovered by Savart that very fine powder does not collect itself at the nodal lines, as does sand in the production of Chladni’s figures, but gathers itself into a cloud which, after hovering for a time, settles itself over the places of maximum vibration’. Rayleigh also identifies two ingredients of acoustic streaming, namely viscous dissipation and nonlinearity: ‘This is not, perhaps, a matter for surprise, when we consider that such currents, involving as they do circulation of the fluid, could not arise in the absence of friction, however great the extent of vibration… we have no chance of reaching an explanation if, as is usual, we limit ourselves to the supposition of infinitely small motion and neglect the squares and higher powers of mathematical symbols by which it is expressed’.

After a few decades, technological developments in the field of sound generation led to the observation of acoustic streaming induced by sound beams propagating in the bulk of the fluid domain. Interestingly, the important paper by Eckart (Reference Eckart1948) puts forward the intimate link between the observed flows and vorticity introduced by the acoustic excitation. Sir J. Lighthill (Reference Lighthill1978) gives an exhaustive picture in an amusing zoology of the different configurations: Stuart streaming, Rayleigh–Nyborg–Westervelt streaming and McIntyre streaming. Lighthill (Reference Lighthill1978) highlights the central part played by sound attenuation and Reynolds stresses. Still more recent technological challenges in the field of cooling (Fand & Kaye Reference Fand and Kaye1960; Michel & Gissinger Reference Michel and Gissinger2021) and lighting (Dreeben & Chini Reference Dreeben and Chini2011), but also thermoacoustics (Daru et al. Reference Daru, Weisman, Baltean-Carlès and Bailliet2021) and microfluidics (Karlsen, Augustsson & Bruus Reference Karlsen, Augustsson and Bruus2016), have shown this framework had to be extended to explain some experimental observations. This lead to the identification of baroclinic streaming, in which gradients in background density play the central role and acoustic attenuation is not so essential. We inherit a field of acoustic streaming fractured into three panels in the following tryptic:

1. (i) baroclinic streaming occurs when a standing acoustic wave is crossed with a strong density gradient (alternatively a compressibility gradient Karlsen et al. Reference Karlsen, Augustsson and Bruus2016);

2. (ii) Rayleigh–Schlichting streaming is observed when a standing wave interacts with a wall; note the comprehensive theoretical description by Hamilton, Ilinskii & Zabolotskaya (Reference Hamilton, Ilinskii and Zabolotskaya2003);

3. (iii) Eckart streaming is observed when acoustic waves progress in the form of an attenuated acoustic beam in the bulk of the fluid (Eckart Reference Eckart1948; Moudjed et al. Reference Moudjed, Botton, Henry, Ben Hadid and Garandet2014; Dubrovski, Friend & Manor Reference Dubrovski, Friend and Manor2023).

Since they describe in a single continuous parameter space both Rayleigh–Schlichting and baroclinic streaming, Mushthaq, Michel & Chini (Reference Mushthaq, Michel and Chini2025) make a decisive step forward into the frontiers between the panels of this tryptic.

2. Three panels in the picture

In their paper, Mushthaq et al. (Reference Mushthaq, Michel and Chini2025) propose a local approach featuring a multi-scale expansion of the different fields appearing in the compressible Navier–Stokes equations. They recall that the forcing yielding acoustic streaming is essentially similar to a Reynolds stress mechanism in which the fluctuating velocity of Reynolds decomposition is replaced by the rapidly oscillating acoustic velocity, and also that some acoustic vorticity is required to get a rotational forcing of the steady flow. Time averaging then enables separation of the slow dynamics of the observed flow from the comparably very fast acoustic scales. Let us adopt a two-dimensional global standpoint to consider the transverse and longitudinal scales at which the acoustic vorticity is expressed. These scales change from one panel of the tryptic to the other (see figure 1). In this global and heuristic standpoint, let us focus on the longitudinal component of the acoustic momentum $\rho _0{u}^\prime$ , with $u^\prime$ the longitudinal acoustic velocity component and $\rho _0$ the background density. The counterpart of the normal Reynolds stress is then the time-average momentum flux through a transverse cross-section over one acoustic period, $\overline {\rho _0 u^{\prime ^2}}h$ , where $h$ is the height of the cross-section considered.

Figure 1.

Acoustic streaming seen as a tryptic: each panel depicts a particular type of streaming. The represented control volume is chosen to fit the scales of longitudinal and transverse variations in the acoustic Reynolds stress $\overline {\rho _0 u^{\prime ^2}}$ . This illustrates how a torque is induced by differences in acoustic momentum flux through the boundaries. Each type of streaming differs by the origins and scales of Reynolds stress variations. (a) Eckart streaming: acoustic waves propagate in the form of a beam. Transverse variations are inherent to the beam shape. Longitudinal variations are due to sound attenuation. (b) Rayleigh–Schlichting streaming: transverse variations are a consequence of the no-slip condition and by viscous diffusion at the boundary-layer scale, $\delta _S$ . Longitudinal variations at a $\lambda /4$ scale are due to the structure of the standing wave (A and N respectively denote antinodes and nodes for the acoustic velocity). (c) Baroclinic streaming occurs in a stratified environment. Longitudinal vibrations yield an acoustic standing wave; assuming that the acoustic momentum profile, $\rho _0\boldsymbol{u}^\prime$ , is nearly uniform along the transverse direction, the acoustic velocity must vary inversely to the density. These variations thus occur at the scale of the cavity, $H$ . Longitudinal variations at a $\lambda /4$ scale are due to the structure of the standing wave (A and N respectively denote antinodes and nodes for the acoustic velocity).

In the first panel (figure 1a) Eckart streaming is induced by an acoustic beam in a cavity. The longitudinal scale is the length of the beam, which is given by the sound intensity attenuation length and can be far greater than an acoustic wavelength (Lighthill Reference Lighthill1978; Moudjed et al. Reference Moudjed, Botton, Henry, Ben Hadid and Garandet2014; Vincent et al. Reference Vincent, Henry, Kumar, Botton, Pothérat and Miralles2024). The transverse scale is that of the cavity. The acoustic wave momentum decreases along the longitudinal direction due to attenuation, generating a forcing in the direction of propagation. The beam-shape structure yields two macroscopic torques which induce two circulating flows at the scale of the upper and lower half-cavities, respectively. A permanent regime is reached when the torques due to external viscosity forces balance these forcing torques.

In the second panel (figure 1b) representing the inner streaming due to a standing wave, the transverse scale is that of the acoustic boundary layer, $\delta _s=\sqrt {{\mu \lambda }/{\rho _0\ c}}$ , where $\mu$ and $c$ respectively denote the dynamic viscosity and the celerity of sound. The longitudinal scale is a quarter of a wavelength $\lambda$ . The longitudinal momentum, $\rho_0\boldsymbol{u}^\prime$ , is maximum at the antinode A of the standing velocity wave and zero at the node N. It also varies in the transverse direction due to viscous diffusion and the no-slip condition at the wall; the momentum flux into the control volume is then stronger in the upper part of the boundary layer than in the lower part. The resulting torque is expected to scale as $\overline {{\rho _0 u^{\prime ^2}}}\delta _S^2$ . Denoting by $u_S$ the order of magnitude of the steady velocity induced at the top of the boundary layer, the external viscous torque at the scale of the inner recirculation is of the order of $({\mu u_s}/{\delta _S}) \lambda \delta _S$ . Balancing one torque by the other yields $u_S\sim \overline {u^{\prime 2}}/c$ , which also gives the scaling of the flow induced in the bulk referred to as Rayleigh streaming. Remarkably, $u_S$ does not depend on the aspect ratio $\delta _S/\lambda$ .

The third panel (figure 1c) is the one of most interest today. Imagine that, prior to the Rayleigh–Schlichting experiment depicted in figure 1(b), heating from above leads to a strong stratification in background density $\rho _0$ (Lin & Farouk Reference Lin and Farouk2008; Michel & Gissinger Reference Michel and Gissinger2021). Introducing longitudinal oscillations would induce a standing wave. In a one-way coupling approximation, the background density is not affected by the wave. Assuming that the fluctuating pressure is approximately uniform across the channel, the longitudinal acoustic momentum, $\rho _0\boldsymbol{u}^\prime$ , is also nearly uniform. The amplitude $u^{\prime }$ of the acoustic velocity must then vary inversely to $\rho _0$ . This induces, at the antinode A, a strong transverse gradient in the time-averaged momentum flux into the control volume. The consequent powerful torque at the scale of the cavity, $H$ , creates a single global recirculation directed towards the antinode A on the side of the heavier fluid. This is a very different picture than the outer recirculations of Rayleigh streaming, flowing towards the nodes near the walls, and which are driven by a smaller recirculation induced at the boundary-layer scale where the acoustic torque applies. Here, the balance by the steady external viscous torque involves the four sides of the control volume and is thus directly dependent on its aspect ratio $\delta =4H/\lambda$ .

These back-of-the-envelope explanations certainly miss several significant effects, for instance the momentum flux associated with the transverse component of the acoustic velocity, $v^\prime$ , at the top of the boundary layer and the modification of the background density by the flow (the coupling actually appears to be a two-way coupling). Our contention, however, is that they give a qualitatively correct picture of the mechanisms at play. The distinction between the mechanisms in figures 1(b) and 1(c) is indeed very thin since baroclinic streaming occurs as soon as the gradient in fluid properties is such that a torque occurs. That is to say that transverse and longitudinal variations in acoustic momentum are combined. As this baroclinic, time-averaged, torque acts at the cavity scale while Rayleigh–Schlichting is created in thin boundary layers, it is very efficient even in a weak stratification. Starting from a state of pure Rayleigh streaming, progressively increasing the heating, i.e. the stratification, until baroclinic streaming becomes significant chases the boundary between figures 1(b) and 1(c) away. This is what Mushthaq et al. (Reference Mushthaq, Michel and Chini2025) do using a local approach for a small level of heating.

3. Baroclinic transition in acoustic streaming: beyond Rayleigh’s paradigm

More specifically, Mushthaq et al. (Reference Mushthaq, Michel and Chini2025) consider standing acoustic wave oscillations of an ideal gas in a differentially heated channel with wall temperatures set to $T_\ast +\Delta \varTheta _\ast$ and $T_\ast$ . They consider Rayleigh streaming and baroclinic streaming as two asymptotic regimes for $\Delta \varTheta _\ast \sim 0$ and $\Delta \varTheta _\ast =O(T_\ast )$ , respectively. Their small parameter is a normalised temperature difference, $\varGamma =\Delta \varTheta _\ast /T_\ast$ , assumed comparable to the acoustic Mach number. As illustrated in their figure 2, their equation (3.56) for the streamfunction describes the steady flow between the node and antinode cross-sections as the superimposition of 3 components:

1. (i) two Rayleigh streaming cells corresponding to the isothermal flow and exhibiting the expected symmetry with respect to the middle plane;

2. (ii) one baroclinic streaming recirculation driven by the heating-induced density gradient, expectedly spanning the entire height of the channel;

3. (iii) a small corrective term depending on the type of forcing, an oscillating body force or an oscillating wall.

As this analytical solution elucidates the separate contributions of viscous torques in the boundary layers and baroclinic forcing in the bulk, the authors can investigate the typical value $\varGamma _c$ over which baroclinic streaming becomes dominant, as shown in their equation (4.8). Remarkably, $\varGamma _c$ does not depend on the forcing amplitude and is composed of three factors; the acoustic Reynolds number (the ratio of the wavelength times the sound celerity to the fluid kinematic viscosity), a term featuring the ideal gas properties only and a function of the aspect ratio $\delta$ only. This typical value $\varGamma _c$ behaves as $\delta ^{-2}$ for narrow chanels and as $\delta ^{1/2}$ for tall channels.

Of course, a lot remains to be investigated. For instance, steady horizontal density gradients induced in this two-way coupling are a motor for buoyancy-driven convection whose magnitude relative to baroclinic streaming should be assessed. This type of forcing could also be situated with respect to thermo-vibrational convection. Eckart acoustic streaming could be coupled with strong background gradients in mechanical properties, etc. Mushthaq et al. (Reference Mushthaq, Michel and Chini2025) significantly improve the understanding of flows driven by sound waves. Experimentalists will be enlightened in designing their set-ups and theoreticians will be inspired by this smart approach. Modern applications of acoustofluidics, think of biomedical applications for instance (Baudoin & Thomas Reference Baudoin and Thomas2020), often combine several types of streaming flows and similarly will benefit from the new insights provided by Mushthaq et al. (Reference Mushthaq, Michel and Chini2025). These insights are not restricted to acoustic streaming, of course, nor to engineering applications: for instance, communities working on other types of steady streaming or on geophysical configurations, think of internal waves (Dauxois et al. Reference Dauxois, Joubaud, Odier and Venaille2018), could well be inspired too. Chasing the boundary between Rayleigh and baroclinic streamings away, Mushthaq et al. (Reference Mushthaq, Michel and Chini2025) definitely take an important step forward.