2.1. Flow configuration

We investigate streaming of an ideal gas in a differentially heated horizontal channel subjected to standing horizontal acoustic-wave oscillations having a wavelength commensurate with the channel height, as illustrated in figure 1. Except for the temperature difference $\Delta \varTheta _*$ that is assumed much smaller, this set-up is similar to that used to investigate baroclinic acoustic streaming in Massih et al. (Reference Massih, Mushthaq, Michel and Chini2024). The dimensional variables and parameters, defined in table 1, are indicated using tildes and asterisks, respectively. Gravity is neglected and only two-dimensional dynamics is considered. Periodic boundary conditions are imposed along the horizontal ( $\tilde {x}$ ) direction, setting the horizontal wavelength $2\pi /k_*$ . The hot and cold walls are modelled as no-slip, no-penetration, isothermal boundaries maintained at constant but different temperatures, respectively $T_* + \Delta \varTheta _*$ and $T_*$ . In the absence of the acoustic wave, heat simply diffuses across the channel: there is no fluid motion, and the steady temperature profile is linear in the wall-normal ( $\tilde {y}$ ) coordinate. A wall-parallel conservative acoustic body force of angular frequency $\omega _*$ excites and sustains the acoustic waves without directly driving a mean flow. The system is governed by the compressible Navier–Stokes equations, supplemented with conservation of mass and internal energy, and with the ideal gas equation of state:

(2.1) \begin{align}& \tilde {\rho } [ \partial _{\tilde {t}} \tilde {\boldsymbol {u}} + ( \tilde {\boldsymbol {u}} \boldsymbol{\cdot} \tilde {\boldsymbol{\boldsymbol{\nabla} }} )\tilde {\boldsymbol {u}} ] = - \tilde {\boldsymbol{\boldsymbol{\nabla} }} \tilde {p} + \mu _* \left[\tilde {\boldsymbol{\boldsymbol{\nabla} }}^2 \tilde {\boldsymbol {u}} + \frac {1}{3}\tilde {\boldsymbol{\boldsymbol{\nabla} }} ( \tilde {\boldsymbol{\boldsymbol{\nabla} }} \boldsymbol{\cdot} \tilde {\boldsymbol {u}} )\right] + \tilde {\mathbb{F}}(\tilde {x}, \tilde {t})\,\boldsymbol{e}_{x}, \end{align}

(2.2) \begin{align}&\qquad\qquad\qquad\qquad\qquad \partial _{\tilde {t}} \tilde {\rho } + \tilde {\boldsymbol{\boldsymbol{\nabla} }} \boldsymbol{\cdot} ( \tilde {\rho } \tilde {\boldsymbol {u}} ) = 0, \end{align}

(2.3) \begin{align}&\qquad\qquad \tilde {\rho } c_v [ \partial _{\tilde {t}} \tilde {T} + ( \tilde {\boldsymbol {u}} \boldsymbol{\cdot} \tilde {\boldsymbol{\boldsymbol{\nabla} }} )\tilde {T} ] = - \tilde {p} ( \tilde {\boldsymbol{\boldsymbol{\nabla} }} \boldsymbol{\cdot} \tilde {u} ) + \kappa _* \tilde {{{\nabla}}}^2 \tilde {T}, \end{align}

(2.4) \begin{align}&\qquad\qquad\qquad\qquad\qquad\qquad \tilde {p} = \tilde {\rho } R_* \tilde {T}, \end{align}

where $\tilde {\rho }$ , $\tilde {p}$ , $\tilde {T}$ and $\tilde {\boldsymbol {u}}=(\tilde {u},\tilde {v})$ denote the density, pressure, temperature and velocity fields, respectively, and $\tilde {\boldsymbol{\nabla} } \equiv (\partial _{\tilde {x}}, \partial _{\tilde {y}})$ . Bulk viscosity and viscous heating are neglected for convenience and analytical tractability, as is the temperature dependence of the viscosity and thermal conductivity (see the discussion in Michel & Chini (Reference Michel and Chini2019)).

Figure 1.

Schematic of the flow configuration. An ideal gas is confined between two horizontal, no-slip and impermeable walls separated by a distance $H_*$ . The temperatures of the cold and hot walls are fixed at $T_*$ and $T_* + \Delta \varTheta _*$ , respectively. Gravity is neglected. In the regime where the mean flow transitions from Rayleigh to baroclinic streaming, a standing acoustic wave of horizontal wavenumber $k_*$ generates a counter-rotating stacked cellular streaming flow, where the cells closer to the hot wall span the majority of the channel height.

Table 1.

Dimensional variables and parameters.

2.2. Scaling and non-dimensionalisation

The scalings and the definition of dimensionless variables and parameters are reported in table 2. Except for the imposed temperature difference, these scalings are similar to those reported in Massih et al. (Reference Massih, Mushthaq, Michel and Chini2024). The acoustic Mach number $\epsilon$ , which is also the inverse of the Strouhal number associated with the oscillatory flow, is assumed small and is used to order the subsequent asymptotic analysis. In this study, the dimensionless temperature differential is assumed to be finite but weak: $\hat {\varGamma } = O(\epsilon )$ . As will become evident, this scaling enables the streaming flow generated by viscous torques in oscillatory BLs and that from baroclinic torques in the interior of the domain to arise at the same order in $\epsilon$ . The remaining parameters, including the aspect ratio $\delta$ , are taken to be $O(1)$ .

Table 2.

Dimensionless variables and parameters, similar to the previous analysis of Massih et al. (Reference Massih, Mushthaq, Michel and Chini2024) except for the dimensionless temperature difference $\hat {\varGamma }$ , which in the present work is asymptotically small.

Note that the Reynolds and Péclet numbers $\textit{Re}=O(1)$ and $\textit{Pe}=O(1)$ here compare the nonlinear terms with viscous and thermal diffusion in the body of the fluid. Note that other definitions of these coefficients can be found in the literature: a ‘wave’ Reynolds number ( $\textit{Re}_w$ ) based on the sound speed is sometimes introduced ( $\textit{Re}_w = \rho _* a_*/ (k_* \mu _*) = {Re}/\epsilon$ , here $\textit{Re}_w \gg 1$ ), which compares unsteady inertia with net viscous forces and confirms that viscous terms do not affect the waves at leading order; or a streaming Reynolds number ( $\textit{Re}_s$ ) based on the appropriate velocity scale for the streaming flow $U_{s_*} = \epsilon ^2 a_*$ ( $\textit{Re}_s = \rho _* \epsilon ^2 a_*/ (k_* \mu _*) = \epsilon {Re}$ , here $\textit{Re}_s \ll 1$ ), showing that nonlinear advection plays no part in the dynamics of the streaming flow in the regimes we investigate (Lighthill Reference Lighthill1978). Using the scalings detailed in table 2, the dimensionless set of equations can be expressed as

(2.5) \begin{align}& \rho \left [ \partial _{t} \boldsymbol {u} + ( \boldsymbol {u} \boldsymbol{\cdot} \boldsymbol{\boldsymbol{\nabla} } )\boldsymbol {u} \right ] = - \frac {1}{\gamma } \boldsymbol{\boldsymbol{\nabla} } p + \frac {\epsilon }{{Re}} \left [ \boldsymbol{\boldsymbol{\nabla} }^2 \boldsymbol {u} + \frac {1}{3}\boldsymbol{\boldsymbol{\nabla} } ( \boldsymbol{\boldsymbol{\nabla} } \boldsymbol{\cdot} \boldsymbol {u} ) \right ] + \mathbb{F} (x, t) \boldsymbol{e}_x, \end{align}

(2.6) \begin{align}&\qquad\qquad\qquad\qquad\qquad \partial _{t} \rho + \boldsymbol{\boldsymbol{\nabla} } \boldsymbol{\cdot} \left ( \rho \boldsymbol {u} \right ) = 0, \end{align}

(2.7) \begin{align}&\qquad\qquad \rho \left [ \partial _{t} T + ( \boldsymbol {u} \boldsymbol{\cdot} \boldsymbol{\boldsymbol{\nabla} } )T \right ] = (1 - \gamma ) p \left ( \boldsymbol{\boldsymbol{\nabla} } \boldsymbol{\cdot} u \right ) + \frac {\gamma \,\epsilon }{Pe} {{\nabla} }^2 T, \end{align}

(2.8) \begin{align}&\qquad\qquad\qquad\qquad\qquad\qquad p = \rho T, \end{align}

where dimensionless variables are not decorated with tildes, $\boldsymbol{\boldsymbol{\nabla} } \equiv (\partial _x , \delta ^{-1} \partial _y )$ and the velocity field $\boldsymbol{u} \equiv (u , \delta v )$ . No-slip and no-penetration boundary conditions are complemented with the thermal boundary conditions $T(x,y=0,t)=1$ and $T(x,y=1,t)=1+\epsilon \varGamma$ .

The scaling of the Reynolds number $\textit{Re}=O(1)$ , implying $\textit{Re}_w=O(1/\epsilon )$ , ensures that viscous forcing is negligible in the interior of the domain for the leading-order acoustic wave dynamics. However, these viscous forces drive oscillatory BLs of typical dimensional width $\delta _{s_*} = \sqrt {\mu _*/(\rho _* \omega _*)}= k_*^{-1}\sqrt {\epsilon /{Re}}$ . The dynamics in these oscillatory BLs near the cold and hot walls, hereafter referred to as C-BL and H-BL, is captured by introducing rescaled wall-normal coordinates $\eta _{c} \equiv \epsilon ^{-1/2} y$ and $\eta _{h} \equiv \epsilon ^{-1/2} (1-y)$ , respectively. In the analysis that follows, the rescaled wall-normal coordinates in both BLs are commonly represented using $\eta$ without a subscript.

3.1. Acoustic and steady dynamics at $O(1)$

At this order the gas is steady (all fluctuating and mean flow speeds are smaller than the speed of sound) and homogeneous (the imposed temperature difference is assumed to be small compared with the cold-wall temperature).

3.2. Steady dynamics at $O(\epsilon )$

No streaming flow arises at this order. The imposed temperature difference between the boundaries maintains the linear temperature and density profiles $\epsilon \varGamma y$ and $\epsilon \overline {\rho }_1$ , with $\overline {\rho }_1\,{=}\,-\varGamma y$ .

3.3. Acoustic dynamics at $O(\epsilon )$

The $O(\epsilon )$ dynamics in the interior satisfies

(3.8) \begin{align}\partial _t u_1^{\prime}+ \gamma ^{-1} \partial _x \pi _1^{\prime} &=0, \end{align}

(3.9) \begin{align}\partial _y \pi _1^{\prime}&=0, \end{align}

(3.10) \begin{align}\partial _t \rho _1^{\prime} + \partial _x u_1^{\prime} & = 0, \end{align}

(3.11) \begin{align}\partial _t \varTheta _1^{\prime} + \left (\gamma -1\right ) \partial _x u_1^{\prime}&=0, \end{align}

(3.12) \begin{align}\pi _1^{\prime} - \rho _1^{\prime} - \varTheta _1^{\prime}&=0, \end{align}

which can be collapsed to the one-dimensional second-order wave equation:

(3.13) \begin{equation} \partial _{tt} \pi _1^{\prime} = \partial _{xx} \pi _1^{\prime},\quad \text{with} \quad \partial _t u_1^{\prime} = - \gamma ^{-1}\partial _x \pi _1^{\prime}, \quad \rho _1^{\prime}= \gamma ^{-1}\pi _1^{\prime}, \quad \varTheta _1^{\prime} = (1-\gamma ^{-1})\pi _1^{\prime}._{_{_{_{\!}}}} \end{equation}

Since the oscillatory flow driven by the plane wave does not satisfy the no-slip and isothermal boundary conditions at the walls, Stokes layers are generated.

The $O(\epsilon )$ dynamics in the BLs is governed by

(3.14) \begin{align}&\qquad\qquad \partial _t u_1^{\prime} +\gamma ^{-1} \partial _x \pi _1^{\prime} - ({Re}\delta ^2)^{-1}\partial _{\eta \eta } u_1^{\prime}=0, \end{align}

(3.15) \begin{align}&\qquad\qquad\qquad\qquad\qquad \partial _{\eta } \pi _1^{\prime}=0, \end{align}

(3.16) \begin{align}&\qquad\qquad\qquad\quad \partial _t \rho _1^{\prime} + \partial _x u_1^{\prime} \pm \partial _{\eta } v_{3/2}^{\prime} = 0, \end{align}

(3.17) \begin{align}& \partial _t \varTheta _1^{\prime} + \left (\gamma -1 \right ) (\partial _x u_1^{\prime} \pm \partial _{\eta } v_{3/2}^{\prime}) - \gamma (Pe \delta ^2)^{-1} \partial _{\eta \eta } \varTheta _1^{\prime}=0, \end{align}

(3.18) \begin{align}&\qquad\qquad\qquad\qquad\,\, \pi _1^{\prime} - \rho _1^{\prime} - \varTheta _1^{\prime}=0, \end{align}

where the first-order wall-normal partial derivatives in the C-BL are $+ \partial _{\eta }$ and those in the H-BL are $- \partial _{\eta }$ .

The complete acoustic wave solution at $O(\epsilon )$ , which satisfies the isothermal, no-slip and no-penetration conditions at the boundaries and asymptotically matches with the horizontal standing-wave solution obtained in the interior, is found to be

(3.19) \begin{align} \pi _1^{\prime} &= \frac {\mathbb{A}}{2}\cos \left (x\right ){\rm e}^{{\rm i}t} + \text{c.c.} \qquad \qquad \qquad \qquad \quad \quad \,\,\text{(interior and BLs)}, \end{align}

(3.20) \begin{align} u_1^{\prime} &= \begin{cases} - {\rm i}\dfrac {\mathbb{A}}{2\gamma }\sin \left (x\right ){\rm e}^{{\rm i}t} + \text{c.c.} &\text{(interior)} \\[8pt] -{\rm i}\dfrac {\mathbb{A}}{2\gamma }(1 - {\rm e}^{-(1 + {\rm i}) \sqrt {\mathbb{R}}\eta })\sin (x){\rm e}^{{\rm i}t} + \text{c.c.} &\text{(BLs),} \end{cases} \end{align}

(3.21) \begin{align} v_{3/2}^{\prime} &= \begin{cases} - \dfrac {\mathbb{A}(1+{\rm i})\,\cos (x){\rm e}^{{\rm i}t}}{4\gamma \sqrt {\mathbb{R}\mathbb{P}}} [(1 - {\rm e}^{-(1 + {\rm i}) \sqrt {\mathbb{R}}\eta}) \sqrt {\mathbb{P}} + (\gamma - 1)(1 - {\rm e}^{-(1 + {\rm i})\sqrt {\mathbb{P}}\eta }) \sqrt {\mathbb{R}}]\\[4pt] \quad + \text{c.c.} \quad\text{(C-BL)}\\[4pt] \dfrac {\mathbb{A} (1+{\rm i})\,\cos (x){\rm e}^{{\rm i}t}}{4\gamma \sqrt {\mathbb{RP}}} [(1 - {\rm e}^{-(1 + {\rm i}) \sqrt {\mathbb{R}}\eta}) \sqrt {\mathbb{P}} + (\gamma - 1)(1 - {\rm e}^{-(1 + {\rm i})\sqrt {\mathbb{P}}\eta }) \sqrt {\mathbb{R}}] \\[4pt] \quad + \text{c.c.} \quad\text{(H-BL),} \end{cases} \end{align}

(3.22) \begin{align} \varTheta _{1}^{\prime} &= \begin{cases} (1-\gamma ^{-1}) \pi _1^{\prime} &\text{(interior)}\\[4pt] (1 - \gamma ^{-1}) \dfrac {\mathbb{A}}{2}(1 - {\rm e}^{-(1 + {\rm i}) \sqrt {\mathbb{P}}\eta})\cos (x){\rm e}^{{\rm i}t} + \text{c.c.} &\text{(BLs),} \end{cases} \end{align}

(3.23) \begin{align} \rho _{1}^{\prime} &= \begin{cases} \gamma ^{-1}\pi _1^{\prime} &\text{(interior)}\\[4pt] \gamma ^{-1}\dfrac {\mathbb{A}}{2} (1 + (\gamma - 1) {\rm e}^{-(1 + {\rm i}) \sqrt {\mathbb{P}} \eta })\cos (x) {\rm e}^{{\rm i}t} + \text{c.c.} &\text{(BLs).} \end{cases}\end{align}

Here, $\mathbb{R} \equiv {Re}\delta ^2/2$ and $\mathbb{P} \equiv Pe \delta ^2/2$ are the aspect-ratio-dependent Reynolds and Péclet numbers and $\mathbb{A}$ is the $O(1)$ acoustic pressure wave amplitude. The expression for $v_{3/2}^{\prime}$ in the interior is derived in the higher-order analysis that follows. At $O(\epsilon )$ , the acoustic velocity field in the interior is irrotational, and acoustic wave vorticity is confined to the BLs.

3.4. Steady dynamics at $O(\epsilon ^{3/2})$

No corrections to the streaming fields arise at this order.

3.5. Acoustic dynamics at $O(\epsilon ^{3/2})$

The vertical velocity that emerges in the leading-order BLs does not vanish as $\eta \rightarrow \infty$ ; see (3.21). The oscillating field it generates would, in the absence of forcing, lead to a slow decay of the leading-order wave amplitude. Here, a stationary state is sustained by the weak external forcing $\mathbb{F}'_{3/2}$ in the interior.

The governing equations at $O(\epsilon ^{3/2})$ in the interior are

(3.24) \begin{align} \partial _t u_{3/2}^{\prime} + \gamma ^{-1} \partial _x \pi _{3/2}^{\prime} & = \mathbb{F}_{3/2}\sin \left (x\right ){\rm e}^{{\rm i}t} + \text{c.c.} , \end{align}

(3.25) \begin{align}\partial _t v_{3/2}^{\prime}+ (\gamma \delta ^2)^{-1} \partial _y \pi _{3/2}^{\prime} & =0, \end{align}

(3.26) \begin{align}\partial _t \rho _{3/2}^{\prime} + \partial _x u_{3/2}^{\prime} + \partial _y v_{3/2}^{\prime} & = 0, \end{align}

(3.27) \begin{align}\partial _t \varTheta _{3/2}^{\prime} + (\gamma -1 )( \partial _x u_{3/2}^{\prime} + \partial _y v_{3/2}^{\prime} )&=0, \end{align}

(3.28) \begin{align}\pi _{3/2}^{\prime} - \rho _{3/2}^{\prime} - \varTheta _{3/2}^{\prime}&=0. \end{align}

In the BLs the waves must satisfy

(3.29) \begin{align}\partial _t u_{3/2}^{\prime} + \gamma ^{-1} \partial _x \pi _{3/2}^{\prime} - ({Re}\delta ^2)^{-1}\partial _{\eta \eta } u_{3/2}^{\prime} & = \mathbb{F}_{3/2}\sin \left (x\right ){\rm e}^{{\rm i}t} + \text{c.c.}, \end{align}

(3.30) \begin{align}\partial _{\eta } \pi _{3/2}^{\prime}&=0, \end{align}

(3.31) \begin{align}\partial _t \rho _{3/2}^{\prime} + \partial _x u_{3/2}^{\prime} \pm \partial _{\eta } v_{2}^{\prime} & = 0, \end{align}

(3.32) \begin{align}\partial _t \varTheta _{3/2}^{\prime} +(\gamma -1) (\partial _x u_{3/2}^{\prime} \pm \partial _{\eta } v_{2}^{\prime}) - \gamma (Pe\delta ^2)^{-1} \partial _{\eta \eta } \varTheta _{3/2}^{\prime}&=0, \end{align}

(3.33) \begin{align}\pi _{3/2}^{\prime} - \rho _{3/2}^{\prime} - \varTheta _{3/2}^{\prime}&=0. \end{align}

This set of equations can be solved analytically with appropriate boundary and matching conditions; see Appendix A. In particular, the balance between external forcing and diffusion is explicit:

(3.34) \begin{equation} \mathbb{F}_{3/2} = (1-i)\,\frac {\mathbb{A}}{2\,\gamma }\left (\frac {\left (\gamma - 1\right )}{\sqrt {\mathbb{P}}} + \frac {1}{\sqrt {\mathbb{R}}}\right ). \end{equation}

At $O(\epsilon ^{3/2})$ , acoustic wave vorticity remains localised in the BLs ( $\boldsymbol{\Omega }_{3/2}^{\prime}= \boldsymbol{\nabla} \times \boldsymbol {u}_{3/2}^{\prime}= \boldsymbol{0}$ in the interior, as shown by taking the curl of the momentum equations (3.24)–(3.25)).

3.6. Steady dynamics at $O(\epsilon ^2)$ (part 1)

The $O(\epsilon ^2)$ equations characterising the steady pressure field $\overline {\pi }_2$ in the interior are

(3.35) \begin{align}\gamma ^{-1} \partial _x \overline {\pi }_2 & = -\overline {\rho _1^{\prime} \partial _t u_1^{\prime}} - \overline {u_1^{\prime} \partial _x u_1^{\prime}} = \overline {(\partial _t\rho _1^{\prime}) u_1^{\prime}} - \overline {u_1^{\prime} \partial _x u_1^{\prime}} = - \partial _x \overline {(u_1^{\prime})^2}, \end{align}

(3.36) \begin{align}\partial _y \overline {\pi }_2 &= 0, \end{align}

where (3.35) has been simplified using (3.10). Thus, to within an arbitrary constant, the mean pressure $\overline {\pi }_2 = - \gamma \overline {(u_1^{\prime})^2}$ .

A streaming flow emerges as the consequence of nonlinearity. Specifically, in the BLs,

(3.37) \begin{equation} ( \boldsymbol {u}' \boldsymbol{\cdot} \boldsymbol{\boldsymbol{\nabla} } )\boldsymbol {u}' = \boldsymbol{\boldsymbol{\nabla}} \left (\frac {1}{2}\,\boldsymbol{u'} \boldsymbol{\cdot} \boldsymbol {u}'\right ) + (\boldsymbol{\boldsymbol{\nabla}} \times \boldsymbol {u}') \times \boldsymbol {u}' \end{equation}

includes an $O(\epsilon ^2)$ component $(\boldsymbol{\boldsymbol{\nabla}} \times \boldsymbol {u}') \times \boldsymbol {u}'$ that (i) is non-zero since the $O(\epsilon )$ acoustic vorticity is non-zero in the BL, (ii) cannot be completely balanced by a pressure gradient and (iii) consists of both an oscillating term proportional to ${\rm e}^{2{\rm i}t}$ that generates harmonics and, crucially, a steady component. As first established by Rayleigh, this term drives a streaming flow in the BL that is viscously communicated to the interior and that can be computed through the momentum equations only. The inertial term in the $x$ -momentum equation can be transformed using (3.16) and recast as

(3.38) \begin{align} \rho _1^{\prime} \partial _t u_1^{\prime} + u_1^{\prime} \partial _x u_1^{\prime} \pm v_{3/2}^{\prime} \partial _\eta u_1^{\prime} &= \partial _t (\rho _1^{\prime} u_1^{\prime}) + \partial _x \big({u'}_{1}^{2}\big) \pm \partial _\eta \big( u_1^{\prime} v_{3/2}^{\prime}\big), \end{align}

and we therefore consider

(3.39) \begin{align} - \gamma ^{-1} \partial _x \overline {\pi }_2 + ({Re}\delta ^2)^{-1} \partial _{\eta \eta } \overline {u}_2&=\partial _x (\overline {u_1^{\prime} u_1^{\prime}}) \pm \partial _\eta (\overline { u_1^{\prime} v_{3/2}^{\prime}}), \end{align}

(3.40) \begin{align} \partial _{\eta } \overline {\pi }_2&=0. \end{align}

This reduced set of equations is unaltered by the imposed temperature difference and can be solved explicitly for $\overline {u}_2$ given the acoustic fields already derived, the expression for $\overline {\pi }_2$ in the interior and the boundary and matching conditions $\overline {u}_2(x,0) =0$ and $\mathrm{lim}_{\eta \rightarrow \infty } \partial _\eta \overline {u}_2(x,\eta ) = 0$ . The full solution is reported in Appendix B and has a non-zero limit as $\eta \rightarrow \infty$ that corresponds to an effective slip velocity $\overline {u}_{\textit{slip}}$ acting on the interior flow, which is independent of the aspect ratio $\delta$ and given by

(3.41) \begin{equation} \overline {u}_{\textit{slip}} = - \frac {3\mathbb{A}^2}{8\gamma ^2(Pe + {Re})}\left (Pe + \frac {2(\gamma -1)}{3}\sqrt {Pe{Re}} + {Re}\right )\sin (2x). \end{equation}

Since $\gamma \gt 1$ , the slip velocity is directed towards the acoustic wave velocity nodes (located at $x=p \pi$ , $p \in \mathbb{Z}$ ) (Lighthill Reference Lighthill1978). Its dimensional expression in the absence of thermal diffusivity is more commonly reported: setting $Pe\rightarrow \infty$ results in a dimensional slip velocity $\overline {u}_{{slip}_*} = - [3U_0^2/(8a_*) ] \sin (2k_* \tilde {x})$ , with $U_0 = (\mathbb{A}/\gamma ) \epsilon a_*$ the amplitude of the leading-order acoustic wave velocity in the interior.

The streaming velocity field $\overline {\boldsymbol {u}}_2$ in the interior is found by solving the time-averaged momentum equation at $O(\epsilon ^3)$ and the continuity equation at $O(\epsilon ^2)$ :

(3.42) \begin{align} -\frac {\boldsymbol{\boldsymbol{\nabla}} \overline {\pi }_3}{\gamma }+ \frac {\boldsymbol{\nabla} ^2 \overline {\boldsymbol {u}}_2}{{Re}}&= \overline {\big(\boldsymbol {u}_1^{\prime} \boldsymbol{\cdot} \boldsymbol{\boldsymbol{\nabla}} \big) \boldsymbol {u}_2^{\prime} + \big(\boldsymbol {u}_{3/2}^{\prime} \boldsymbol{\cdot} \boldsymbol{\boldsymbol{\nabla}} \big) \boldsymbol {u}_{3/2}^{\prime}+ \big(\boldsymbol {u}_2^{\prime} \boldsymbol{\cdot} \boldsymbol{\boldsymbol{\nabla}} \big) \boldsymbol {u}_1^{\prime}+ \overline {\rho }_1\big(\boldsymbol {u}_1^{\prime} \boldsymbol{\cdot} \boldsymbol{\boldsymbol{\nabla}} \big)\boldsymbol {u}_1^{\prime}}\nonumber \\&\quad \overline { +\, \rho _1^{\prime}\partial _t \boldsymbol {u}_2^{\prime} + \rho _{3/2}^{\prime} \partial _t \boldsymbol {u}_{3/2}^{\prime}+ \rho _2^{\prime} \partial _t \boldsymbol {u}_1^{\prime}}, \end{align}

(3.43) \begin{align}&\qquad\qquad\qquad\quad \boldsymbol{\boldsymbol{\nabla}} \boldsymbol{\cdot} \overline {\boldsymbol {u}}_2=0, \end{align}

where the nonlinear term $\partial _x (\overline {\rho _1^{\prime} u_1^{\prime}})$ has not been included in (3.43) because $\overline {\rho _1^{\prime} u_1^{\prime}}=0$ . Most of the nonlinear terms in (3.42) can be balanced by a pressure gradient and those driving streaming flows are readily identified by taking the curl of this equation. Since acoustic vorticity vanishes in the interior at $O(\epsilon )$ and $O(\epsilon ^{3/2})$ , substantial simplifications occur, detailed in Appendix C.1. With $\overline {\boldsymbol {u}}_2 = -\, \boldsymbol{\boldsymbol{\nabla}} \times (\overline {\psi }_2 \boldsymbol{e}_z)$ (i.e. $(\overline {u}_2, \overline {v}_2) = \delta ^{-1}(-\partial _y\overline {\psi }_2, \partial _x\overline {\psi }_2)$ ), where $\boldsymbol{e}_z\equiv \boldsymbol{e}_x\times \boldsymbol{e}_y$ , the stream function $\overline {\psi }_2$ can be obtained by solving the following equation:

(3.44) \begin{align} \frac {1}{{Re}}\,\boldsymbol{\nabla} ^4 \overline {\psi }_2\boldsymbol{e}_z &= \overline {\boldsymbol{\boldsymbol{\nabla}} \times \left ( \boldsymbol{\Omega }_{2}^{\prime} \times \boldsymbol {u}_{1}^{\prime} \right ) + (\boldsymbol{\boldsymbol{\nabla}} \overline {\rho }_1/2) \times \left [\boldsymbol{\boldsymbol{\nabla}} \left(\boldsymbol {u}_{1}^{\prime} \boldsymbol{\cdot} \boldsymbol {u}_{1}^{\prime}\right)\right ] +\boldsymbol {u}_{3/2}^{\prime} \times (\partial _t \mathbb{F}'_{3/2}\boldsymbol{e}_x)} \nonumber \\&\quad \,\overline { +\, \boldsymbol{\boldsymbol{\nabla}} \times \left [ \rho _1^{\prime}\partial _t \boldsymbol {u}_2^{\prime} + \rho _2^{\prime} \partial _t \boldsymbol {u}_1^{\prime} \right ]} . \end{align}

The $O(\epsilon ^2)$ acoustic fields are required to proceed.

3.7. Acoustic dynamics at $O(\epsilon ^{2})$

The momentum equations governing the $O(\epsilon ^{2})$ acoustic dynamics in the interior are

(3.45) \begin{align}\partial _t u_2^{\prime}+ \gamma ^{-1}\partial _x \pi _2^{\prime} & =\left(\varGamma y - \rho _1^{\prime}\right) \partial _t u_1^{\prime} + \overline {\rho _1^{\prime} \partial _t u_1^{\prime}} - u_1^{\prime} \partial _x u_1^{\prime} + \overline { u_1^{\prime} \partial _x u_1^{\prime}} \nonumber \\&\quad\,+\,(3{Re})^{-1}4\partial _{x x} u_1^{\prime} + \mathbb{F}_{2}(x,t), \end{align}

(3.46) \begin{align}\partial _t v_{2}^{\prime} +(\gamma \delta ^2)^{-1}\partial _{y} \pi _2^{\prime}&=0, \end{align}

(3.47) \begin{align}\partial _t \rho _2^{\prime} + \partial _x u_2^{\prime} + \partial _{y} v_{2}^{\prime} &=- \partial _x \left [\left (\rho _1^{\prime} - \varGamma y\right )u_1^{\prime}\right ]+ \overline {\partial _x \left (\rho _1^{\prime} u_1^{\prime}\right )}, \end{align}

(3.48) \begin{align} \partial _t \varTheta _2^{\prime}+ (\gamma -1 ) \left( \partial _x u_2^{\prime} + \partial _{y} v_{2}^{\prime}\right) &=\left (\varGamma y-\rho _1^{\prime}\right )\partial _t \varTheta _1^{\prime}+\overline {\rho _1^{\prime} \partial _t \varTheta _1^{\prime}}- u_1^{\prime} \partial _x \varTheta _1^{\prime}+\overline {u_1^{\prime} \partial _x \varTheta _1^{\prime}}\nonumber \\& \quad +\, \gamma Pe^{-1}\partial _{x x} \varTheta _1^{\prime} + (1-\gamma ) \big( \pi _1^{\prime} \partial _x u_1^{\prime}- \overline {\pi _1^{\prime} \partial _x u_1^{\prime}}\big) , \end{align}

(3.49) \begin{align}\pi _2^{\prime}-\rho _2^{\prime}-\varTheta _2^{\prime} & = \left(\varTheta _1^{\prime} + \varGamma y \right) \rho _1^{\prime} -\overline {\varTheta _1^{\prime} \rho _1^{\prime} } - \varGamma y \varTheta _1^{\prime}\,. \end{align}

Since the leading-order acoustic fields in the interior do not depend on $y$ , using (3.45) and (3.46) it can be seen that acoustic wave vorticity $\Omega '_{2} = \delta \partial _x v_2^{\prime} - \delta ^{-1} \partial _y u_2^{\prime}$ is generated according to

(3.50) \begin{equation} \partial _t \Omega _2^{\prime} = - \delta ^{-1} \partial _y \big(\varGamma y \partial _{t} u_1^{\prime}\big) = -\delta ^{-1}\varGamma \partial _t u_1^{\prime} \rightarrow \boldsymbol{\Omega }_2^{\prime} = -\frac {\varGamma u_1^{\prime}}{\delta } \boldsymbol{e}_z. \end{equation}

In contrast to corresponding results at $O(\epsilon )$ and $O(\epsilon ^{3/2})$ , at $O(\epsilon ^2)$ the wave vorticity is non-zero as a result of the imposed temperature difference $\varGamma$ . This solution can be used to evaluate certain nonlinear forcing terms already introduced, e.g.

(3.51) \begin{equation} \overline {(\boldsymbol{\boldsymbol{\nabla}} \overline {\rho }_1/2) \times \left [\boldsymbol{\boldsymbol{\nabla}} \big(\boldsymbol {u}_{1}^{\prime} \boldsymbol{\cdot} \boldsymbol {u}_{1}^{\prime}\big)\right ] + \boldsymbol{\boldsymbol{\nabla}} \times \left ( \boldsymbol{\Omega }_{2}^{\prime} \times \boldsymbol {u}_{1}^{\prime} \right ) } = - \overline {\big(\partial _x u_1^{\prime}\big) \boldsymbol{\Omega }_2^{\prime}}+\overline {\partial _x \big(u_1^{\prime} \boldsymbol{\Omega }_2^{\prime}\big)} = \overline {u_1^{\prime} \partial _x \boldsymbol{\Omega }_2^{\prime}}._{_{_{_{\!}}}} \end{equation}

Moreover, as detailed in Appendix C.2, using these results we can show that

(3.52) \begin{equation} \overline {\boldsymbol{\boldsymbol{\nabla}} \times (\rho _1^{\prime} \partial _t \boldsymbol {u}_2^{\prime}+ \rho _2^{\prime} \partial _t \boldsymbol {u}_1^{\prime}) } = \boldsymbol{0}. \end{equation}

Figure 2.

Visualisation of the strength and orientation of the streaming velocity field in the channel interior. (a) The total velocity field $\overline {\boldsymbol {u}}_2$ , (b) the homogeneous Rayleigh streaming component $\overline {\boldsymbol {u}}_{2\mathit{H}}$ , (c) the baroclinic contribution $\overline {\boldsymbol {u}}_{2\mathit{B}}$ and (d) the inhomogeneous component $\overline {\boldsymbol {u}}_{2\mathit{F}}$ resulting from the specified external body force. Components $\overline {\boldsymbol {u}}_{2\mathit{H}}$ and $\overline {\boldsymbol {u}}_{2\mathit{F}}$ have stacked multicellular structure while a single pair of cells that spans the channel interior is manifest in $\overline {\boldsymbol {u}}_{2\mathit{B}}$ . The components $\overline {\boldsymbol {u}}_{2\mathit{H}}$ , $\overline {\boldsymbol {u}}_{2\mathit{F}}$ and $\overline {\boldsymbol {u}}_{2\mathit{B}}$ possess wall-normal symmetry about the mid-plane, which is not reflected in the total velocity field $\overline {\boldsymbol {u}}_{2}$ . The parameters correspond to $A=1$ , $\textit{Re}=500$ , $Pr=0.71$ , $\delta =1$ , $\gamma =1.4$ and $\varGamma =\varGamma _{c} = 0.278$ ( $\varGamma _{c}$ is defined in § 4.2).

3.8. Steady streaming at $O(\epsilon ^{2})$ (part 2)

Now that the forcing terms involving the $O(\epsilon ^2)$ acoustic fields have been computed, the streaming flow can be evaluated. The equation for the stream function reduces to

(3.53) \begin{equation} \frac {1}{{Re}}\boldsymbol{\nabla} ^4\overline {\psi }_2 = \underbrace {\overline {u_1^{\prime} \partial _x {\Omega }_2^{\prime}}}_{B} - \underbrace {\overline {\delta \,v_{3/2}^{\prime} \partial _t \mathbb{F}_{3/2}^{\prime}}}_{F}\, ,\end{equation}

with the baroclinic and external forcing terms respectively defined as

(3.54) \begin{equation} B \equiv \overline {u_1^{\prime} \partial _x \Omega _2^{\prime}} = - \frac {\mathbb{A}^2\varGamma }{4\delta \gamma ^2}\sin {(2x)}, \quad F \equiv - \overline {\delta v_{3/2}^{\prime} \partial _t \mathbb{F}_{3/2}^{\prime}} = - \delta \left ( y- \frac {1}{2}\right ) \left\vert \mathbb{F}_{3/2} \right\vert ^2 \sin (2x). \end{equation}

This equation is supplemented with the following no-penetration and slip velocity conditions:

(3.55) \begin{align} \partial _x \overline {\psi }_2\left (x, y=0\right ) = \partial _x \overline {\psi }_2\left (x, y=1\right ) = 0 , \quad \partial _y \overline {\psi }_2\left (x, y=0\right ) = \partial _y \overline {\psi }_2\left (x, y=1\right ) = \delta \overline {u}_{\textit{slip}},_{_{_{_{\!}}}} \end{align}

with $\overline {u}_{\textit{slip}}$ reported in (3.41). An explicit solution for $\overline {\psi }_2$ can be computed; a representative velocity field is shown in figure 2(a) for a specific set of dimensionless parameters. To gain physical insight into the various contributions generating this streaming flow, it is instructive to split the stream function into three components:

(3.56) \begin{align} \overline {\psi }_2 = \overline {\psi }_{2\mathit{H}} + \overline {\psi }_{2\mathit{B}}+\overline {\psi }_{2\mathit{F}} , \end{align}

where these functions (explicitly reported in Appendix D) have the following properties.

1. (i) Component $\overline {\psi }_{2\mathit{H}}$ is the solution of the homogeneous partial differential equation $\boldsymbol{\nabla} ^4\overline {\psi }_{2\mathit{H}}=0$ with the full boundary conditions (3.55). This component corresponds to the extension to $O(1)$ aspect ratio of the solution obtained by Rayleigh for fluids of uniform (mean) temperature. As shown in figure 2(b), the velocity field consists of stacked counter-rotating cells and is symmetric with respect to $y=1/2$ . This contribution is independent of the imposed temperature difference $\varGamma$ .

2. (ii) Component $\overline {\psi }_{2\mathit{B}}$ is the solution of the inhomogeneous partial differential equation $\boldsymbol{\nabla} ^4\overline {\psi }_{2\mathit{B}}={Re}\, \mathrm{B}$ , with no-penetration and no-slip boundary conditions ( $\partial _x \overline {\psi }_{2\mathit{B}} (x, y=0 ) = \partial _x \overline {\psi }_{2\mathit{B}} (x, y=1 )=\partial _y \overline {\psi }_{2\mathit{B}} (x, y=0 ) = \partial _y \overline {\psi }_{2\mathit{B}} (x, y=1 ) = 0$ ). It is the only component $\overline {\psi }_{2\mathit{B}}$ that involves the temperature difference $\varGamma$ and therefore describes the baroclinic component of the streaming flow. This component consists of a cellular flow that spans the entire height of the channel; see figure 2(c).

3. (iii) Component $\overline {\psi }_{2\mathit{F}}$ is the solution of the inhomogeneous partial differential equation $\boldsymbol{\nabla} ^4\overline {\psi }_{2\mathit{F}}={Re}\, \mathrm{F}$ , subject to no-penetration and no-slip boundary conditions. This component of the flow depends on the details of the wave forcing mechanism, and would differ if the acoustic wave were generated by an oscillating wall rather than by an external body force. (A periodic motion of a solid boundary also generates streaming (Lighthill Reference Lighthill1978).) However, this contribution is found to be negligible for the set of dimensionless parameters considered here, as for instance evident in figure 2(d).

4.1. Comparison with previous studies

The streaming flow derived in § 3.8 is now compared with other theoretical and numerical solutions reported in the literature. In the absence of an external temperature difference and thermal diffusion, the solution converges to that derived by Rayleigh in the limit of narrow channels. Specifically, as discussed in § 3.6, the slip velocity in the absence of thermal diffusion reads $\overline {u}_{{slip}_*} = - [3 U_0^2/(8a_*) ] \sin (2k_* \tilde {x})$ . Since, in the limit $\delta \rightarrow 0$ , $\overline {\psi }_{2\mathit{H}} \rightarrow -\, \delta \, \overline {u}_{\textit{slip}}\,y\,(2y^2-3y+1)$ , the corresponding leading-order dimensional streaming velocity becomes

(4.1) \begin{align}&\quad \tilde{\overline{u}}_{H} \sim \frac {3 U_0^2}{16a_*} \sin (2 k_* \tilde {x})\left [1-3 \left (\frac {(H_*/2) - \tilde {y}}{(H_*/2)} \right )^2 \right ]\kern-2.5pt, \end{align}

(4.2) \begin{align}& \tilde{\overline{v}}_{H}\sim \frac {3 U_0^2}{16a_*} 2k_*\cos (2 k_* \tilde {x})\left [\frac {H_*}{2} - \tilde {y} - \frac {(H_*/2 - \tilde {y})^3}{(H_*/2)^2} \right]\kern-2.5pt, \end{align}

which are the same as (93)–(94) in Rayleigh (Reference Rayleigh1884) (except for the signs since Rayleigh considered an acoustic velocity field $\propto \cos (k_* \tilde {x})$ whereas it is here $\propto \sin (k_* \tilde {x})$ ).

Figure 3.

Comparison of the present analytical solution with the (a) $x$ and (b) $y$ components of the streaming velocity field (at $\tilde {x} = \pi /(4 k_*)$ and $\tilde {x} = 0$ respectively) from the DNS performed by Lin & Farouk (Reference Lin and Farouk2008) (their cases 1A and 1B). The full composite solution representing the dynamics in the BLs as well as the interior is shown by the solid curves. The parameters correspond to $A=6.37$ , $\textit{Re}=631$ , $\textit{Pe}=448$ , $\delta =0.2252$ , $\gamma =5/3$ , $\epsilon =10^{-2}$ and $\varGamma = \lbrace 0,6.67\rbrace$ . For these parameters, the critical temperature difference $\varGamma _{c}=4.45$ , i.e. $\Delta\varTheta _{*_{c}}\approx 13.4\,^{\circ }\mathrm{C}$ ( $\varGamma _{c}$ is defined in § 4.2).

Figure 4.

Comparison of the present analytical solution with the (a) $x$ and (b) $y$ components of the streaming velocity field (at $x = \pi /4$ and $x = 0$ respectively) obtained using the two-way coupled numerical algorithm of Massih et al. (Reference Massih, Mushthaq, Michel and Chini2024) that assumes $\hat {\varGamma } = O(1)$ (instead of $\hat {\varGamma } = O(\epsilon )$ for the present analysis). The parameters correspond to $A=0.01$ , $\textit{Re}=2500$ , $\textit{Pe}=1775$ , $\delta =1$ , $\gamma =1.4$ , $\epsilon =3 \times 10^{-4}$ and $\hat {\varGamma } = \epsilon \varGamma = 0.03 \gg \epsilon \varGamma _{c}$ ( $\varGamma _{c}$ , defined in § 4.2, is $0.056$ for this set of parameters).

When a finite temperature difference is imposed ( $\varGamma \neq 0$ ), the solution derived in the current work can be compared with DNS of the compressible Navier–Stokes equations performed by Lin & Farouk (Reference Lin and Farouk2008); see figure 3. To account for the finite width of the oscillatory BLs of the DNS, a composite analytical solution (Van Dyke Reference Van Dyke1969) is obtained by adding the streaming velocity profile in the BLs (§ 3.6) with that in the interior (§ 3.8). Reasonable agreement is observed, some discrepancy being expected since the DNS employs a different forcing mechanism (an oscillating wall) and incorporates variations of viscosity and thermal conductivity with temperature. In this regime, a qualitative comparison also is subsequently made with the experimental observations of Nabavi et al. (Reference Nabavi, Siddiqui and Dargahi2008).

For even larger imposed temperature differences $\hat {\varGamma } = \Delta \varTheta _*/T_*$ , we compare the solution derived in this work assuming $\hat {\varGamma } = \epsilon \varGamma$ , with $\varGamma = O(1)$ , with the numerical results obtained by Massih et al. (Reference Massih, Mushthaq, Michel and Chini2024) in the asymptotic limit $\hat {\varGamma } = O(1)$ . For this comparison, we consider a finite temperature difference of $\hat {\varGamma }=0.03$ , treated as $O(1)$ in the numerical algorithm of Massih et al. (Reference Massih, Mushthaq, Michel and Chini2024) and as $\varGamma =0.03/\epsilon$ in the current framework, all the other dimensionless parameters being identical ( $A=0.01$ , $\textit{Re}=2500$ , $\textit{Pe}=1775$ , $\delta = 1$ and $\gamma = 1.4$ ). The small parameter $\epsilon$ is arbitrarily set to $3 \times 10^{-4}$ . The wall-normal profiles of the x and y streaming velocity components in the interior of the domain are shown in figure 4. The new asymptotic model captures the numerical results, with the accuracy improving in the limit $\epsilon \rightarrow 0$ , as expected. In this strongly stratified limit, the streaming velocity field is largely dominated by the baroclinic component $\overline {\boldsymbol {u}}_{2\mathit{B}}$ , with the other components being orders of magnitude smaller for the chosen parameter values.

4.2. Critical non-dimensional temperature difference $\varGamma _{c}$

Figure 5 shows the interior streamlines of the streaming flow, with the magnitude of the streaming velocity $\vert \overline {\boldsymbol {u}}_2\vert$ shown in colour, for domains of aspect ratio $\delta = 1$ and $\delta = 10$ . The analytical solution captures the evolution of the structure and intensity of the streaming flow as the temperature difference increases. A parametric threshold can be defined to quantify the transition from the stacked multicellular Rayleigh streaming to the unicellular baroclinic acoustic streaming, i.e. from $\overline {\boldsymbol {u}}_{2H}= \boldsymbol{\boldsymbol{\nabla}} \times (\overline {\psi }_{2\mathit{H}} \boldsymbol{e}_z)$ to $\overline {\boldsymbol {u}}_{2B}= \boldsymbol{\boldsymbol{\nabla}} \times (\overline {\psi }_{2\mathit{B}} \boldsymbol{e}_z)$ . To that end, we define a kinetic energy parameter for each component of the streaming flow:

(4.3) \begin{align} &\mathit{K}_{\mathit{H}} \equiv \frac {1}{\pi } \int _{0}^{2\pi } \int _{0}^{1} \left \vert \overline {\boldsymbol {u}}_{2 \mathit{H}} \right \vert ^2{\rm d}x\,{\rm d}y = \frac {1}{\pi }\int _{0}^{2\pi }\int _{0}^{1}\left \vert \boldsymbol{\boldsymbol{\nabla}} \overline {\psi }_{2\mathit{H}} \right \vert ^2{\rm d}x{\rm d}y, \end{align}

(4.4) \begin{align} &\mathit{K}_{\mathit{B}} \equiv \frac {1}{\pi } \int _{0}^{2\pi } \int _{0}^{1} \left \vert \overline {\boldsymbol {u}}_{2 \mathit{B}} \right \vert ^2{\rm d}x\,{\rm d}y = \frac {1}{\pi }\int _{0}^{2\pi }\int _{0}^{1}\left \vert \boldsymbol{\boldsymbol{\nabla}} \overline {\psi }_{2\mathit{B}} \right \vert ^2{\rm d}x{\rm d}y, \end{align}

(4.5) \begin{align} &\mathit{K}_{\mathit{F}} \equiv \frac {1}{\pi } \int _{0}^{2\pi } \int _{0}^{1} \left \vert \overline {\boldsymbol {u}}_{2 \mathit{F}} \right \vert ^2{\rm d}x\,{\rm d}y = \frac {1}{\pi }\int _{0}^{2\pi }\int _{0}^{1}\left \vert \boldsymbol{\boldsymbol{\nabla}} \overline {\psi }_{2\mathit{F}} \right \vert ^2{\rm d}x{\rm d}y. \end{align}

Figure 5.

Streaming flow for $A=1$ , $\textit{Re}=500$ , $Pr=0.71$ , $\gamma =1.4$ , $\delta = \lbrace 1,10 \rbrace$ and (a) $\varGamma = 0$ , (b) $\varGamma =0.07$ , (c) $\varGamma = 0.28$ , (d) $\varGamma = 0.5$ , (e) $\varGamma = 1$ and (f) $\varGamma = 5$ . The streaming velocity increases with the imposed temperature difference, while the cells closer to the hot boundary expand vertically and those closer to the cold wall shrink. For this set of parameters, $\varGamma _{c}(\delta = 1)=0.28$ and $\varGamma _{c}(\delta = 10)=0.07$ . The ( $x$ ) width of the domain plotted corresponds to one half of an acoustic wave wavelength.

The analytical expressions for these coefficients are reported in Appendix E. Note that $\left \vert \overline {\boldsymbol {u}}_{2} \right \vert ^2\neq \left \vert \overline {\boldsymbol {u}}_{2 \mathit{H}} \right \vert ^2 + \left \vert \overline {\boldsymbol {u}}_{2 \mathit{B}} \right \vert ^2+\left \vert \overline {\boldsymbol {u}}_{2 \mathit{F}} \right \vert ^2$ . Since only $K_B$ depends on $\varGamma$ , a critical temperature difference $\varGamma _{c}$ can be defined such that

(4.6) \begin{equation} \mathit{K}_{\mathit{H}}\left ( \delta ,{Re},Pr,\gamma ,A\right ) = \mathit{K}_{\mathit{B}}\left (\varGamma = \varGamma _{c}, \delta ,{Re},Pr,\gamma ,A\right ). \end{equation}

For the set of parameters reported in figure 5, $\varGamma _{c}=0.28$ for $\delta = 1$ and $\varGamma _{c}=0.07$ for $\delta = 10$ , which qualitatively corresponds to a hot cell that spans around 80 % of the height of the channel. The dependence of $\varGamma _{c}$ on other independent parameters can also be investigated. Using the expressions reported in Appendix E,

(4.7) \begin{align} \mathit{K}_{\mathit{H}} &= \mathbb{A}^4\gamma ^{-4} \left [1+Pr+2(\gamma -1)\sqrt {Pr}/3 \right ]^2(1+Pr)^{-2} F_{\mathit{H}}(\delta ), \nonumber \\ \mathit{K}_{\mathit{B}} &= \mathbb{A}^4\varGamma ^{2} {Re}^2 \gamma ^{-4}F_{\mathit{B}}(\delta ), \end{align}

and therefore

(4.8) \begin{equation} \hat {\varGamma }_{c} = \epsilon \,\varGamma _{c} = \frac {1}{{Re}_w}\,\left (1 + \frac {2(\gamma -1)\sqrt {Pr}}{3(1+Pr)}\right )\,F_{c}(\delta ) \end{equation}

is notably independent of the wave amplitude $\mathbb{A}$ and inversely proportional to $\textit{Re}_w = \rho _*\,a_*/(k_*\,\mu _*)$ . The terms grouped in parentheses represent the dependence of $\varGamma _{c}$ on the fluid properties. The dependence of $\varGamma _{c}$ on the channel geometry ( $\delta$ ) is non-monotonic: $F_{c}$ has a minimum for $\delta =\delta _{min}=4.74$ and satisfies

(4.9) \begin{equation} F_{c}(\delta _{min})= 26.68, \qquad F_{c}(\delta ) \underset {\delta \ll 1}{\simeq } 18\sqrt {42}\,\delta ^{-2},\qquad F_{c}(\delta ) \underset {\delta \gg 1}{\simeq } 6\sqrt {2}\,\sqrt {\delta } . \end{equation}

Figure 6.

(a) Aspect-ratio dependence of the function $F_{c}$ to which the critical temperature difference $\varGamma _{c}$ is proportional (see (4.8)). As the imposed temperature difference increases, the transition from Rayleigh to baroclinic streaming first occurs at $\delta =\delta _{min}=4.74$ . In the narrow channel limit the critical temperature difference $\varGamma _{c}=O(\delta ^{-2})$ , while in the tall channel limit the critical temperature difference $\varGamma _{c}=O(\sqrt {\delta })$ (see (4.9)). (b) Kinetic energy parameters $\mathit{K}_{\mathit{H}}$ , $\mathit{K}_{\mathit{F}}$ and $\mathit{K}_{\mathit{B}}$ as functions of the aspect ratio $\delta$ ( $A=1$ , $\varGamma =0.5$ , $\textit{Re}=300$ , $Pr=0.71$ and $\gamma =1.4$ ).

The dependence of $F_{c}$ and each of the kinetic energy parameters on the aspect ratio  $\delta$ is shown in figure 6. In particular, the kinetic energy parameter corresponding to the specific acoustic wave forcing mechanism, $\mathit{K}_{\mathit{F}}$ , is found to be several orders of magnitude smaller than the others. The most consequential result is that $\varGamma _{c}$ is minimum for $\delta _{min}=4.74$ independently of all the other dimensionless parameters and significantly increases as $\delta$ departs from this value. This knowledge should prove valuable for the design of an experimental set-up aiming to show this transition near $\delta _{min}$ . In cases where streaming flows are undesirable, our result highlights the extreme sensitivity of channels of aspect ratios near this value to external temperature differences.

To further emphasise the role of $\varGamma _{c}$ in dictating the flow morphology, we qualitatively analyse the observations of Nabavi et al. (Reference Nabavi, Siddiqui and Dargahi2008), who report experimental measurements of the streaming flow in an acoustic resonator subject to external temperature differences. For the set of dimensionless parameters considered by those authors ( $A=7$ , $\textit{Re}=1236.5$ , $\textit{Pe}=953$ , $\delta =0.7$ , $\gamma =1.4$ and $\epsilon =10^{-3}$ ), we compute $\varGamma _{c} = 0.214$ , corresponding to a dimensional critical temperature difference $\Delta \varTheta _{c_*} = 0.06\,^{\circ }\text{C}$ across the channel. Although the experimental regime (particularly $\textit{Re} = 1236.5 \approx 1/\epsilon$ and $Pe = 953 \approx 1/\epsilon$ ) seemingly falls outside the range of asymptotic validity of the current analysis, the analytical results presented here nevertheless capture the transition that was found to occur between  $\Delta \varTheta _* = 0\, ^{\circ }\text{C}$ and $\Delta \varTheta _* = 0.3\, ^{\circ }\text{C}$ . In fact, knowledge of the critical temperature difference $\varGamma _{c}$ could be used to inform future experiments aiming to precisely track the transition in flow morphology. Figure 7(a) compares the wall-normal structure of the interior horizontal streaming velocity $\overline {u}_2$ for different temperature differences $\Delta \varTheta _* = \lbrace 0,0.06,0.3\rbrace\, ^{\circ }\text{C}$ . The corresponding normalised velocity fields are also shown in figure 7(b). The fields obtained for $\Delta \varTheta _* = 0\, ^{\circ }\text{C}$ and $\Delta \varTheta _* = 0.3\, ^{\circ }\text{C}$ compare well with the experimental results reported in Nabavi et al. (Reference Nabavi, Siddiqui and Dargahi2008).

Figure 7.

Streaming flow predicted for the dimensionless parameters reported in Nabavi et al. (Reference Nabavi, Siddiqui and Dargahi2008): $A=7$ , $\textit{Re}=1236.5$ , $\textit{Pe}=953$ , $\delta =0.7$ , $\gamma =1.4$ and $\epsilon =10^{-3}$ . (a) Vertical profile of the interior horizontal streaming velocity $\overline {u}_2(x=\pi /4,\,y)$ and (b) normalised streaming velocity field $|\overline {\boldsymbol {u}}_2|/|\overline {\boldsymbol {u}}_2|_{max}$ and velocity vectors for $\varGamma \,{=}\,0$ ( $\Delta \varTheta _*=0\, ^{\circ }\text{C}$ ), $\varGamma =\varGamma _{c}=0.214$ ( $\Delta \varTheta _{c_*}=0.06\, ^{\circ }\text{C}$ ) and $\varGamma = 1$ ( $\Delta \varTheta _*=0.3\,^{\circ }\text{C}$ ). For this set of parameters, $\varGamma _{c} = 0.214$ ( $\Delta \varTheta _{c_*}=0.06\, ^{\circ }\text{C}$ ) lies in the transition range evident in figure 4 of Nabavi et al. (Reference Nabavi, Siddiqui and Dargahi2008).