2.1 Basic equations and boundary conditions

At the outset, we summarize the basic equations for compressible Newtonian fluids obeying the ideal gas law, gravity being ignored. They consist of the equations of continuity, of motion, i.e. Navier–Stokes equation and of energy, which are given, respectively, by

(2.1) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{1}{{\it\rho}}\frac{\text{D}{\it\rho}}{\text{D}t}+\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{v}=0, & \displaystyle\end{eqnarray}$$

(2.2) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\rho}\frac{\text{D}\boldsymbol{v}}{\text{D}t}=-\boldsymbol{{\rm\nabla}}p+\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\left[2{\it\mu}\left(\unicode[STIX]{x1D65A}-{\textstyle \frac{1}{3}}\unicode[STIX]{x1D644}\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{v}\right)\right]+\boldsymbol{{\rm\nabla}}({\it\mu}_{v}\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{v}), & \displaystyle\end{eqnarray}$$

(2.3) $$\begin{eqnarray}\displaystyle {\it\rho}c_{p}\frac{\text{D}T}{\text{D}t}=\frac{\text{D}p}{\text{D}t}+\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }(k\boldsymbol{{\rm\nabla}}T)+{\it\Phi}, & & \displaystyle\end{eqnarray}$$

where $\text{D}/\text{D}t$ stands for the Lagrangian derivative defined by $\partial /\partial t+\boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}$ , $t$ being the time; ${\it\rho}$ , $\boldsymbol{v}$ , $p$ and $T$ denote, respectively, density, velocity vector, pressure and absolute temperature, while ${\it\mu}$ , ${\it\mu}_{v}$ and $k$ denote, respectively, a shear viscosity, a bulk viscosity and a thermal conductivity, $c_{p}$ being the specific heat at constant pressure; $\unicode[STIX]{x1D65A}$ and $\unicode[STIX]{x1D644}$ denote, respectively, the rate of stain tensor and the unit tensor, and the components of $\unicode[STIX]{x1D65A}$ will be given later in terms of those of $\boldsymbol{v}$ ; ${\it\Phi}$ denotes the viscous dissipation function defined by

(2.4) $$\begin{eqnarray}\displaystyle {\it\Phi}=\text{tr}({\bf\sigma}\unicode[STIX]{x1D65A})=2{\it\mu}\text{tr}(\unicode[STIX]{x1D65A}\unicode[STIX]{x1D65A})+({\it\mu}_{v}-2{\it\mu}/3)[\text{tr}(\unicode[STIX]{x1D65A})]^{2}, & & \displaystyle\end{eqnarray}$$

with $\text{tr}(\unicode[STIX]{x1D65A})=\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{v}$ , where $\text{tr}(\cdots \,)$ designates taking the trace of the tensor, and the viscous stress tensor ${\bf\sigma}$ is related to $\unicode[STIX]{x1D65A}$ through

(2.5) $$\begin{eqnarray}\displaystyle {\bf\sigma}=2{\it\mu}[\unicode[STIX]{x1D65A}-\text{tr}(\unicode[STIX]{x1D65A})\unicode[STIX]{x1D644}/3]+{\it\mu}_{v}\text{tr}(\unicode[STIX]{x1D65A})\unicode[STIX]{x1D644}. & & \displaystyle\end{eqnarray}$$

The temperature dependence of ${\it\mu}$ , ${\it\mu}_{v}$ and $k$ is taken into account by a power law of the form

(2.6) $$\begin{eqnarray}\displaystyle \frac{{\it\mu}}{{\it\mu}_{0}}=\frac{{\it\mu}_{v}}{{\it\mu}_{v0}}=\frac{k}{k_{0}}=\left(\frac{T}{T_{0}}\right)^{{\it\beta}}, & & \displaystyle\end{eqnarray}$$

where quantities with a subscript 0 imply values in a reference state, and ${\it\beta}$ is a positive constant between 0.5 and 0.6 for air.

In addition to these equations, the ideal gas is stipulated by the equation of state given by

(2.7) $$\begin{eqnarray}\displaystyle p=\mathscr{R}{\it\rho}T, & & \displaystyle\end{eqnarray}$$

where $\mathscr{R}$ is the gas constant. Using this, the equation of continuity is rewritten as

(2.8) $$\begin{eqnarray}\displaystyle \frac{1}{p}\frac{\text{D}p}{\text{D}t}-\frac{1}{T}\frac{\text{D}T}{\text{D}t}+\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{v}=0. & & \displaystyle\end{eqnarray}$$

On the other hand, boundary conditions on the surface of solid wall require that

(2.9a,b ) $$\begin{eqnarray}\displaystyle \boldsymbol{v}=\mathbf{0}\quad \text{and}\quad T=T_{w}\quad \text{on wall surface}, & & \displaystyle\end{eqnarray}$$

where $T_{w}$ represents the temperature on the wall surface. It is assumed that the heat capacity of the solid wall is so large that no temperature variations take place in the solid (for the effects of heat conduction, see Sugimoto & Hyodo (Reference Sugimoto and Hyodo2012)). Hence $T_{w}$ is given and fixed. Boundary conditions at both ends of a channel and a pore are left open, assuming that they are long axially.

3.1 Dimensionless parameters and approximation

We consider a gas-filled channel of width $2H$ confined between two flat and parallel walls, as shown in figure 1. The $x$ -axis is taken along the channel, while the $y$ -axis is taken normal to it with the origin at a midpoint between the plates. This illustration depicts not only the channel but also a circular pore of radius $R$ , to be treated later.

Figure 1.

Illustration of a gas-filled channel of width $2H$ between two flat and parallel walls subjected axially to a non-uniform temperature distribution, where the $x$ -axis is taken along the channel and the $y$ -axis is taken normal to it. This illustration includes a gas-filled pore of radius $R$ if $y$ and $\pm H$ are replaced, respectively, with the radial coordinate $r$ and the radius $R$ . Here $T_{w}$ denotes the wall temperature, while $T_{e}$ and ${\it\rho}_{e}$ denote, respectively, the gas temperature and density in a quiescent state at uniform pressure $p_{0}$ ; $p^{\prime }$ , ${\it\rho}^{\prime }$ and $T^{\prime }$ denote, respectively, excess quantities in pressure, density and temperature, while $u^{\prime }$ and $v^{\prime }$ denote, respectively, axial and spanwise (radial) components of the velocity vector, $s$ and $q$ designating, respectively, the shear stress acting on the gas in contact with the wall surface and the heat flux flowing into it.

Supposing that the span is so narrow that the diffusion layer is thick enough to fill the channel spanwise, we develop a weakly nonlinear theory for disturbances to the quiescent state. Since gravity is neglected, the pressure in this state is uniform throughout the gas, and is denoted by $p_{0}$ . In the first place, we remark that the present theory is based on the narrow-tube approximation stipulated by $H/L\ll 1$ , where $L$ represents the typical length in the $x$ -direction. The theory takes account of quantities of order $H/L$ but neglects those of $(H/L)^{2}$ . Consider the wall temperature $T_{w}(x)$ , which varies in the $x$ direction so gradually that the following inequalities are satisfied:

(3.1) $$\begin{eqnarray}\displaystyle \frac{H^{2}}{T_{w}}\left|\frac{\text{d}^{2}T_{w}}{\text{d}x^{2}}\right|\ll \frac{H}{T_{w}}\left|\frac{\text{d}T_{w}}{\text{d}x}\right|\ll 1. & & \displaystyle\end{eqnarray}$$

Because $T_{w}|\text{d}T_{w}/\text{d}x|^{-1}$ specifies a typical axial length of the temperature gradient, the middle term is small, of order $H/L$ , while the term on the left-hand side is of order $(H/L)^{2}$ and is smaller.

When the second-order derivative of $T_{w}$ is ignored, it is shown that the gas temperature in the quiescent state, denoted by $T_{e}$ , may be regarded as being spanwise uniform and may be identified with $T_{w}$ (Sugimoto Reference Sugimoto2010). Then distinction between $T_{w}$ and $T_{e}$ is unnecessary and $T_{e}$ is used instead of $T_{w}$ hereafter. When $T_{e}$ is spanwise uniform, so also is the gas density ${\it\rho}_{e}$ because ${\it\rho}_{e}T_{e}$ must be equal to ${\it\rho}_{0}T_{0}$ by Charles’ law. Hereafter, the quantities with subscript 0 represent those at temperature $T_{0}$ in the quiescent state under pressure $p_{0}$ , while quantities with the subscript $e$ represent those in the quiescent state determined in terms of $T_{e}$ so that they are functions of $x$ .

Besides the narrow-tube approximation, we are concerned with the slow dynamical process of a small, but finite magnitude, pressure disturbance. Letting a typical angular frequency of the disturbance be ${\it\omega}$ , the process is characterized by the following three parameters:

(3.2a-c ) $$\begin{eqnarray}\displaystyle \frac{H}{\sqrt{{\it\nu}/{\it\omega}}}\equiv \frac{1}{{\it\delta}}\ll 1,\quad \frac{{\it\omega}L}{a}\equiv \frac{1}{{\it\chi}}\ll 1\quad \text{and}\quad \frac{{\rm\Delta}p}{p_{0}}\equiv {\it\varepsilon}\ll 1, & & \displaystyle\end{eqnarray}$$

where ${\it\nu}$ , $a$ and ${\rm\Delta}p$ represent, respectively, typical values of kinematic viscosity ${\it\mu}/{\it\rho}$ , adiabatic sound speed and the magnitude of the excess pressure $p^{\prime }\,(=p-p_{0})$ . In the case of the circular pore, $H$ should be read as $R$ . The first parameter ${\it\delta}$ measures the thickness of the viscous diffusion layer $\sqrt{{\it\nu}/{\it\omega}}$ relative to the half-width of the channel. There also appears the thermal diffusion layer of thickness $\sqrt{{\it\kappa}/{\it\omega}}$ , ${\it\kappa}\,(=\sqrt{k/{\it\rho}c_{p}})$ being a typical thermal diffusivity. Because the Prandtl number $Pr\,(={\it\nu}/{\it\kappa})$ is of order unity, for example $0.72$ for air, both diffusion layers may be regarded as being comparable in thickness, though the thermal one is a little thicker.

The parameter ${\it\delta}^{2}$ may also be regarded as the ratio of the typical period of slow temporal variations ${\it\omega}^{-1}$ relative to the time for the viscous diffusion to spread spanwise over $H$ , i.e.  $H^{2}/{\it\nu}$ . When the diffusion layer is thick, disturbances are propagated neither with the adiabatic sound speed nor with the isothermal one, though the isothermal process takes place. Hence both the adiabatic and isothermal sound speeds have no physical significance in this situation. When the temperature gradient is present, as was already shown in Sugimoto (Reference Sugimoto2010), disturbances are not only diffused but also propagated axially in the sense of the positive gradient. The second parameter ${\it\chi}$ measures this slow speed of propagation relative to the adiabatic sound speed. The last parameter ${\it\varepsilon}$ measures the magnitude of the disturbances. This is the parameter measuring the order of nonlinearity. Relationships of these parameters with the Reynolds number and the Mach number will be discussed later.

Because ${\it\nu}$ and $a$ depend on temperature, ${\it\delta}$ and ${\it\chi}$ may be defined based on local values of ${\it\nu}_{e}(={\it\mu}_{e}/{\it\rho}_{e})$ and $a_{e}\,[=\sqrt{{\it\gamma}p_{0}/{\it\rho}_{e}}=\sqrt{{\it\gamma}\mathscr{R}T_{e}}=\sqrt{({\it\gamma}-1)c_{p}T_{e}}]$ , where ${\it\mu}_{e}$ denotes the viscosity at the temperature $T_{e}$ at a position of $x$ and ${\it\gamma}\,(\equiv c_{p}/c_{v})$ denotes the ratio of specific heats, $c_{v}$ being the specific heat at constant volume. These local values of ${\it\delta}$ and ${\it\chi}$ may be designated by attaching the subscript $e$ . When ${\it\delta}$ and ${\it\chi}$ are defined in the reference state at $p_{0}$ and $T_{0}$ , their local values are related through

(3.3a,b ) $$\begin{eqnarray}\displaystyle \frac{{\it\delta}_{e}}{{\it\delta}}=\left(\frac{T_{e}}{T_{0}}\right)^{(1+{\it\beta})/2}\quad \text{and}\quad \frac{{\it\chi}_{e}}{{\it\chi}}=\left(\frac{T_{e}}{T_{0}}\right)^{1/2}. & & \displaystyle\end{eqnarray}$$

Although ${\it\delta}_{e}$ and ${\it\chi}_{e}$ change with position, they may be regarded as being of order ${\it\delta}$ and ${\it\chi}$ , unless $T_{e}/T_{0}$ is extremely large or small. In the following, $T_{e}/T_{0}$ is assumed to be of order unity so that no subscript $e$ is attached to ${\it\delta}$ and ${\it\chi}$ .

3.2 Lowest-order approximation

We consider the lowest-order approximation in the limits of ${\it\delta}\rightarrow \infty$ , ${\it\chi}\rightarrow \infty$ and ${\it\varepsilon}\rightarrow 0$ . In these limits, fluid inertia is negligible and the wall friction dominates to counteract the pressure gradient. On the other hand, since the solid wall has a large heat capacity, the heat conduction suppresses gas temperature fluctuations from the local wall temperature so that $T^{\prime }\,(=T-T_{e})$ vanishes. Then the viscosity and thermal conductivity at position $x$ take the values ${\it\mu}_{e}$ , ${\it\mu}_{ve}$ and $k_{e}$ at $T=T_{e}$ .

Designating a disturbance to the quiescent state by attaching a prime $^{\prime }$ to a variable, and assuming the disturbance to depend on $x$ , $y$ and $t$ , the equation of continuity (2.1) is linearised around the quiescent state as

(3.4) $$\begin{eqnarray}\displaystyle \frac{\partial }{\partial t}\left(\frac{{\it\rho}^{\prime }}{{\it\rho}_{e}}\right)-\frac{u^{\prime }}{T_{e}}\frac{\text{d}T_{e}}{\text{d}x}+\frac{\partial u^{\prime }}{\partial x}+\frac{\partial v^{\prime }}{\partial y}=0, & & \displaystyle\end{eqnarray}$$

where $\boldsymbol{v}$ is thus set to $\boldsymbol{v}^{\prime }\,[=(u^{\prime },v^{\prime })]$ and use has been made of the relation ${\it\rho}_{e}^{-1}\,\text{d}{\it\rho}_{e}/\text{d}x=-T_{e}^{-1}\,\text{d}T_{e}/\text{d}x$ , since ${\it\rho}_{e}T_{e}$ is constant in $x$ . From this equation, $v^{\prime }$ is estimated to be smaller than $u^{\prime }$ by $H/L$ .

When the temperature dependence of the viscosity is taken into account, the $x$ - and $y$ -components of the equation of motion (2.2) are given, respectively, by (A 1) and (A 2) in appendix A. Equation (A 1) is approximated by neglecting the inertia terms to be

(3.5) $$\begin{eqnarray}\displaystyle 0=-\frac{\partial p^{\prime }}{\partial x}+\frac{\partial }{\partial y}\left({\it\mu}_{e}\frac{\partial u^{\prime }}{\partial y}\right), & & \displaystyle\end{eqnarray}$$

where the inertia and the other linear terms neglected are small, of order $1/{\it\delta}^{2}$ and $(H/L)^{2}$ , respectively, in comparison with the terms retained, and ${\it\mu}$ is taken to be the value at $T=T_{e}$ . It is estimated from this that $u^{\prime }$ is comparable with $H^{2}p^{\prime }/{\it\mu}L$ . Using this, (A 2) for the $y$ -component is similarly approximated by neglecting $(H/L)^{2}$ to be

(3.6) $$\begin{eqnarray}\displaystyle 0=-\frac{\partial p^{\prime }}{\partial y}. & & \displaystyle\end{eqnarray}$$

Thus $p^{\prime }$ is uniform over the channel width so that $u^{\prime }$ is obtained from (3.5) as

(3.7) $$\begin{eqnarray}\displaystyle u^{\prime }=-\frac{1}{2{\it\mu}_{e}}\frac{\partial p^{\prime }}{\partial x}(H^{2}-y^{2}). & & \displaystyle\end{eqnarray}$$

This parabolic distribution in $y$ is simply the velocity profile due to Poiseuille flow.

When approximating the equation of energy (2.3), $T^{\prime }/T_{0}$ is assumed momentarily to be comparable with $p^{\prime }/p_{0}$ by the linearised equation of state (2.7), i.e.  $p^{\prime }/p_{0}={\it\rho}^{\prime }/{\it\rho}_{e}+T^{\prime }/T_{e}$ , though $T^{\prime }$ is anticipated to be negligible physically because the gas is always in thermally perfect contact with the wall. Then (2.3) is approximated to be

(3.8) $$\begin{eqnarray}\displaystyle 0=\frac{\partial }{\partial y}\left(k_{e}\frac{\partial T^{\prime }}{\partial y}\right). & & \displaystyle\end{eqnarray}$$

In deriving this, ${\it\rho}c_{p}\partial T/\partial t$ and $\partial p/\partial t$ are negligibly small, of order $1/{\it\delta}^{2}$ , and the other linear terms are of order $(H/L)^{2}$ relative to (3.8). The advective term $u^{\prime }\,\text{d}T_{e}/\text{d}x$ is of ${\it\chi}^{2}/{\it\delta}^{2}$ relative to $\partial T^{\prime }/\partial t$ , but its magnitude is unknown at the present stage because the relation between ${\it\delta}$ and ${\it\chi}$ is unknown. As shown later, however, this is negligible. It then follows from the boundary conditions at $x=\pm H$ that $T^{\prime }=0$ , as expected. Thus the equation of state yields the isothermal relation locally

(3.9) $$\begin{eqnarray}\displaystyle \frac{p^{\prime }}{p_{0}}=\frac{{\it\rho}^{\prime }}{{\it\rho}_{e}}. & & \displaystyle\end{eqnarray}$$

It is found from this and (3.4) that $u^{\prime }$ is of order ${\it\varepsilon}{\it\omega}L$ . Because $u^{\prime }$ is stipulated by (3.5) to be of order $H^{2}p^{\prime }/{\it\mu}L$ and $p^{\prime }/p_{0}$ is regarded as being of order ${\it\varepsilon}$ , it follows that

(3.10) $$\begin{eqnarray}\displaystyle \frac{p_{0}H^{2}}{{\it\mu}{\it\omega}L^{2}}\sim \frac{a^{2}H^{2}}{{\it\gamma}{\it\nu}{\it\omega}L^{2}}\sim \frac{{\it\chi}^{2}}{{\it\delta}^{2}}\sim O(1), & & \displaystyle\end{eqnarray}$$

where ${\sim}$ implies equality in order of magnitude and the numerical factor ${\it\gamma}$ (and $Pr$ later) are regarded as being of order unity. Thus ${\it\chi}$ turns out to be comparable with  ${\it\delta}$ .

If (3.7) and (3.9) are substituted into (3.4), $v^{\prime }$ will be obtained. Before doing this, integration of (3.4) over the channel width with use of the boundary condition $v^{\prime }=0$ at $y=\pm H$ yields

(3.11) $$\begin{eqnarray}\displaystyle \frac{\partial \overline{{\it\rho}^{\prime }}}{\partial t}+\frac{\partial }{\partial x}({\it\rho}_{e}\overline{u^{\prime }})=0, & & \displaystyle\end{eqnarray}$$

where the overline stands for the average over the channel width defined, for example, for $u^{\prime }$ by

(3.12) $$\begin{eqnarray}\displaystyle \overline{u^{\prime }}\equiv \frac{1}{2H}\int _{-H}^{H}u^{\prime }\,\text{d}y. & & \displaystyle\end{eqnarray}$$

Using (3.7) and (3.9), $p^{\prime }$ is governed by the following diffusion wave equation:

(3.13) $$\begin{eqnarray}\displaystyle \frac{\partial p^{\prime }}{\partial t}-\frac{\partial }{\partial x}\left({\it\alpha}_{e}\frac{\partial p^{\prime }}{\partial x}\right)+\frac{{\it\alpha}_{e}}{T_{e}}\frac{\text{d}T_{e}}{\text{d}x}\frac{\partial p^{\prime }}{\partial x}=0, & & \displaystyle\end{eqnarray}$$

where ${\it\alpha}_{e}$ is the diffusivity of the gas confined in the channel and is given by

(3.14) $$\begin{eqnarray}\displaystyle {\it\alpha}_{e}=\frac{a_{e}^{2}H^{2}}{3{\it\gamma}{\it\nu}_{e}}=\frac{p_{0}H^{2}}{3{\it\mu}_{e}}. & & \displaystyle\end{eqnarray}$$

It is found that because ${\it\chi}\sim {\it\delta}$ , the first term in (3.13) balances with the second term. For a moderate temperature gradient, the third term is also of the same order as the second term. If the temperature dependence of the shear viscosity is neglected, then ${\it\alpha}_{e}$ is a constant independent of $x$ . Equation (3.13) is the same as the one derived from the thermoacoustic wave equation for a thick diffusion layer (Sugimoto Reference Sugimoto2010). This equation means that $p^{\prime }$ is not only diffused by the second term but also is propagated by the third term with a velocity $({\it\alpha}_{e}/T_{e})\,\text{d}T_{e}/\text{d}x$ .

The explicit form of $v^{\prime }$ is obtained by integrating (3.4) with respect to $y$ as

(3.15) $$\begin{eqnarray}\displaystyle v^{\prime }=-\frac{\partial }{\partial t}\left(\frac{p^{\prime }}{p_{0}}\right)y+\left[\frac{\partial }{\partial x}\left(\frac{1}{{\it\mu}_{e}}\frac{\partial p^{\prime }}{\partial x}\right)-\frac{1}{{\it\mu}_{e}T_{e}}\frac{\text{d}T_{e}}{\text{d}x}\frac{\partial p^{\prime }}{\partial x}\right]\frac{(3H^{2}-y^{2})y}{6}, & & \displaystyle\end{eqnarray}$$

where (3.13) has been used. Thanks to this, it is confirmed that $v^{\prime }$ satisfies the boundary conditions $v^{\prime }=0$ at $y=\pm H$ . Furthermore $v^{\prime }$ is expressed in a compact form as

(3.16) $$\begin{eqnarray}\displaystyle v^{\prime }=\frac{1}{2p_{0}H^{2}}\frac{\partial p^{\prime }}{\partial t}(H^{2}-y^{2})y, & & \displaystyle\end{eqnarray}$$

where $v^{\prime }$ is odd in $y$ and $|v^{\prime }|$ has peaks at $y=\pm H/\sqrt{3}$ . Although $u^{\prime }$ has the same profile as that of Poiseuille flow of incompressible fluids, note that $v^{\prime }$ exists due to the compressibility of the gas. It is verified by (3.15) that $v^{\prime }$ is of order ${\it\varepsilon}{\it\omega}H$ and is smaller than $u^{\prime }$ by the first order of $H/L$ . What is also to be noted is that $u^{\prime }$ is determined by the spatial pressure gradient as expected, whereas $v^{\prime }$ is determined by the temporal pressure gradient.

With all lowest-order terms available, we check the relations among them. The acoustic Mach number $Ma$ of the axial velocity relative to the local adiabatic sound speed $a_{e}$ is defined by

(3.17) $$\begin{eqnarray}\displaystyle Ma=\frac{u^{\prime }}{a_{e}}\sim \frac{{\it\varepsilon}{\it\omega}L}{a_{e}}\sim \frac{{\it\varepsilon}}{{\it\chi}}. & & \displaystyle\end{eqnarray}$$

Thus the acoustic Mach number is much smaller than ${\it\varepsilon}$ . The quantity $p^{\prime }/u^{\prime }$ defines an acoustic impedance, and this is found to be of order ${\it\chi}{\it\rho}_{e}a_{e}$ . Hence the characteristic acoustic impedance is much larger than ${\it\rho}_{e}a_{e}$ for a travelling wave in a lossless medium. Next we check the Reynolds number $Re$ . Taking a typical speed and length to be ${\it\varepsilon}{\it\omega}L$ and $H$ , respectively, it is given by

(3.18) $$\begin{eqnarray}\displaystyle Re=\frac{{\it\varepsilon}{\it\omega}LH}{{\it\nu}_{e}}=\frac{{\it\varepsilon}}{{\it\delta}^{2}}\frac{L}{H}. & & \displaystyle\end{eqnarray}$$

Hence the Reynolds number is determined by the competition among the three small parameters $1/{\it\delta}$ , ${\it\varepsilon}$ and $H/L$ .

Finally, we evaluate the components of the rate of strain tensor $\unicode[STIX]{x1D65A}$ . These are calculated straightforwardly as

(3.19a ) $$\begin{eqnarray}\displaystyle e_{xx} & \equiv & \displaystyle \displaystyle \frac{\partial u^{\prime }}{\partial x}=-\frac{3}{2p_{0}H^{2}}\frac{\partial }{\partial x}\left({\it\alpha}_{e}\frac{\partial p^{\prime }}{\partial x}\right)(H^{2}-y^{2})\sim {\it\varepsilon}{\it\omega},\end{eqnarray}$$

(3.19b ) $$\begin{eqnarray}\displaystyle e_{yy} & \equiv & \displaystyle \displaystyle \frac{\partial v^{\prime }}{\partial y}=\frac{1}{2p_{0}H^{2}}\frac{\partial p^{\prime }}{\partial t}(H^{2}-3y^{2})\sim {\it\varepsilon}{\it\omega},\end{eqnarray}$$

(3.19c ) $$\begin{eqnarray}\displaystyle e_{xy} & = & \displaystyle \displaystyle e_{yx}\equiv \frac{1}{2}\left(\frac{\partial u^{\prime }}{\partial y}+\frac{\partial v^{\prime }}{\partial x}\right)\nonumber\\ \displaystyle & = & \displaystyle \displaystyle \frac{1}{2p_{0}H^{2}}\left[3{\it\alpha}_{e}\frac{\partial p^{\prime }}{\partial x}+\frac{1}{2}\frac{\partial ^{2}p^{\prime }}{\partial x\partial t}(H^{2}-y^{2})\right]y\sim \frac{L}{H}{\it\varepsilon}{\it\omega}.\end{eqnarray}$$

$e_{xy}$

$e_{yx}$

$H/L$

(3.20) $$\begin{eqnarray}\displaystyle e_{xx}+e_{yy}=-\frac{1}{p_{0}}\frac{\partial p^{\prime }}{\partial t}-\frac{1}{2{\it\mu}_{e}T_{e}}\frac{\text{d}T_{e}}{\text{d}x}\frac{\partial p^{\prime }}{\partial x}(H^{2}-y^{2}). & & \displaystyle\end{eqnarray}$$

Since this is equivalent to the last two terms of (3.4), (3.20) is consistent with (3.4) for an isothermal process with ${\it\rho}^{\prime }/{\it\rho}_{e}=p^{\prime }/p_{0}$ .

3.3 Higher-order approximation

We now proceed to take account of the finiteness of the small parameters ${\it\delta}^{-1}$ , ${\it\chi}^{-1}$ and ${\it\varepsilon}$ . We start with the $y$ -component of the equation of motion and see how the uniformity of pressure over the channel width is affected. The equation is given by reviving the neglected terms in (A 2) in appendix A as

(3.21) $$\begin{eqnarray}\displaystyle \frac{\partial p^{\prime }}{\partial y}={\it\mu}_{e}\frac{\partial ^{2}v^{\prime }}{\partial y^{2}}+\frac{\text{d}{\it\mu}_{e}}{\text{d}x}\frac{\partial u^{\prime }}{\partial y}+\left(\frac{{\it\mu}_{e}}{3}+{\it\mu}_{ve}\right)\frac{\partial }{\partial y}\left(\frac{\partial u^{\prime }}{\partial x}+\frac{\partial v^{\prime }}{\partial y}\right), & & \displaystyle\end{eqnarray}$$

where ${\it\mu}$ and ${\it\mu}_{v}$ are expanded around $T_{e}$ as

(3.22a,b ) $$\begin{eqnarray}\displaystyle {\it\mu}={\it\mu}_{e}\left(1+\frac{{\it\beta}T^{\prime }}{T_{e}}+\cdots \,\right)\quad \text{and}\quad {\it\mu}_{v}={\it\mu}_{ve}\left(1+\frac{{\it\beta}T^{\prime }}{T_{e}}+\cdots \,\right), & & \displaystyle\end{eqnarray}$$

but since $T^{\prime }$ vanishes to the lowest order, ${\it\mu}$ and ${\it\mu}_{v}$ have been set equal to ${\it\mu}_{e}$ and ${\it\mu}_{ve}$ , respectively. The term $(\partial /\partial x)({\it\mu}\partial v/\partial x)$ in (A 2) is small, of order $(H/L)^{2}$ , relative to the first term on the right-hand side of (3.21), while the inertial term ${\it\rho}_{e}\partial v^{\prime }/\partial t$ is smaller by $1/{\it\delta}^{2}$ and the nonlinear advection terms are even smaller by ${\it\varepsilon}/{\it\delta}^{2}$ .

With the right-hand side of (3.21) evaluated using the lowest-order expressions available, it is solved for $p^{\prime }$ . Its solution, denoted by $p_{2}^{\prime }$ , is given by

(3.23) $$\begin{eqnarray}\displaystyle p_{2}^{\prime }=p^{\prime }-\frac{3{\it\mu}_{e}}{p_{0}H^{2}}\frac{\partial p^{\prime }}{\partial t}\frac{y^{2}}{2}+\left(\frac{1}{3}+{\it\beta}+\frac{{\it\mu}_{ve}}{{\it\mu}_{e}}\right)\frac{1}{T_{e}}\frac{\text{d}T_{e}}{\text{d}x}\frac{\partial p^{\prime }}{\partial x}\frac{y^{2}}{2}, & & \displaystyle\end{eqnarray}$$

where $p^{\prime }(x,t)$ , which is an arbitrary integration constant and uniform over a cross-section, corresponds to the lowest-order solution. It is found from (3.23) that the pressure becomes non-uniform spanwise and exhibits a parabolic distribution in $y$ . The second term is estimated relative to $p^{\prime }$ to be

(3.24) $$\begin{eqnarray}\displaystyle \frac{3{\it\mu}_{e}}{p_{0}H^{2}p^{\prime }}\frac{\partial p^{\prime }}{\partial t}\frac{y^{2}}{2}\sim \frac{{\it\nu}_{e}{\it\omega}}{a_{e}^{2}}\sim \frac{{\it\nu}/{\it\omega}}{H^{2}}\left(\frac{{\it\omega}L}{a}\right)^{2}\left(\frac{H}{L}\right)^{2}\sim \frac{{\it\delta}^{2}}{{\it\chi}^{2}}\left(\frac{H}{L}\right)^{2}. & & \displaystyle\end{eqnarray}$$

The third term is obviously of order $(H/L)^{2}$ . Hence, as long as the ratio $(H/L)^{2}$ is negligible, $p_{2}^{\prime }$ is regarded as being uniform over the channel. In the following analysis, we still maintain this assumption as the narrow-tube approximation that the pressure is uniform over the cross-section.

Next we look into the $x$ -component of (A 1) to revive neglected terms in (3.5). Among these, terms next to be included are given as follows:

(3.25) $$\begin{eqnarray}\displaystyle \frac{\partial }{\partial y}\left({\it\mu}_{e}\frac{\partial u^{\prime }}{\partial y}\right) & = & \displaystyle \frac{\partial p^{\prime }}{\partial x}+{\it\rho}_{e}\frac{\partial u^{\prime }}{\partial t}+{\it\rho}^{\prime }\frac{\partial u^{\prime }}{\partial t}+{\it\rho}_{e}u^{\prime }\frac{\partial u^{\prime }}{\partial x}+{\it\rho}_{e}v^{\prime }\frac{\partial u^{\prime }}{\partial y}\nonumber\\ \displaystyle & & \displaystyle -\,\frac{\partial }{\partial x}\left(\!{\it\mu}_{e}\frac{\partial u^{\prime }}{\partial x}\!\right)\!+\frac{\text{d}{\it\mu}_{e}}{\text{d}x}\frac{\partial v^{\prime }}{\partial y}-\frac{\partial }{\partial x}\left[\!\left(\frac{{\it\mu}_{e}}{3}+{\it\mu}_{ve}\right)\left(\frac{\partial u^{\prime }}{\partial x}+\frac{\partial v^{\prime }}{\partial y}\right)\!\right].\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

With the lowest-order expressions of $u^{\prime }$ and $v^{\prime }$ and with $T^{\prime }=0$ , the right-hand side is evaluated in terms of these and (3.25) is solved for $u^{\prime }$ . On the first line of the right-hand side of (3.25), the second term is estimated to be of order $1/{\it\delta}^{2}$ relative to the first pressure gradient, while the three nonlinear terms due to inertia are evaluated to be small, of order of ${\it\varepsilon}/{\it\delta}^{2}$ or ${\it\varepsilon}/{\it\chi}^{2}$ , relative to it. All terms on the second line are neglected because they are of order $(H/L)^{2}$ relative to the left-hand side. It is noted that, although the bulk viscosity is often neglected from the outset, it does not come into play even if included. When the nonlinear terms are included, whereas the second line is ignored, this means that

(3.26) $$\begin{eqnarray}\displaystyle {\it\delta}^{2}\left(\frac{H}{L}\right)^{2}\sim {\it\chi}^{2}\left(\frac{H}{L}\right)^{2}\ll {\it\varepsilon}. & & \displaystyle\end{eqnarray}$$

Since $(H/L)^{2}$ is neglected, the relation is satisfied. In actual situations this will be fulfilled because ${\it\varepsilon}$ is of $10^{-2}{-}10^{-1}$ , while $H/L$ is of $10^{-2}{-}10^{-3}$ and ${\it\delta}$ of $10{-}10^{2}$ .

Under such a condition, (3.25) is solved for $u^{\prime }$ with the boundary conditions $u^{\prime }=0$ at $y=\pm H$ . It then follows straightforwardly that

(3.27) $$\begin{eqnarray}\displaystyle u^{\prime } & = & \displaystyle -\frac{1}{{\it\mu}_{e}}\frac{\partial p^{\prime }}{\partial x}{\it\phi}_{2}+\frac{1}{{\it\rho}_{e}{\it\nu}_{e}^{2}}\frac{\partial ^{2}p^{\prime }}{\partial t\partial x}{\it\phi}_{4}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{1}{{\it\rho}_{e}{\it\nu}_{e}^{2}}\left[\frac{p^{\prime }}{p_{0}}\frac{\partial ^{2}p^{\prime }}{\partial t\partial x}{\it\phi}_{4}-\frac{\partial p^{\prime }}{\partial x}\frac{\partial }{\partial x}\left(\frac{1}{{\it\mu}_{e}}\frac{\partial p^{\prime }}{\partial x}\right){\it\phi}_{6}-\frac{1}{p_{0}H^{2}}\frac{\partial p^{\prime }}{\partial t}\frac{\partial p^{\prime }}{\partial x}{\it\phi}_{64}\right],\end{eqnarray}$$

with ${\it\phi}_{i}$ $(i=2,4,6)$ defined, respectively, by

(3.28a ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\phi}_{2}(y)={\textstyle \frac{1}{2}}(H^{2}-y^{2}), & \displaystyle\end{eqnarray}$$

(3.28b ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\phi}_{4}(y)={\textstyle \frac{1}{24}}(5H^{2}-y^{2})(H^{2}-y^{2}), & \displaystyle\end{eqnarray}$$

(3.28c ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\phi}_{6}(y)={\textstyle \frac{1}{120}}(11H^{4}-4H^{2}y^{2}+y^{4})(H^{2}-y^{2}), & \displaystyle\end{eqnarray}$$

${\it\phi}_{i}$

$\text{d}^{2}{\it\phi}_{i}/\text{d}y^{2}=-{\it\phi}_{2}^{i/2-1}$

${\it\phi}_{i}(\pm H)=0$

${\it\phi}_{64}$

$H^{2}{\it\phi}_{4}-2{\it\phi}_{6}$

(3.29) $$\begin{eqnarray}\displaystyle {\it\phi}_{64}(y)={\textstyle \frac{1}{120}}(3H^{4}+3H^{2}y^{2}-2y^{4})(H^{2}-y^{2}), & & \displaystyle\end{eqnarray}$$

which is determined by the solution of $\text{d}^{2}{\it\phi}_{64}/\text{d}y^{2}=-(H^{2}-y^{2})y^{2}/2$ with ${\it\phi}_{64}(\pm H)=0$ .

Using (3.27), we estimate the order of each term relative to the leading term. The second term on the first line is due to inertia and is of order $1/{\it\delta}^{2}$ . This reflects the finite effect of the span length, The terms on the second line are due to nonlinear advection and small, of ${\it\varepsilon}/{\it\delta}^{2}$ or ${\it\varepsilon}{\it\chi}^{2}/{\it\delta}^{4}$ . Thus it is found that the nonlinear effect remains secondary in (3.27) in comparison with the finite effect of span length. From the standpoint of the asymptotic expansion, this implies that the expressions of the nonlinear terms may be inaccurate. If the lowest-order approximation is regarded as the zeroth-order approximation, the second term in (3.27) belongs to the first-order one but the nonlinear terms belong to the second-order one. To validate them, therefore, we must proceed to the second-order approximation. This will be discussed later.

Next we examine the equation of energy to specify a small temperature variation by taking higher-order terms into account. Noting that $T^{\prime }$ vanishes at the lowest order, quadratic terms neglected in (3.8) are revived as

(3.30) $$\begin{eqnarray}\displaystyle \frac{\partial }{\partial y}\left(k_{e}\frac{\partial T^{\prime }}{\partial y}\right)={\it\rho}_{e}c_{p}u^{\prime }\frac{\text{d}T_{e}}{\text{d}x}+c_{p}{\it\rho}^{\prime }u^{\prime }\frac{\text{d}T_{e}}{\text{d}x}-\frac{\partial p^{\prime }}{\partial t}-u^{\prime }\frac{\partial p^{\prime }}{\partial x}-{\it\Phi}. & & \displaystyle\end{eqnarray}$$

On the right-hand side, the leading terms in magnitude are ${\it\rho}_{e}c_{p}u^{\prime }\,\text{d}T_{e}/\text{d}x$ and $-\partial p^{\prime }/\partial t$ , while ${\it\rho}_{e}c_{p}\partial T^{\prime }/\partial t$ is neglected because it is smaller than the left-hand side by $1/{\it\delta}^{2}$ and $T^{\prime }$ vanishes to the lowest order. Also neglected are $(\text{d}/\text{d}x)(k_{e}\,\text{d}T_{e}/\text{d}x)$ and $(\partial /\partial x)(k_{e}\partial T^{\prime }/\partial x)$ , which are smaller than the terms retained by $(H/L)^{2}$ . The nonlinear terms arise from $c_{p}{\it\rho}^{\prime }u^{\prime }\,\text{d}T_{e}/\text{d}x$ due to the density change advected under the temperature gradient, the advective term of pressure $u^{\prime }\partial p^{\prime }/\partial x$ and the viscous dissipation function ${\it\Phi}$ . The latter two terms are free from the temperature gradient. The function ${\it\Phi}$ is evaluated in terms of the components (2.4) and is contributed to mainly by $e_{xy}$ and $e_{yx}$ . It is given by (3.19c ) as

(3.31) $$\begin{eqnarray}\displaystyle {\it\Phi}\approx 2{\it\mu}_{e}(e_{xy}^{2}+e_{yx}^{2})\approx {\it\mu}_{e}\left(\frac{\partial u^{\prime }}{\partial y}\right)^{2}=\frac{1}{{\it\mu}_{e}}\left(\frac{\partial p^{\prime }}{\partial x}\right)^{2}y^{2}, & & \displaystyle\end{eqnarray}$$

to the lowest order.

Substituting the lowest-order relations in (3.30) and solving for $T^{\prime }$ , it follows that

(3.32) $$\begin{eqnarray}\displaystyle T^{\prime }=\frac{1}{k_{e}}\frac{\partial p^{\prime }}{\partial t}{\it\phi}_{2}+\frac{Pr}{{\it\rho}_{e}{\it\nu}_{e}^{2}}\frac{\text{d}T_{e}}{\text{d}x}\frac{\partial p^{\prime }}{\partial x}{\it\phi}_{4}+\frac{Pr}{{\it\rho}_{e}{\it\nu}_{e}^{2}}\frac{\text{d}T_{e}}{\text{d}x}\frac{p^{\prime }}{p_{0}}\frac{\partial p^{\prime }}{\partial x}{\it\phi}_{4}-\frac{1}{2k_{e}{\it\mu}_{e}}\left(\frac{\partial p^{\prime }}{\partial x}\right)^{2}{\it\phi}_{2}^{2}, & & \displaystyle\end{eqnarray}$$

where ${\it\rho}_{e}c_{p}/k_{e}{\it\mu}_{e}=Pr/{\it\rho}_{e}{\it\nu}_{e}^{2}$ with $Pr={\it\nu}_{e}/{\it\kappa}_{e}$ , ${\it\kappa}_{e}$ being $k_{e}/{\it\rho}_{e}c_{p}$ and where $Pr$ is independent of $T_{e}$ . Here we check the order of $T^{\prime }$ relative to $T_{e}$ . The first term is estimated to be

(3.33) $$\begin{eqnarray}\displaystyle \frac{1}{k_{e}T_{e}}\frac{\partial p^{\prime }}{\partial t}{\it\phi}_{2}\sim \frac{({\it\gamma}-1)Pr}{a_{e}^{2}{\it\mu}_{e}}{\it\omega}p^{\prime }H^{2}\sim \frac{({\it\gamma}-1)Pr}{{\it\gamma}{\it\delta}^{2}}\frac{p^{\prime }}{p_{0}}\sim \frac{{\it\varepsilon}}{{\it\delta}^{2}}, & & \displaystyle\end{eqnarray}$$

where $1/k_{e}T_{e}=({\it\gamma}-1)Pr/a_{e}^{2}{\it\mu}_{e}=[({\it\gamma}-1)Pr/{\it\gamma}]/{\it\nu}_{e}p_{0}$ , while the second term is estimated to be

(3.34) $$\begin{eqnarray}\displaystyle \frac{Pr}{{\it\rho}_{e}{\it\nu}_{e}^{2}T_{e}}\frac{\text{d}T_{e}}{\text{d}x}\frac{\partial p^{\prime }}{\partial x}{\it\phi}_{4}\sim \frac{Pr}{{\it\rho}_{e}{\it\nu}_{e}^{2}L}\frac{p^{\prime }}{L}H^{4}\sim \frac{{\it\varepsilon}{\it\chi}^{2}}{{\it\delta}^{4}}. & & \displaystyle\end{eqnarray}$$

For ${\it\delta}\sim {\it\chi}$ , the second term in (3.32) is comparable with the first term. In a similar fashion, both the third and fourth terms are estimated to be of order ${\it\varepsilon}^{2}{\it\chi}^{2}/{\it\delta}^{4}$ . Hence it is found that the first two terms are comparably small, of order ${\it\varepsilon}/{\it\delta}^{2}$ , relative to $T_{e}$ , which confirms that $T^{\prime }$ vanishes at the lowest order, and that the temperature shows a parabolic distribution plus a quartic one in $y$ . The third term in (3.32) results from the second term on the right-hand side of (3.30), while the last term results from the third and fourth terms. The last term in (3.32) is independent of the temperature gradient and is always negative. Because ${\it\Phi}$ is positive and $T^{\prime }$ vanishes at $y=\pm H$ , $-{\it\Phi}$ contributes in (3.30) to making $T^{\prime }$ positive, whereas because the advective term $u^{\prime }\partial p^{\prime }/\partial x$ is negative, it cancels ${\it\Phi}$ to make $T^{\prime }$ negative in the form of the last term. With the sign of the third term in (3.32) unknown yet, these nonlinear terms are of higher order than the first two terms by ${\it\varepsilon}$ . As in the case of $u^{\prime }$ , the nonlinear terms must be validated by evaluating higher-order terms.

To facilitate further reduction of equations in proceeding to the second-order approximation, the order of magnitude of all variables is summarized. On the basis of $p^{\prime }/p_{0}$ of order ${\it\varepsilon}$ , it follows that

(3.35a-c ) $$\begin{eqnarray}\displaystyle \frac{p^{\prime }}{p_{0}}\sim \frac{{\it\rho}^{\prime }}{{\it\rho}_{e}}\sim \frac{u^{\prime }}{{\it\omega}L}\sim {\it\varepsilon},\quad \frac{v^{\prime }}{{\it\omega}L}\sim \frac{u^{\prime }}{{\it\omega}L}\frac{H}{L}\sim {\it\varepsilon}\frac{H}{L}\quad \text{and}\quad \frac{T^{\prime }}{T_{e}}\sim \frac{{\it\varepsilon}}{{\it\delta}^{2}}\sim \frac{{\it\varepsilon}{\it\chi}^{2}}{{\it\delta}^{4}}. & & \displaystyle\end{eqnarray}$$

Now that $T^{\prime }$ has been specified, we reconsider (3.25) to the second order and check by using (3.27) and (3.32) (up to the first two terms) whether or not there are any other terms missing, but comparable with the terms taken. Because of the finite temperature disturbance, the temperature dependence of the viscosity should be included so that ${\it\mu}_{e}$ on the left-hand side of (3.25) should be modified. From (A 1) with (3.22), it is taken into account as

(3.36) $$\begin{eqnarray}\displaystyle \frac{\partial }{\partial y}\left({\it\mu}_{e}\frac{\partial u^{\prime }}{\partial y}+{\it\beta}{\it\mu}_{e}\frac{T^{\prime }}{T_{e}}\frac{\partial u^{\prime }}{\partial y}\right). & & \displaystyle\end{eqnarray}$$

The second term is comparable with the advection terms. This term is evaluated by using $u^{\prime }$ in (3.7) and the lowest-order terms of $T^{\prime }$ in (3.32). In addition, there arises a linear term of higher order emerging from the second term of (3.25) by taking $u^{\prime }$ in (3.27) up to the second term. Thus $u^{\prime }$ in (3.27) is corrected to include the second-order terms as

(3.37) $$\begin{eqnarray}\displaystyle u^{\prime } & = & \displaystyle -\frac{1}{{\it\mu}_{e}}\frac{\partial p^{\prime }}{\partial x}{\it\phi}_{2}+\frac{1}{{\it\rho}_{e}{\it\nu}_{e}^{2}}\frac{\partial ^{2}p^{\prime }}{\partial t\partial x}{\it\phi}_{4}-\frac{1}{{\it\rho}_{e}{\it\nu}_{e}^{3}}\frac{\partial ^{3}p^{\prime }}{\partial t^{2}\partial x}{\it\phi}_{60}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{1}{{\it\rho}_{e}{\it\nu}_{e}^{2}}\left[\frac{p^{\prime }}{p_{0}}\frac{\partial ^{2}p^{\prime }}{\partial t\partial x}{\it\phi}_{4}-\frac{\partial p^{\prime }}{\partial x}\frac{\partial }{\partial x}\left(\frac{1}{{\it\mu}_{e}}\frac{\partial p^{\prime }}{\partial x}\right){\it\phi}_{6}-\frac{1}{p_{0}H^{2}}\frac{\partial p^{\prime }}{\partial t}\frac{\partial p^{\prime }}{\partial x}{\it\phi}_{64}\right]\nonumber\\ \displaystyle & & \displaystyle +\,\frac{{\it\beta}}{{\it\mu}_{e}}\left[\frac{1}{k_{e}T_{e}}\frac{\partial p^{\prime }}{\partial t}\frac{\partial p^{\prime }}{\partial x}\frac{{\it\phi}_{2}^{2}}{2}+\frac{Pr}{{\it\rho}_{e}{\it\nu}_{e}^{2}}\frac{1}{T_{e}}\frac{\text{d}T_{e}}{\text{d}x}\left(\frac{\partial p^{\prime }}{\partial x}\right)^{2}{\it\phi}_{62}\right],\end{eqnarray}$$

where ${\it\phi}_{60}$ is defined by a solution to $\text{d}^{2}{\it\phi}_{60}/\text{d}y^{2}=-{\it\phi}_{4}$ with ${\it\phi}_{60}(\pm H)=0$ and given by $(5H^{2}{\it\phi}_{4}-{\it\phi}_{64})/12$ , while ${\it\phi}_{62}$ is defined through $\text{d}^{2}{\it\phi}_{62}/\text{d}y^{2}=$ $(\text{d}/\text{d}y)({\it\phi}_{4}\,\text{d}{\it\phi}_{2}/\text{d}y)$ with ${\it\phi}_{62}(\pm H)=0$ , and they are given, respectively, by

(3.38a ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\phi}_{60}(y)={\textstyle \frac{1}{720}}(61H^{4}-14H^{2}y^{2}+y^{4})(H^{2}-y^{2}), & \displaystyle\end{eqnarray}$$

(3.38b ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\phi}_{62}(y)={\textstyle \frac{1}{144}}(7H^{2}-y^{2})(H^{2}-y^{2})^{2}. & \displaystyle\end{eqnarray}$$

$u^{\prime }$

$1/{\it\delta}^{4}$

${\it\varepsilon}/{\it\delta}^{2}$

${\it\varepsilon}{\it\chi}^{2}/{\it\delta}^{4}$

Next to be checked is the equation of energy (3.30). Because the heat conductivity also changes with the temperature, $k$ is also expanded in the form of (3.22). Then there comes a term ${\it\beta}k_{e}(T^{\prime }/T_{e})\partial T^{\prime }/\partial y$ into the parentheses on the left-hand side of (3.30). Yet it is found that this term is smaller than the nonlinear terms taken and is negligible. On the right-hand side, however, ${\it\rho}_{e}c_{p}\partial T^{\prime }/\partial t$ must be revived, and $u^{\prime }$ in the first term should be taken up to the second term in (3.27). Then (3.32) is corrected as

(3.39) $$\begin{eqnarray}\displaystyle T^{\prime } & = & \displaystyle \frac{1}{k_{e}}\frac{\partial p^{\prime }}{\partial t}{\it\phi}_{2}+\frac{Pr}{{\it\rho}_{e}{\it\nu}_{e}^{2}}\frac{\text{d}T_{e}}{\text{d}x}\frac{\partial p^{\prime }}{\partial x}{\it\phi}_{4}+\frac{Pr}{{\it\rho}_{e}{\it\nu}_{e}^{2}}\frac{\text{d}T_{e}}{\text{d}x}\frac{p^{\prime }}{p_{0}}\frac{\partial p^{\prime }}{\partial x}{\it\phi}_{4}-\frac{1}{2k_{e}{\it\mu}_{e}}\left(\frac{\partial p^{\prime }}{\partial x}\right)^{2}{\it\phi}_{2}^{2}\nonumber\\ \displaystyle & & \displaystyle -\,\frac{Pr}{k_{e}{\it\nu}_{e}}\frac{\partial ^{2}p^{\prime }}{\partial t^{2}}{\it\phi}_{4}-\frac{(1+Pr)Pr}{{\it\rho}_{e}{\it\nu}_{e}^{3}}\frac{\text{d}T_{e}}{\text{d}x}\frac{\partial ^{2}p^{\prime }}{\partial t\partial x}{\it\phi}_{60}.\end{eqnarray}$$

The second line gives the second-order corrections to the two linear terms on the first line. Thus $T^{\prime }$ is specified correctly up to the order of ${\it\varepsilon}/{\it\delta}^{4}$ , ${\it\varepsilon}{\it\chi}^{2}/{\it\delta}^{6}$ and ${\it\varepsilon}^{2}{\it\chi}^{2}/{\it\delta}^{4}$ relative to $T_{e}$ .

With $u^{\prime }$ and $T^{\prime }$ specified, we finally consider the equation of continuity expressed in the form of (2.8). Noting that $p^{\prime }$ is spanwise uniform and the nonlinear advection terms of $T^{\prime }$ are smaller than $u^{\prime }\,\text{d}T_{e}/\text{d}x$ , (2.8) is approximated to include the nonlinear terms of the lowest order as

(3.40) $$\begin{eqnarray}\displaystyle \frac{1}{(p_{0}+p^{\prime })}\left(\frac{\partial p^{\prime }}{\partial t}+u^{\prime }\frac{\partial p^{\prime }}{\partial x}\right)-\frac{1}{(T_{e}+T^{\prime })}\left(\frac{\partial T^{\prime }}{\partial t}+u^{\prime }\frac{\text{d}T_{e}}{\text{d}x}\right)+\frac{\partial u^{\prime }}{\partial x}+\frac{\partial v^{\prime }}{\partial y}=0. & & \displaystyle\end{eqnarray}$$

Expanding $1/(p_{0}+p^{\prime })$ and $1/(T_{e}+T^{\prime })$ with respect to the disturbances, and taking the mean of (3.40) over the channel width, it follows that

(3.41) $$\begin{eqnarray}\displaystyle \frac{1}{p_{0}}\left(\frac{\partial p^{\prime }}{\partial t}+\overline{u^{\prime }}\frac{\partial p^{\prime }}{\partial x}\right)-\frac{p^{\prime }}{p_{0}^{2}}\frac{\partial p^{\prime }}{\partial t}-\frac{1}{T_{e}}\left(\frac{\partial \overline{T^{\prime }}}{\partial t}+\overline{u^{\prime }}\frac{\text{d}T_{e}}{\text{d}x}\right)+\frac{\partial \overline{u^{\prime }}}{\partial x}=0, & & \displaystyle\end{eqnarray}$$

to quadratic order in the disturbances. Here $(u^{\prime }/p_{0})\partial p^{\prime }/\partial x$ and $(p^{\prime }/p_{0}^{2})\partial p^{\prime }/\partial t$ are comparable in magnitude of the lowest order, whereas $(T^{\prime }/T_{e}^{2})\partial T^{\prime }/\partial t$ and $(T^{\prime }/T_{e}^{2})u^{\prime }\,\text{d}T_{e}/\text{d}x$ have been neglected because they are smaller than the lowest-order ones by $1/{\it\delta}^{2}$ and $1/{\it\delta}^{4}$ , respectively.

To close this equation for $p^{\prime }$ , $\overline{u^{\prime }}$ and $\overline{T^{\prime }}$ are necessary. They are calculated by (3.37) and (3.39) and using the averages: $\overline{{\it\phi}_{2}}=H^{2}/3$ , $\overline{{\it\phi}_{4}}=2H^{4}/15$ , $\overline{{\it\phi}_{2}^{2}}=2H^{4}/15$ , $\overline{{\it\phi}_{6}}=2H^{6}/35$ , $\overline{{\it\phi}_{64}}=2H^{6}/105$ , $\overline{{\it\phi}_{60}}=17H^{6}/315$ and $\overline{{\it\phi}_{62}}=8H^{6}/315$ as

(3.42) $$\begin{eqnarray}\displaystyle \overline{u^{\prime }} & = & \displaystyle -\frac{H^{2}}{3{\it\mu}_{e}}\frac{\partial p^{\prime }}{\partial x}+\frac{2}{15}\frac{H^{4}}{{\it\rho}_{e}{\it\nu}_{e}^{2}}\frac{\partial ^{2}p^{\prime }}{\partial t\partial x}-\frac{17}{315}\frac{H^{6}}{{\it\rho}_{e}{\it\nu}_{e}^{3}}\frac{\partial ^{3}p^{\prime }}{\partial t^{2}\partial x}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{H^{4}}{105{\it\rho}_{e}{\it\nu}_{e}^{2}p_{0}}\left[14p^{\prime }\frac{\partial ^{2}p^{\prime }}{\partial t\partial x}-18\frac{\partial p^{\prime }}{\partial x}\frac{\partial }{\partial x}\left({\it\alpha}_{e}\frac{\partial p^{\prime }}{\partial x}\right)-2\frac{\partial p^{\prime }}{\partial t}\frac{\partial p^{\prime }}{\partial x}\right]\nonumber\\ \displaystyle & & \displaystyle +\,\frac{{\it\beta}Pr\,H^{4}}{105{\it\rho}_{e}{\it\nu}_{e}^{2}p_{0}}\left[\frac{7({\it\gamma}-1)}{{\it\gamma}}\frac{\partial p^{\prime }}{\partial t}\frac{\partial p^{\prime }}{\partial x}+\frac{8{\it\alpha}_{e}}{T_{e}}\frac{\text{d}T_{e}}{\text{d}x}\left(\frac{\partial p^{\prime }}{\partial x}\right)^{2}\right],\end{eqnarray}$$

and

(3.43) $$\begin{eqnarray}\displaystyle \overline{T^{\prime }} & = & \displaystyle \frac{H^{2}}{3k_{e}}\frac{\partial p^{\prime }}{\partial t}+\frac{2}{15}\frac{PrH^{4}}{{\it\rho}_{e}{\it\nu}_{e}^{2}}\frac{\text{d}T_{e}}{\text{d}x}\frac{\partial p^{\prime }}{\partial x}+\frac{2}{15}\frac{PrH^{4}}{{\it\rho}_{e}{\it\nu}_{e}^{2}}\frac{\text{d}T_{e}}{\text{d}x}\frac{p^{\prime }}{p_{0}}\frac{\partial p^{\prime }}{\partial x}-\frac{H^{4}}{15k_{e}{\it\mu}_{e}}\left(\frac{\partial p^{\prime }}{\partial x}\right)^{2}\nonumber\\ \displaystyle & & \displaystyle -\,\frac{2}{15}\frac{PrH^{4}}{k_{e}{\it\nu}_{e}}\frac{\partial ^{2}p^{\prime }}{\partial t^{2}}-\frac{17}{315}\frac{(1+Pr)PrH^{6}}{{\it\rho}_{e}{\it\nu}_{e}^{3}}\frac{\text{d}T_{e}}{\text{d}x}\frac{\partial ^{2}p^{\prime }}{\partial t\partial x}.\end{eqnarray}$$

Substituting these into (3.41), it is found that the nonlinear terms involved in (3.42) and (3.43) make smaller contributions than the leading nonlinear terms by $1/{\it\delta}^{2}$ or ${\it\chi}^{2}/{\it\delta}^{4}$ . Although inclusion of these terms is straightforward, we retain only the nonlinear terms of the lowest order to derive the equation higher in order than (3.13) as

(3.44) $$\begin{eqnarray}\displaystyle \frac{\partial p^{\prime }}{\partial t} & - & \displaystyle \frac{\partial }{\partial x}\left({\it\alpha}_{e}\frac{\partial p^{\prime }}{\partial x}\right)+\frac{{\it\alpha}_{e}}{T_{e}}\frac{\text{d}T_{e}}{\text{d}x}\frac{\partial p^{\prime }}{\partial x}-\frac{({\it\gamma}-1)Pr{\it\alpha}_{e}}{a_{e}^{2}}\frac{\partial ^{2}p^{\prime }}{\partial t^{2}}\nonumber\\ \displaystyle & + & \displaystyle \frac{2}{5}\left[\frac{\partial }{\partial x}\left(\frac{{\it\alpha}_{e}H^{2}}{{\it\nu}_{e}}\frac{\partial ^{2}p^{\prime }}{\partial t\partial x}\right)-\frac{(1+Pr){\it\alpha}_{e}H^{2}}{{\it\nu}_{e}T_{e}}\frac{\text{d}T_{e}}{\text{d}x}\frac{\partial ^{2}p^{\prime }}{\partial t\partial x}\right]-\frac{p^{\prime }}{p_{0}}\frac{\partial p^{\prime }}{\partial t}-\frac{{\it\alpha}_{e}}{p_{0}}\left(\frac{\partial p^{\prime }}{\partial x}\right)^{2}=0.\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

This is the desired higher-order equation next to (3.13). Here it is noted that, as far as (3.44) is concerned, the higher-order expressions of (3.37) and (3.39) are unnecessary. However, it will turn out that they are required when mean values over one period of oscillations are discussed for the acoustic and thermoacoustic streaming in § 5.

As the first three terms of (3.44) represent the lowest-order equation, the fourth and fifth terms represent the finite effect of the span length of $1/{\it\delta}^{2}$ relative to the lowest-order terms, while the last two terms represent the finite (nonlinear) effect of the pressure disturbance of ${\it\varepsilon}$ or $({\it\chi}^{2}/{\it\delta}^{2}){\it\varepsilon}$ . Using the lowest-order balance, the first term in the square bracket may be replaced by

(3.45) $$\begin{eqnarray}\displaystyle \frac{\partial }{\partial x}\left(\frac{{\it\alpha}_{e}H^{2}}{{\it\nu}_{e}}\frac{\partial ^{2}p^{\prime }}{\partial t\partial x}\right)=\frac{H^{2}}{{\it\nu}_{e}}\frac{\partial ^{2}p^{\prime }}{\partial t^{2}}-\frac{{\it\beta}{\it\alpha}_{e}H^{2}}{{\it\nu}_{e}T_{e}}\frac{\text{d}T_{e}}{\text{d}x}\frac{\partial ^{2}p^{\prime }}{\partial t\partial x}, & & \displaystyle\end{eqnarray}$$

to the approximation concerned. Then (3.44) is expressed as

(3.46) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\partial p^{\prime }}{\partial t}-\frac{\partial }{\partial x}\left({\it\alpha}_{e}\frac{\partial p^{\prime }}{\partial x}\right)+\frac{{\it\alpha}_{e}}{T_{e}}\frac{\text{d}T_{e}}{\text{d}x}\frac{\partial p^{\prime }}{\partial x}+\left[\frac{6}{5}{\it\gamma}-({\it\gamma}-1)Pr\right]\frac{{\it\alpha}_{e}}{a_{e}^{2}}\frac{\partial ^{2}p^{\prime }}{\partial t^{2}}\nonumber\\ \displaystyle & & \displaystyle \quad -\,\frac{2}{5}(1+{\it\beta}+Pr)\frac{{\it\alpha}_{e}H^{2}}{{\it\nu}_{e}T_{e}}\frac{\text{d}T_{e}}{\text{d}x}\frac{\partial ^{2}p^{\prime }}{\partial t\partial x}-\frac{p^{\prime }}{p_{0}}\frac{\partial p^{\prime }}{\partial t}-\frac{{\it\alpha}_{e}}{p_{0}}\left(\frac{\partial p^{\prime }}{\partial x}\right)^{2}=0.\end{eqnarray}$$

It is noted that the nonlinear terms are independent of the temperature gradient $\text{d}T_{e}/\text{d}x$ , and also of the temperature, if ${\it\alpha}_{e}$ were taken as constant for ${\it\beta}=0$ . Hence it is revealed that the temperature and its gradient do not impact nonlinear terms in evolution of the pressure, and that the temperature dependence of the viscosity and heat conductivity appear only through ${\it\alpha}_{e}$ and the coefficient ${\it\beta}$ .

With the higher-order relations available, $v^{\prime }$ is specified to the next higher order than (3.16) by integrating (3.40) with respect to $y$ from $y=0$ . Executing the integrals of the functions in $y$ involved, i.e.  $1$ , ${\it\phi}_{2}$ and ${\it\phi}_{4}$ , it follows that

(3.47) $$\begin{eqnarray}\displaystyle v^{\prime } & = & \displaystyle -\left(1-\frac{p^{\prime }}{p_{0}}\right)\frac{\partial }{\partial t}\left(\frac{p^{\prime }}{p_{0}}\right)y\nonumber\\ \displaystyle & & \displaystyle +\,\left[\frac{\partial }{\partial x}\left(\frac{1}{{\it\mu}_{e}}\frac{\partial p^{\prime }}{\partial x}\right)-\frac{1}{{\it\mu}_{e}T_{e}}\frac{\text{d}T_{e}}{\text{d}x}\frac{\partial p^{\prime }}{\partial x}+\frac{1}{k_{e}T_{e}}\frac{\partial ^{2}p^{\prime }}{\partial t^{2}}+\frac{1}{{\it\mu}_{e}p_{0}}\left(\frac{\partial p^{\prime }}{\partial x}\right)^{2}\right]\frac{(3H^{2}-y^{2})y}{6}\nonumber\\ \displaystyle & & \displaystyle -\,\left[\frac{\partial }{\partial x}\left(\frac{1}{{\it\rho}_{e}{\it\nu}_{e}^{2}}\frac{\partial ^{2}p^{\prime }}{\partial t\partial x}\right)-\frac{(1+Pr)}{{\it\rho}_{e}{\it\nu}_{e}^{2}T_{e}}\frac{\text{d}T_{e}}{\text{d}x}\frac{\partial ^{2}p^{\prime }}{\partial t\partial x}\right]\frac{(5H^{2}-y^{2})^{2}y}{120},\end{eqnarray}$$

where $v^{\prime }$ vanishes at $y=\pm H$ owing to (3.44). Here it should be noted that for $\partial u^{\prime }/\partial x$ in (3.40), only the first two terms in (3.37) have been used because $v^{\prime }$ is smaller than $u^{\prime }$ by $H/L$ . Using (3.44), (3.47) may be expressed in an alternative form as

(3.48) $$\begin{eqnarray}\displaystyle v^{\prime } & = & \displaystyle \left(1-\frac{p^{\prime }}{p_{0}}\right)\frac{\partial }{\partial t}\left(\frac{p^{\prime }}{p_{0}}\right)\frac{{\it\phi}_{2}}{H^{2}}y\nonumber\\ \displaystyle & & \displaystyle -\,\frac{1}{{\it\nu}_{e}p_{0}}\left[\frac{\partial ^{2}p^{\prime }}{\partial t^{2}}-(1+{\it\beta}+Pr)\frac{{\it\alpha}_{e}}{T_{e}}\frac{\text{d}T_{e}}{\text{d}x}\frac{\partial ^{2}p^{\prime }}{\partial t\partial x}\right]\frac{{\it\phi}_{2}^{2}}{10H^{2}}y.\end{eqnarray}$$

We have demonstrated a way to carry the theory up to higher order in ${\it\varepsilon}$ . As was seen, the leading nonlinear terms result from $p^{-1}\text{D}p/\text{D}t$ in (2.8) so that the first term and the nonlinear terms in (3.41) may be expressed without expansion with respect to $p^{\prime }/p_{0}$ as

(3.49) $$\begin{eqnarray}\displaystyle \left(\frac{\partial }{\partial t}+\overline{u^{\prime }}\frac{\partial }{\partial x}\right)\log \left(1+\frac{p^{\prime }}{p_{0}}\right). & & \displaystyle\end{eqnarray}$$

This form may be useful if a higher-order equation than (3.46) is required. In taking the average of $\boldsymbol{v}^{\prime }\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\log (1+p^{\prime }/p_{0})$ over the channel width, $\overline{u^{\prime }}$ should be replaced by (3.42), and $p^{\prime }$ is assumed to be independent of $y$ . On top of these, nonlinear terms arise from $T^{-1}\text{D}T/\text{D}t$ written as

(3.50) $$\begin{eqnarray}\displaystyle \left(\frac{\partial }{\partial t}+\boldsymbol{v}^{\prime }\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\right)\log (T_{e}+T^{\prime }). & & \displaystyle\end{eqnarray}$$

On taking the average, (3.43) is useful but spanwise variations in $T^{\prime }$ need to be included together with $v^{\prime }$ . The calculations are expected to be complicated but higher-order theory could be developed step-by-step without any difficulties in principle.

5.1 Relations for mean values

Denoting a time period of the oscillations in $p^{\prime }$ by ${\it\tau}$ , i.e.  $p^{\prime }(x,t)\,[=p^{\prime }(x,t+{\it\tau})]$ , a mean value of $p^{\prime }$ is defined by attaching a tilde as

(5.2) $$\begin{eqnarray}\displaystyle \widetilde{p^{\prime }}\equiv \frac{1}{{\it\tau}}\int _{t}^{t+{\it\tau}}p^{\prime }(x,{\it\theta})\,\text{d}{\it\theta}. & & \displaystyle\end{eqnarray}$$

It is then straightforward to show that

(5.3a,b ) $$\begin{eqnarray}\displaystyle \widetilde{\frac{\partial p^{\prime }}{\partial x}}=\frac{\partial \widetilde{p^{\prime }}}{\partial x}\quad \text{and}\quad \widetilde{p^{\prime }\frac{\partial p^{\prime }}{\partial x}}=\frac{\partial }{\partial x}\left(\frac{\widetilde{p^{\prime 2}}}{2}\right). & & \displaystyle\end{eqnarray}$$

Taking the mean of (5.1), it follows that

(5.4) $$\begin{eqnarray}\displaystyle -\frac{\partial }{\partial x}\left({\it\alpha}_{e}\widetilde{\frac{\partial p^{\prime }}{\partial x}}\right)+\frac{{\it\alpha}_{e}}{T_{e}}\frac{\text{d}T_{e}}{\text{d}x}\widetilde{\frac{\partial p^{\prime }}{\partial x}}-\frac{{\it\alpha}_{e}}{p_{0}}\widetilde{\left(\frac{\partial p^{\prime }}{\partial x}\right)^{2}}=0, & & \displaystyle\end{eqnarray}$$

where the terms associated with differentiation with respect to $t$ drop out because of the periodicity. This relates the spatial derivatives of the mean pressure $\widetilde{p^{\prime }}$ to the mean of the pressure gradient squared. Note that the finite effect of the span length does not contribute to (5.4). Because (5.4) is written as

(5.5) $$\begin{eqnarray}\displaystyle \frac{{\it\alpha}_{e}}{T_{e}}\widetilde{\left(\frac{\partial p^{\prime }}{\partial x}\right)^{2}}=-\frac{\partial }{\partial x}\left(\frac{{\it\alpha}_{e}}{T_{e}}\widetilde{\frac{\partial p^{\prime }}{\partial x}}\right)p_{0}, & & \displaystyle\end{eqnarray}$$

the mean of the pressure gradient squared and multiplied with ${\it\alpha}_{e}/T_{e}$ on the left-hand side is expressed in terms of the spatial derivative of the mean pressure gradient multiplied with $-{\it\alpha}_{e}p_{0}/T_{e}$ . Alternatively this implies that the gradient of the mean pressure is small, of quadratic order. In fact, $\partial \widetilde{p^{\prime }}/\partial x$ is expressed as

(5.6) $$\begin{eqnarray}\displaystyle \frac{\partial \widetilde{p^{\prime }}}{\partial x}=-\frac{T_{e}}{{\it\alpha}_{e}}\int ^{x}\frac{{\it\alpha}_{e}}{p_{0}T_{e}}\widetilde{\left(\frac{\partial p^{\prime }}{\partial x}\right)^{2}}\,\text{d}x. & & \displaystyle\end{eqnarray}$$

When the nonlinear terms are neglected, the mean pressure gradient vanishes so that $\widetilde{p^{\prime }}$ is taken to be zero. However, when the nonlinear terms are taken into account, it is obvious that the mean pressure decreases with $x$ .

Next multiplying (5.1) with $p^{\prime }$ and taking the mean, it follows that

(5.7) $$\begin{eqnarray}\displaystyle -\frac{\partial }{\partial x}\left({\it\alpha}_{e}\widetilde{p^{\prime }\frac{\partial p^{\prime }}{\partial x}}\right)+{\it\alpha}_{e}\widetilde{\left(\frac{\partial p^{\prime }}{\partial x}\right)^{2}}+\frac{{\it\alpha}_{e}}{T_{e}}\frac{\text{d}T_{e}}{\text{d}x}\widetilde{p^{\prime }\frac{\partial p^{\prime }}{\partial x}}+\widetilde{\mathscr{P}}=0, & & \displaystyle\end{eqnarray}$$

where $\widetilde{\mathscr{P}}(x)$ stands for the higher-order terms given by

(5.8) $$\begin{eqnarray}\displaystyle \widetilde{\mathscr{P}} & = & \displaystyle -[n{\it\gamma}-({\it\gamma}-1)Pr]\frac{{\it\alpha}_{e}}{a_{e}^{2}}\widetilde{\left(\frac{\partial p^{\prime }}{\partial t}\right)^{2}}+m(1+{\it\beta}+Pr)\frac{{\it\alpha}_{e}H^{2}}{{\it\nu}_{e}T_{e}}\frac{\text{d}T_{e}}{\text{d}x}\widetilde{\frac{\partial p^{\prime }}{\partial t}\frac{\partial p^{\prime }}{\partial x}}\nonumber\\ \displaystyle & & \displaystyle -\,\frac{{\it\alpha}_{e}}{p_{0}}\widetilde{p^{\prime }\left(\frac{\partial p^{\prime }}{\partial x}\right)^{2}},\end{eqnarray}$$

where the tilde over the last term is effective with $p^{\prime }$ inclusive. The first three terms in (5.7) are of the same order and each term in $\widetilde{\mathscr{P}}$ is small, of the order of $1/{\it\chi}^{2}$ , $1/{\it\delta}^{2}$ and ${\it\varepsilon}$ relative to the leading terms. Elimination of the second term in (5.7) by use of (5.4) and rearrangement of the terms yield

(5.9) $$\begin{eqnarray}\displaystyle \frac{\partial }{\partial x}\left[\frac{{\it\alpha}_{e}}{T_{e}}\left(\widetilde{\frac{\partial p^{\prime }}{\partial x}}+\widetilde{\frac{p^{\prime }}{p_{0}}\frac{\partial p^{\prime }}{\partial x}}\right)\right]=\frac{\widetilde{\mathscr{P}}}{p_{0}T_{e}}. & & \displaystyle\end{eqnarray}$$

When the right-hand side is neglected, it follows that

(5.10) $$\begin{eqnarray}\displaystyle \widetilde{p^{\prime }}+\frac{\widetilde{p^{\prime 2}}}{2p_{0}}=c_{1}\int ^{x}\frac{T_{e}}{{\it\alpha}_{e}}\,\text{d}x+c_{2}, & & \displaystyle\end{eqnarray}$$

where $c_{1}$ and $c_{2}$ are arbitrary constants in $x$ . Because the right-hand side is independent of $p^{\prime }$ , and is present even in the linear case, it is reasonable to take $c_{1}$ and $c_{2}$ to vanish. Then it is found that the mean pressure $\widetilde{p^{\prime }}$ is negative and small, of order ${\it\varepsilon}^{2}p_{0}$ . The terms neglected in (5.10) (due to $\widetilde{\mathscr{P}}$ ) are even smaller, of order ${\it\varepsilon}^{2}/{\it\chi}^{2}$ , ${\it\varepsilon}^{2}/{\it\delta}^{2}$ and ${\it\varepsilon}^{3}$ relative to $p_{0}$ .

When (5.1) is multiplied with ${\it\alpha}_{e}\partial p^{\prime }/\partial x$ and the mean is taken, the mean of the product of the temporal gradient of $p^{\prime }$ and its spatial gradient is expressed in terms of the spatial derivative of the spatial gradient squared and multiplied with ${\it\alpha}_{e}^{2}/2T_{e}^{2}$ as

(5.11) $$\begin{eqnarray}\displaystyle \frac{{\it\alpha}_{e}}{T_{e}^{2}}\widetilde{\frac{\partial p^{\prime }}{\partial t}\frac{\partial p^{\prime }}{\partial x}}=\frac{\partial }{\partial x}\left[\frac{{\it\alpha}_{e}^{2}}{2T_{e}^{2}}\widetilde{\left(\frac{\partial p^{\prime }}{\partial x}\right)^{2}}\right], & & \displaystyle\end{eqnarray}$$

to the lowest order. Similarly when (5.1) is multiplied with $\partial p^{\prime }/\partial t$ and the mean is taken, it follows that

(5.12) $$\begin{eqnarray}\displaystyle \widetilde{\left(\frac{\partial p^{\prime }}{\partial t}\right)^{2}}=\frac{\partial }{\partial x}\left({\it\alpha}_{e}\widetilde{\frac{\partial p^{\prime }}{\partial t}\frac{\partial p^{\prime }}{\partial x}}\right)-\frac{{\it\alpha}_{e}}{T_{e}}\frac{\text{d}T_{e}}{\text{d}x}\widetilde{\frac{\partial p^{\prime }}{\partial t}\frac{\partial p^{\prime }}{\partial x}}. & & \displaystyle\end{eqnarray}$$

This is rewritten as

(5.13) $$\begin{eqnarray}\displaystyle \frac{1}{T_{e}^{2}}\widetilde{\left(\frac{\partial p^{\prime }}{\partial t}\right)^{2}}=\frac{1}{T_{e}}\frac{\partial }{\partial x}\left(\frac{{\it\alpha}_{e}}{T_{e}}\widetilde{\frac{\partial p^{\prime }}{\partial t}\frac{\partial p^{\prime }}{\partial x}}\right)=\frac{1}{T_{e}}\frac{\partial }{\partial x}\left\{T_{e}\frac{\partial }{\partial x}\left[\frac{{\it\alpha}_{e}^{2}}{2T_{e}^{2}}\widetilde{\left(\frac{\partial p^{\prime }}{\partial x}\right)^{2}}\right]\right\}. & & \displaystyle\end{eqnarray}$$

Here it should be remembered that the relations (5.11) to (5.13) hold to the lowest order only, whereas (5.5) is derived rigorously from (5.1). It is found from (5.11) that the mean of the product of the spatial and temporal gradients of $p^{\prime }$ is expressed, besides the factors ${\it\alpha}_{e}$ and $T_{e}$ , in terms of the spatial derivative of the mean of the pressure gradient squared, which is related to the gradient of the mean excess pressure through (5.5). Similarly, it is found from (5.13) that the mean of the temporal gradient squared is expressed in terms of the second-order derivative of the mean of the pressure gradient squared. Hence the means of the products of the spatial and/or temporal gradients of $p^{\prime }$ are related to each other through spatial differentiation and ultimately they are reduced to the spatial derivatives of the mean pressure $\widetilde{p^{\prime }}$ by (5.5).

5.2 Mean shear stress and mean heat flux on walls

Knowing the relations among the mean values, we now consider mean values of the shear stress $s$ acting on the gas and of the heat flux $q$ flowing into the gas on the walls. Noting that the shear viscosity and the heat conductivity on the wall are, respectively, ${\it\mu}_{e}$ and $k_{e}$ because $T^{\prime }$ vanishes there, $s$ and $q$ are defined for the channel by

(5.14a,b ) $$\begin{eqnarray}\displaystyle \left.s=2{\it\mu}_{e}\frac{\partial u^{\prime }}{\partial y}\right|_{y=H}\quad \text{and}\quad \left.q=2k_{e}\frac{\partial T^{\prime }}{\partial y}\right|_{y=H}, & & \displaystyle\end{eqnarray}$$

where the factor $2$ reflects the contribution from the lower wall by using even functions of $u^{\prime }$ and $T^{\prime }$ in $y$ , and defined for the pore by

(5.15) $$\begin{eqnarray}\displaystyle \left.s={\it\mu}_{e}\frac{\partial u^{\prime }}{\partial r}\right|_{r=R}\quad \text{and}\quad \left.q=k_{e}\frac{\partial T^{\prime }}{\partial r}\right|_{r=R}. & & \displaystyle\end{eqnarray}$$

Substituting into these expressions $u^{\prime }$ and $T^{\prime }$ obtained in §§ 3 and 4, $s$ and $q$ are expressed in terms of $p^{\prime }$ . For the channel, $u^{\prime }$ and $T^{\prime }$ are given, respectively, by (3.37) and (3.39), while for the pore, they are given, respectively, by (4.10) and (4.15).

By doing this and taking the mean, $\widetilde{s}$ and $\widetilde{q}$ are obtained for the respective cases as follows:

(5.16) $$\begin{eqnarray}\displaystyle \frac{\widetilde{s}}{2H}=\widetilde{\frac{\partial p^{\prime }}{\partial x}}+\frac{mH^{2}}{{\it\nu}_{e}p_{0}}\left[2\widetilde{\frac{\partial p^{\prime }}{\partial t}\frac{\partial p^{\prime }}{\partial x}}+\frac{{\it\alpha}_{e}}{T_{e}}\frac{\text{d}T_{e}}{\text{d}x}\widetilde{\left(\frac{\partial p^{\prime }}{\partial x}\right)^{2}}\right], & & \displaystyle\end{eqnarray}$$

and

(5.17) $$\begin{eqnarray}\displaystyle \frac{\widetilde{q}}{2H}=-\left(\frac{{\it\gamma}}{{\it\gamma}-1}\right)\frac{{\it\alpha}_{e}}{T_{e}}\frac{\text{d}T_{e}}{\text{d}x}\left(\widetilde{\frac{\partial p^{\prime }}{\partial x}}+\widetilde{\frac{p^{\prime }}{p_{0}}\frac{\partial p^{\prime }}{\partial x}}\right), & & \displaystyle\end{eqnarray}$$

where note that $mPrk_{e}H^{2}/n{\it\rho}_{e}{\it\nu}_{e}^{2}=[{\it\gamma}/({\it\gamma}-1)]{\it\alpha}_{e}/T_{e}$ and ${\it\gamma}/({\it\gamma}-1)=c_{p}/\mathscr{R}=c_{p}{\it\rho}_{e}T_{e}/p_{0}$ . For the case of the pore, $2H$ on the left-hand sides of (5.16) and (5.17) is replaced by $R/2$ , whereas $H$ on the right-hand side is replaced by $R$ . The quantities on the left-hand sides of (5.16) and (5.17) represent, respectively, the total shear force and the heat flux along the wetted perimeter divided by the cross-sectional area. For a channel of unit depth perpendicular to the $(x,y)$ -plane, the wetted perimeter in contact with the gas is $2$ (which has already been included in (5.14)) and the cross-sectional area is $2H$ .

It is found that the mean shear stress is contributed to not only by the gradient of the mean pressure but also by the means of the products of the spatial and/or temporal pressure gradients. However, using (5.11) and (3.7) (or (4.3) in the case of the pore), (5.16) is rewritten as

(5.18) $$\begin{eqnarray}\displaystyle \frac{\widetilde{s}}{2H}=\widetilde{\frac{\partial p^{\prime }}{\partial x}}+\frac{mH^{2}}{p_{0}}\frac{\partial }{\partial x}\left[\frac{{\it\alpha}_{e}}{{\it\nu}_{e}}\widetilde{\left(\frac{\partial p^{\prime }}{\partial x}\right)^{2}}\right]=\widetilde{\frac{\partial p^{\prime }}{\partial x}}+\frac{\partial }{\partial x}({\it\rho}_{e}\widetilde{\overline{u^{\prime 2}}}), & & \displaystyle\end{eqnarray}$$

where $\overline{u^{\prime 2}}=(m^{2}H^{4}/n{\it\mu}_{e}^{2})(\partial p^{\prime }/\partial x)^{2}$ to the lowest-order approximation. The first and last terms represent, to quadratic order, the balance of the $x$ component of the momentum flux density vector averaged over the span length. This form is rather more obvious physically than (5.16). Because of (5.10), the gradient of the mean pressure is small and comparable with $\widetilde{(p^{\prime }\partial p^{\prime }/\partial x)}/p_{0}$ , yet the remaining nonlinear terms are smaller than this term by $1/{\it\delta}^{2}$ and ${\it\chi}^{2}/{\it\delta}^{4}$ . This means that the shear stress is mainly caused by the pressure gradient and the contribution from ${\it\rho}_{e}u^{\prime 2}$ remains small.

On the other hand, (5.17) is derived directly by averaging (2.3) over the span length. The terms on the right-hand side stem from the advection under the temperature gradient, $c_{p}({\it\rho}_{e}+{\it\rho}^{\prime })u^{\prime }\,\text{d}T_{e}/\text{d}x$ , while the sum $u^{\prime }\partial p^{\prime }/\partial x+{\it\Phi}$ vanishes on averaging. The relation (5.17) shows that the mean heat flux flows only when the gradient is present. To examine this, we start with the lowest-order approximation. Because $\widetilde{p^{\prime }}$ vanishes and so does the mean pressure gradient, there is no mean heat flux. This result is reasonable because $T^{\prime }$ vanishes everywhere. Hence the mean heat flux is brought about by the finite effects of $1/{\it\delta}$ , $1/{\it\chi}$ and ${\it\varepsilon}$ in the presence of a temperature gradient.

When the higher-order terms in (5.9) are taken into account, it is integrated as

(5.19) $$\begin{eqnarray}\displaystyle \widetilde{\frac{\partial p^{\prime }}{\partial x}}+\widetilde{\frac{p^{\prime }}{p_{0}}\frac{\partial p^{\prime }}{\partial x}}=\frac{T_{e}}{{\it\alpha}_{e}}\int ^{x}\frac{\widetilde{\mathscr{P}}}{p_{0}T_{e}}\,\text{d}x. & & \displaystyle\end{eqnarray}$$

Hence $\widetilde{q}$ in (5.17) is rewritten as

(5.20) $$\begin{eqnarray}\displaystyle \frac{\widetilde{q}}{2H}=-\left(\frac{{\it\gamma}}{{\it\gamma}-1}\right)\frac{\text{d}T_{e}}{\text{d}x}\int ^{x}\frac{\widetilde{\mathscr{P}}}{p_{0}T_{e}}\,\text{d}x. & & \displaystyle\end{eqnarray}$$

The mean heat flux is thus available after integration of the right-hand side. Taking a reference heat flux to be $p_{0}{\it\omega}H$ , it is estimated from (5.20) that $\widetilde{q}$ is of order ${\it\varepsilon}^{2}/{\it\delta}^{2}$ , $({\it\chi}^{2}/{\it\delta}^{4}){\it\varepsilon}^{2}$ and $({\it\chi}^{2}/{\it\delta}^{2}){\it\varepsilon}^{3}$ relative to the reference value. No further reduction appears to be possible at this stage, but integration of $\widetilde{\mathscr{P}}/T_{e}$ is possible in special cases. This will be discussed later.

5.3 Mean mass flux

Next we consider a mean mass flux due to the periodic oscillations, which gives rise to the acoustic streaming. (Strictly speaking, the mass flux in the present context should accompany with the term ‘density vector’ as the mass flux density vector, but this will be omitted below for brevity not only for the mass flux but also for the energy flux.) The mass flux ${\it\rho}\boldsymbol{v}$ is split into

(5.21) $$\begin{eqnarray}\displaystyle {\it\rho}\boldsymbol{v}={\it\rho}_{e}\boldsymbol{v}^{\prime }+{\it\rho}^{\prime }\boldsymbol{v}^{\prime }. & & \displaystyle\end{eqnarray}$$

The excess density ${\it\rho}^{\prime }$ is expressed by using the equation of state (2.7), i.e.  $p^{\prime }/p_{0}={\it\rho}^{\prime }/{\it\rho}_{e}+T^{\prime }/T_{e}+{\it\rho}^{\prime }T^{\prime }/{\it\rho}_{e}T_{e}$ , as

(5.22) $$\begin{eqnarray}\displaystyle {\it\rho}^{\prime }\boldsymbol{v}^{\prime }={\it\rho}_{e}\left(\frac{p^{\prime }}{p_{0}}-\frac{T^{\prime }}{T_{e}}-\frac{{\it\rho}^{\prime }}{{\it\rho}_{e}}\frac{T^{\prime }}{T_{e}}\right)\boldsymbol{v}^{\prime }={\it\rho}_{e}\frac{p^{\prime }}{p_{0}}\boldsymbol{v}^{\prime }-\frac{h^{\prime }}{h_{e}}{\it\rho}\boldsymbol{v}^{\prime }, & & \displaystyle\end{eqnarray}$$

with a specific enthalpy designated by $h~(=c_{p}T)$ , where $h_{e}~(=c_{p}T_{e})$ and $h^{\prime }~(=h-h_{e}=c_{p}T^{\prime })$ denote, respectively, a local specific enthalpy in the quiescent state and an excess specific enthalpy from it. Note that since $h^{\prime }/h_{e}~(=T^{\prime }/T_{e})$ is smaller than $p^{\prime }/p_{0}$ , the first term dominates on the right-hand side of (5.22). At any rate, it is found from this that ${\it\rho}^{\prime }\boldsymbol{v}^{\prime }$ is related to the acoustic energy flux $p^{\prime }\boldsymbol{v}^{\prime }$ and the excess enthalpy flux ${\it\rho}h^{\prime }\boldsymbol{v}^{\prime }$ .

For the case of the channel, the $x$ -component of the steady mass flux is calculated by using (3.37) and (3.39) as

(5.23) $$\begin{eqnarray}\displaystyle \widetilde{{\it\rho}u^{\prime }} & = & \displaystyle -\frac{1}{{\it\nu}_{e}}\left(\widetilde{\frac{\partial p^{\prime }}{\partial x}}+\widetilde{\frac{p^{\prime }}{p_{0}}\frac{\partial p^{\prime }}{\partial x}}\right){\it\phi}_{2}-\frac{1}{{\it\nu}_{e}^{2}p_{0}}\left[\widetilde{\frac{\partial p^{\prime }}{\partial t}\frac{\partial p^{\prime }}{\partial x}}{\it\phi}_{46}+\frac{{\it\alpha}_{e}}{T_{e}}\frac{\text{d}T_{e}}{\text{d}x}\widetilde{\left(\frac{\partial p^{\prime }}{\partial x}\right)^{2}}\frac{n{\it\phi}_{6}}{mH^{2}}\right]\nonumber\\ \displaystyle & & \displaystyle +\,\frac{Pr}{{\it\nu}_{e}^{2}p_{0}}\left[\frac{({\it\gamma}-1)}{{\it\gamma}}\widetilde{\frac{\partial p^{\prime }}{\partial t}\frac{\partial p^{\prime }}{\partial x}}{\it\phi}_{2}^{2}+\frac{{\it\alpha}_{e}}{T_{e}}\frac{\text{d}T_{e}}{\text{d}x}\widetilde{\left(\frac{\partial p^{\prime }}{\partial x}\right)^{2}}\frac{n{\it\phi}_{2}{\it\phi}_{4}}{mH^{2}}\right]\nonumber\\ \displaystyle & & \displaystyle +\,\frac{{\it\beta}Pr}{{\it\nu}_{e}^{2}p_{0}}\left[\frac{({\it\gamma}-1)}{{\it\gamma}}\widetilde{\frac{\partial p^{\prime }}{\partial t}\frac{\partial p^{\prime }}{\partial x}}\frac{{\it\phi}_{2}^{2}}{2}+\frac{{\it\alpha}_{e}}{T_{e}}\frac{\text{d}T_{e}}{\text{d}x}\widetilde{\left(\frac{\partial p^{\prime }}{\partial x}\right)^{2}}\frac{n{\it\phi}_{62}}{mH^{2}}\right],\end{eqnarray}$$

with $n/m=3$ and ${\it\phi}_{46}=2{\it\phi}_{4}+[(n/m){\it\phi}_{6}+{\it\phi}_{64}]/H^{2}=(86H^{4}-19H^{2}y^{2}+y^{4})(H^{2}-y^{2})/120H^{2}$ where the lowest-order relation (3.13) has been used. Here the second and third lines with $Pr$ and ${\it\beta}Pr$ are contributed to, respectively, by the excess enthalpy flux and the temperature dependence of the viscosity. For the case of the pore with $n/m=8$ and $n=8/6$ , $H$ and ${\it\phi}$ in (5.23) are replaced by $R$ and ${\it\psi}$ , respectively, and ${\it\psi}_{46}=2{\it\psi}_{4}+(8{\it\psi}_{6}+{\it\psi}_{64})/R^{2}=(103R^{4}-41R^{2}r^{2}+4r^{4})(R^{2}-r^{2})/576R^{2}$ .

On the other hand, however, the $y$ -component is not calculated by (3.48) accurately, because $v^{\prime }$ has not been specified up to the approximation consistent with (3.37). To do this, it is necessary to derive the higher-order equation than (3.44) by taking account of the nonlinear advection terms (the second and third lines in (3.42)) in (3.41) and to obtain the expression of $v^{\prime }$ higher than (3.48) from (3.40). The procedure is the same as was used and is simple in principle but becomes very awkward. To circumvent this, it is helpful to note that the mean mass flux satisfies the divergence-free condition i.e.  $\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\widetilde{({\it\rho}\boldsymbol{v}^{\prime })}=0$ . Noting that the integral of $\widetilde{{\it\rho}u^{\prime }}$ over the span length is constant in $x$ ,

(5.24) $$\begin{eqnarray}\displaystyle \frac{\text{d}}{\text{d}x}\int _{-H}^{H}\widetilde{{\it\rho}u^{\prime }}\,\text{d}y=0, & & \displaystyle\end{eqnarray}$$

$\widetilde{{\it\rho}v^{\prime }}$ is calculated by the following integral of (5.23) as

(5.25) $$\begin{eqnarray}\displaystyle \widetilde{{\it\rho}v^{\prime }}=-\frac{\partial }{\partial x}\int _{0}^{y}\widetilde{{\it\rho}u^{\prime }}\,\text{d}y, & & \displaystyle\end{eqnarray}$$

where, since $\widetilde{{\it\rho}u^{\prime }}$ is even in $y$ , while $\widetilde{{\it\rho}v^{\prime }}$ is odd so that it vanishes at $y=0$ , (5.25) satisfies the boundary conditions $\widetilde{{\it\rho}v^{\prime }}=0$ at $y=\pm H$ owing to (5.24). For the case of the pore, $\widetilde{{\it\rho}u^{\prime }}$ and $\widetilde{{\it\rho}v^{\prime }}$ are replaced, respectively, by $r\widetilde{{\it\rho}u^{\prime }}$ and $r\widetilde{{\it\rho}v^{\prime }}$ , while $y$ and $H$ are replaced, respectively, by $r$ and $R$ . In any case, it is found that $\widetilde{{\it\rho}v^{\prime }}$ contains the spatial derivatives of the terms with the tilde in (5.23).

Here we estimate the order of the mass flux. It is noted that the sum in the first parentheses in (5.23) which is proportional to ${\it\phi}_{2}$ vanishes to the lowest order as in the case of the mean heat flux, and therefore it must be evaluated by using (5.9). Taking a reference magnitude of the mass flux to be ${\it\rho}_{e}{\it\omega}L$ , and using (5.19) with (5.8), it is found that

(5.26) $$\begin{eqnarray}\displaystyle \frac{1}{{\it\nu}_{e}}\left(\widetilde{\frac{\partial p^{\prime }}{\partial x}}+\widetilde{\frac{p^{\prime }}{p_{0}}\frac{\partial p^{\prime }}{\partial x}}\right){\it\phi}_{2}\sim {\it\rho}_{0}{\it\omega}L\left(\frac{{\it\varepsilon}^{2}}{{\it\delta}^{2}},\frac{{\it\varepsilon}^{2}{\it\chi}^{2}}{{\it\delta}^{4}},\frac{{\it\varepsilon}^{3}{\it\chi}^{2}}{{\it\delta}^{2}}\right), & & \displaystyle\end{eqnarray}$$

since ${\it\phi}_{2}\sim H^{2}$ . Next, the first terms in the three square brackets of (5.23) are estimated to be

(5.27) $$\begin{eqnarray}\displaystyle \frac{1}{{\it\nu}_{e}^{2}p_{0}}\widetilde{\frac{\partial p^{\prime }}{\partial t}\frac{\partial p^{\prime }}{\partial x}}H^{4}\sim {\it\rho}_{0}{\it\omega}L\frac{{\it\varepsilon}^{2}{\it\chi}^{2}}{{\it\delta}^{4}}, & & \displaystyle\end{eqnarray}$$

since ${\it\phi}_{46}\sim {\it\phi}_{2}^{2}\sim H^{4}$ , and the second terms are of the same order as the first terms, i.e.  ${\it\chi}^{2}/{\it\delta}^{2}$ relative to the first ones. Hence $\widetilde{{\it\rho}u^{\prime }}$ is found to be of order

(5.28) $$\begin{eqnarray}\displaystyle \frac{\widetilde{{\it\rho}u^{\prime }}}{{\it\rho}_{0}{\it\omega}L}\sim \frac{{\it\varepsilon}^{2}}{{\it\delta}^{2}}\left(1,\frac{{\it\chi}^{2}}{{\it\delta}^{2}},\frac{{\it\chi}^{4}}{{\it\delta}^{4}},{\it\varepsilon}{\it\chi}^{2}\right). & & \displaystyle\end{eqnarray}$$

For $\widetilde{{\it\rho}v^{\prime }}$ given by (5.25), on the other hand, the integration of $\widetilde{{\it\rho}u^{\prime }}$ with respect to $y$ and then differentiation with respect to $x$ lowers the order by $H/L$ . Thus $\widetilde{{\it\rho}v^{\prime }}$ is estimated to be

(5.29) $$\begin{eqnarray}\displaystyle \widetilde{{\it\rho}v^{\prime }}\sim \widetilde{{\it\rho}u^{\prime }}\frac{H}{L}. & & \displaystyle\end{eqnarray}$$

It is noted that $\widetilde{{\it\rho}v^{\prime }}$ includes the spatial derivatives of the factors with the tilde in (5.23) by differentiation with respect to $x$ .

Hence when the three parameters $1/{\it\delta}^{2}$ , $1/{\it\chi}^{2}$ and ${\it\varepsilon}$ are regarded as being comparably small, the magnitude of the $x$ -component of the mass flux is of order ${\it\varepsilon}^{3}{\it\rho}_{0}{\it\omega}L$ , while the $y$ -component is smaller than it by $H/L$ . If the mean pressure gradient is known by any means, it is straightforward to evaluate the means of the spatial and/or temporal gradients by (5.11) and (5.13). Then the vector field of the mean of the mass flux can be analysed by (5.23) and (5.25).

5.4 Mean energy flux

It has been revealed that the mean acoustic energy flux and excess enthalpy flux are engaged in the mean mass flux. Here we consider the mean energy flux. While the energy equation is given by (2.3), it is originally derived from the following conservation form:

(5.30) $$\begin{eqnarray}\displaystyle \frac{\partial \mathscr{E}}{\partial t}+\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\mathbf{\mathscr{H}}=0, & & \displaystyle\end{eqnarray}$$

where $\mathscr{E}$ stands for the total energy density ${\it\rho}(e+\boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{v}/2)$ , $e$ being a specific internal energy $c_{v}T$ and $\mathbf{\mathscr{H}}$ stands for the energy flux density vector given by

(5.31) $$\begin{eqnarray}\displaystyle \mathbf{\mathscr{H}}={\it\rho}({\textstyle \frac{1}{2}}\boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{v}+h)\boldsymbol{v}-k\boldsymbol{{\rm\nabla}}T-\boldsymbol{v}\boldsymbol{\cdot }{\bf\sigma}. & & \displaystyle\end{eqnarray}$$

Taking the mean of (5.30) over one period, it follows that $\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\widetilde{\mathbf{\mathscr{H}}}=0$ . As $\mathbf{\mathscr{H}}$ consists of four terms, we look at the magnitude of each term.

The first term ${\it\rho}(\boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{v})\boldsymbol{v}/2$ is due to transfer of the kinetic energy. This may be approximated by ${\it\rho}_{e}u^{\prime 2}\boldsymbol{v}^{\prime }/2$ because the contribution from $v^{\prime 2}/2$ is smaller by the order $H^{2}/L^{2}$ relative to $u^{\prime 2}/2$ . Taking a typical magnitude of the energy flux to be ${\it\rho}_{0}h_{0}{\it\omega}L$ and noting that $h_{0}=c_{p}T_{0}=a_{0}^{2}/({\it\gamma}-1)$ , the transfer of kinetic energy is estimated to be of order ${\it\varepsilon}^{3}{\it\chi}^{4}/{\it\delta}^{6}$ relative to the typical one. Because we are concerned with the energy flux at the same level of approximation as the mass flux relative to ${\it\rho}_{0}{\it\omega}L$ , the transfer of the kinetic energy is neglected.

The second term represents the enthalpy flux ${\it\rho}h\boldsymbol{v}^{\prime }$ , which is transported substantially with the mass flux. It may be viewed as the sum of the flux carrying the local value $h_{e}$ and the excess enthalpy $h^{\prime }$ . Thus it is broken into

(5.32) $$\begin{eqnarray}\displaystyle {\it\rho}h\boldsymbol{v}=h_{e}{\it\rho}\boldsymbol{v}^{\prime }+h^{\prime }{\it\rho}\boldsymbol{v}^{\prime }=h_{e}({\it\rho}_{e}\boldsymbol{v}^{\prime }+{\it\rho}^{\prime }\boldsymbol{v}^{\prime })+h^{\prime }{\it\rho}\boldsymbol{v}^{\prime }. & & \displaystyle\end{eqnarray}$$

Using (5.22) to eliminate ${\it\rho}^{\prime }\boldsymbol{v}^{\prime }$ , the enthalpy flux is written as

(5.33) $$\begin{eqnarray}\displaystyle {\it\rho}h\boldsymbol{v}=h_{e}{\it\rho}_{e}\boldsymbol{v}^{\prime }+h_{e}{\it\rho}_{e}\frac{p^{\prime }}{p_{0}}\boldsymbol{v}^{\prime }. & & \displaystyle\end{eqnarray}$$

Comparing this with the mass flux (5.21) (with (5.22)) multiplied with $h_{e}$ , the only difference is due to the term $h^{\prime }{\it\rho}\boldsymbol{v}^{\prime }$ .

The third term in (5.31) represents the heat conduction and the leading term is given by the spanwise component. Using (3.39) and noticing that the linear terms associated with differentiation with respect to $t$ drop out on taking the mean and the sum of the second and third terms becomes smaller than the fourth term by (5.26), it follows that

(5.34) $$\begin{eqnarray}\displaystyle -k\frac{\partial \widetilde{T^{\prime }}}{\partial y}\approx {\it\mu}_{e}\frac{\partial }{\partial y}\left(\frac{\widetilde{u^{\prime 2}}}{2}\right), & & \displaystyle\end{eqnarray}$$

where for the case of the pore, $y$ is replaced with $r$ . The term (5.34) is estimated to be of order ${\it\varepsilon}^{2}{\it\delta}^{2}H/{\it\chi}^{2}L$ relative to ${\it\rho}_{0}h_{0}{\it\omega}L$ and is found to be smaller.

The fourth term represents the flux due to the viscous power. Because the leading term is due to the $y$ - or $r$ -component, it is given for the case of the channel by

(5.35) $$\begin{eqnarray}\displaystyle -\widetilde{\boldsymbol{v}\boldsymbol{\cdot }{\bf\sigma}}|_{y}\approx -u^{\prime }{\it\sigma}_{xy}\approx -{\it\mu}_{e}\frac{\partial }{\partial y}\left(\frac{\widetilde{u^{\prime 2}}}{2}\right), & & \displaystyle\end{eqnarray}$$

and for the case of the pore, $y$ is replaced by $r$ . Hence it is found that the fourth term cancels the third term.

The mean energy flux is thus given by

(5.36) $$\begin{eqnarray}\displaystyle \widetilde{\mathbf{\mathscr{H}}}=h_{e}\left({\it\rho}_{e}\widetilde{\boldsymbol{v}^{\prime }}+\frac{{\it\rho}_{e}}{p_{0}}\widetilde{p^{\prime }\boldsymbol{v}^{\prime }}\right). & & \displaystyle\end{eqnarray}$$

Designating $\widetilde{\mathbf{\mathscr{H}}}$ by $(\widetilde{\mathscr{H}_{x}},\widetilde{\mathscr{H}_{y}})$ , the $x$ -component takes the form

(5.37) $$\begin{eqnarray}\displaystyle \widetilde{\mathscr{H}_{x}} & = & \displaystyle -\frac{h_{e}}{{\it\nu}_{e}}\left(\widetilde{\frac{\partial p^{\prime }}{\partial x}}+\widetilde{\frac{p^{\prime }}{p_{0}}\frac{\partial p^{\prime }}{\partial x}}\right){\it\phi}_{2}-\frac{h_{e}}{{\it\nu}_{e}^{2}p_{0}}\left[\widetilde{\frac{\partial p^{\prime }}{\partial t}\frac{\partial p^{\prime }}{\partial x}}{\it\phi}_{46}+\frac{{\it\alpha}_{e}}{T_{e}}\frac{\text{d}T_{e}}{\text{d}x}\widetilde{\left(\frac{\partial p^{\prime }}{\partial x}\right)^{2}}\frac{n{\it\phi}_{6}}{mH^{2}}\right]\nonumber\\ \displaystyle & & \displaystyle +\,\frac{{\it\beta}Prh_{e}}{{\it\nu}_{e}^{2}p_{0}}\left[\frac{({\it\gamma}-1)}{{\it\gamma}}\widetilde{\frac{\partial p^{\prime }}{\partial t}\frac{\partial p^{\prime }}{\partial x}}\frac{{\it\phi}_{2}^{2}}{2}+\frac{{\it\alpha}_{e}}{T_{e}}\frac{\text{d}T_{e}}{\text{d}x}\widetilde{\left(\frac{\partial p^{\prime }}{\partial x}\right)^{2}}\frac{n{\it\phi}_{62}}{mH^{2}}\right].\end{eqnarray}$$

On the other hand, the $y$ -component must also be obtained by the divergence-free condition of the mean energy flux $\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\widetilde{\mathbf{\mathscr{H}}}=0$ . The integral of $\widetilde{\mathscr{H}_{x}}$ over the span yields

(5.38) $$\begin{eqnarray}\displaystyle \frac{\text{d}}{\text{d}x}\int _{-H}^{H}\widetilde{\mathscr{H}_{x}}\,\text{d}y=\widetilde{q}, & & \displaystyle\end{eqnarray}$$

where $q$ is given by (5.14). Noting that $\widetilde{\mathscr{H}}_{x}$ is even in $y$ , while $\widetilde{\mathscr{H}_{y}}$ is odd so that it vanishes at $y=0$ , $\widetilde{\mathscr{H}_{y}}$ is obtained by

(5.39) $$\begin{eqnarray}\displaystyle \widetilde{\mathscr{H}_{y}} & = & \displaystyle -\frac{\partial }{\partial x}\int _{0}^{y}\widetilde{\mathscr{H}_{x}}\,\text{d}y.\end{eqnarray}$$

for the case of the channel, where $\widetilde{\mathscr{H}_{y}}$ takes $-\widetilde{q}/2$ at $y=H$ . For the case of the pore, $\mathscr{H}_{x}$ and $\mathscr{H}_{y}$ are replaced, respectively, by $r\mathscr{H}_{x}$ and $r\mathscr{H}_{y}$ , while $y$ and $H$ are replaced, respectively, by $r$ and $R$ . Comparing (5.37) with (5.23), the difference is the second line in (5.23), which results from $-h^{\prime }{\it\rho}\boldsymbol{v}/h_{e}$ in (5.22). As in the case of the mass flux, the vector field of the mean energy flux can be analysed by (5.37) and (5.39).

Finally, the magnitude of the mean energy flux is summarized. Referring to the typical magnitude ${\it\rho}_{0}h_{0}{\it\omega}L$ , $\widetilde{\mathscr{H}_{x}}$ is found to be of order

(5.40) $$\begin{eqnarray}\displaystyle \frac{\widetilde{\mathscr{H}_{x}}}{{\it\rho}_{0}h_{0}{\it\omega}L}\sim \frac{{\it\varepsilon}^{2}}{{\it\delta}^{2}}\left(1,\frac{{\it\chi}^{2}}{{\it\delta}^{2}},\frac{{\it\chi}^{4}}{{\it\delta}^{4}},{\it\varepsilon}{\it\chi}^{2}\right), & & \displaystyle\end{eqnarray}$$

while $\widetilde{\mathscr{H}_{y}}$ is found to be of order

(5.41) $$\begin{eqnarray}\displaystyle \widetilde{\mathscr{H}_{y}}\sim \widetilde{\mathscr{H}_{x}}\frac{H}{L}. & & \displaystyle\end{eqnarray}$$

5.5 Integration of (5.9)

It has been revealed that the mean values of the shear stress and heat flux on the wall and the mean vector fields of the mass and energy fluxes are expressed in terms of the mean values of the products of the spatial and/or temporal gradients of the excess pressure, i.e. (5.5), (5.11) and (5.13). These are expressed through (5.5) by the spatial derivatives of the mean pressure gradient. Among them, the heat flux (5.17) and the mass and energy fluxes given, respectively, by (5.23) and (5.37) involve the integration of (5.9). We consider how to evaluate them.

The right-hand side of (5.9) is of higher-order terms and may be rewritten by using the lowest-order approximation for $p^{\prime }$ . Noticing that (5.1) is expressed as

(5.42) $$\begin{eqnarray}\displaystyle \frac{{\it\alpha}_{e}}{T_{e}^{2}}\frac{\partial p^{\prime }}{\partial t}-\frac{{\it\alpha}_{e}}{T_{e}}\frac{\partial }{\partial x}\left(\frac{{\it\alpha}_{e}}{T_{e}}\frac{\partial p^{\prime }}{\partial x}\right)=0, & & \displaystyle\end{eqnarray}$$

to the lowest order, a new variable ${\it\xi}$ is introduced instead of $x$ through the following relation as

(5.43) $$\begin{eqnarray}\displaystyle \frac{\text{d}{\it\xi}}{\text{d}x}=\frac{T_{e}}{{\it\alpha}_{e}}\quad \text{i.e. }\frac{{\it\alpha}_{e}}{T_{e}}\frac{\partial p^{\prime }}{\partial x}=\frac{\partial p^{\prime }}{\partial {\it\xi}}. & & \displaystyle\end{eqnarray}$$

Note that ${\it\xi}$ has a physical dimension of $\text{m}^{-1}~\text{s}~\text{K}$ . To make ${\it\xi}$ have the dimension of length, the right-hand side might better be multiplied with a reference value ${\it\alpha}_{0}/T_{0}$ . Using ${\it\xi}$ , (5.42) is written simply as the diffusion equation in the ${\it\xi}$ -space as

(5.44) $$\begin{eqnarray}\displaystyle \frac{{\it\alpha}_{e}}{T_{e}^{2}}\frac{\partial p^{\prime }}{\partial t}-\frac{\partial ^{2}p^{\prime }}{\partial {\it\xi}^{2}}=0. & & \displaystyle\end{eqnarray}$$

Similarly (5.5) may be written as

(5.45) $$\begin{eqnarray}\displaystyle \widetilde{\left(\frac{\partial p^{\prime }}{\partial {\it\xi}}\right)^{2}}=-\frac{\partial ^{2}\widetilde{p^{\prime }}}{\partial {\it\xi}^{2}}p_{0}. & & \displaystyle\end{eqnarray}$$

The relation (5.11) is written as

(5.46) $$\begin{eqnarray}\displaystyle \frac{{\it\alpha}_{e}}{T_{e}^{2}}\widetilde{\frac{\partial p^{\prime }}{\partial t}\frac{\partial p^{\prime }}{\partial {\it\xi}}}=\frac{1}{2}\frac{\partial }{\partial {\it\xi}}\widetilde{\left(\frac{\partial p^{\prime }}{\partial {\it\xi}}\right)^{2}}, & & \displaystyle\end{eqnarray}$$

and (5.13) is written as

(5.47) $$\begin{eqnarray}\displaystyle \frac{{\it\alpha}_{e}}{T_{e}^{2}}\widetilde{\left(\frac{\partial p^{\prime }}{\partial t}\right)^{2}}=\frac{\partial }{\partial {\it\xi}}\left(\widetilde{\frac{\partial p^{\prime }}{\partial t}\frac{\partial p^{\prime }}{\partial {\it\xi}}}\right). & & \displaystyle\end{eqnarray}$$

Multiplying both sides of (5.9) with ${\it\alpha}_{e}/T_{e}$ , it follows that

(5.48) $$\begin{eqnarray}\displaystyle \frac{\partial }{\partial {\it\xi}}\left(\widetilde{\frac{\partial p^{\prime }}{\partial {\it\xi}}}+\widetilde{\frac{p^{\prime }}{p_{0}}\frac{\partial p^{\prime }}{\partial {\it\xi}}}\right)=\frac{{\it\alpha}_{e}}{p_{0}T_{e}^{2}}\widetilde{\mathscr{P}}. & & \displaystyle\end{eqnarray}$$

Setting

(5.49) $$\begin{eqnarray}\displaystyle \left(\frac{\partial p^{\prime }}{\partial {\it\xi}}\right)^{2}=G, & & \displaystyle\end{eqnarray}$$

the right-hand side of (5.48) is written as

(5.50) $$\begin{eqnarray}\displaystyle -\frac{X}{2{\it\gamma}\mathscr{R}p_{0}}\left[\frac{\partial }{\partial {\it\xi}}\left(T_{e}\frac{\partial \widetilde{G}}{\partial {\it\xi}}\right)+(1+{\it\beta})\frac{\text{d}T_{e}}{\text{d}{\it\xi}}\frac{\partial \widetilde{G}}{\partial {\it\xi}}\right]+\frac{nY}{2m\mathscr{R}p_{0}}\frac{\text{d}T_{e}}{\text{d}{\it\xi}}\frac{\partial \widetilde{G}}{\partial {\it\xi}}-\frac{1}{p_{0}^{2}}\widetilde{p^{\prime }G}, & & \displaystyle\end{eqnarray}$$

with $\text{d}T_{e}/\text{d}{\it\xi}=({\it\alpha}_{e}/T_{e})\,\text{d}T_{e}/\text{d}x$ , where $X$ and $Y$ denote, respectively, the coefficients $n{\it\gamma}-({\it\gamma}-1)Pr$ and $m(1+{\it\beta}+Pr)$ involved in (5.1), and use has been made of the relations $a_{e}^{2}={\it\gamma}\mathscr{R}T_{e}$ and $H^{2}T_{e}/{\it\alpha}_{e}{\it\nu}_{e}=n/m\mathscr{R}$ . The term $p^{\prime }G$ in (5.50) is written by using (5.44) as

(5.51) $$\begin{eqnarray}\displaystyle \frac{p^{\prime }}{p_{0}^{2}}\left(\frac{\partial p^{\prime }}{\partial {\it\xi}}\right)^{2}=\frac{1}{2p_{0}^{2}}\frac{\partial }{\partial {\it\xi}}\left(p^{\prime 2}\frac{\partial p^{\prime }}{\partial {\it\xi}}\right)-\frac{{\it\alpha}_{e}}{6T_{e}^{2}p_{0}^{2}}\frac{\partial p^{\prime 3}}{\partial t}. & & \displaystyle\end{eqnarray}$$

Taking the mean, the second term vanishes so that the last term in (5.50) is equivalent to the mean of the first term of (5.51). With the right-hand side of (5.48) specified, however, (5.48) is generally not integrated.

If we consider a special case that $\text{d}T_{e}/\text{d}{\it\xi}$ is constant, then (5.48) may be integrated once. Such a case occurs when the temperature distribution obeys

(5.52) $$\begin{eqnarray}\displaystyle \frac{T_{e}}{T_{0}}=\left(1-\frac{x}{x_{\infty }}\right)^{-1/{\it\beta}}, & & \displaystyle\end{eqnarray}$$

for a finite value of ${\it\beta}$ , with $T_{e}(0)=T_{0}$ where $x_{\infty }$ is arbitrary and $\text{d}T_{e}/\text{d}{\it\xi}={\it\alpha}_{e}(0)/{\it\beta}x_{\infty }$ . If ${\it\beta}$ is taken to vanish so that ${\it\alpha}_{e}$ is independent of $x$ , then $T_{e}$ obeys an exponential distribution as

(5.53) $$\begin{eqnarray}\displaystyle \frac{T_{e}}{T_{0}}=\exp \left(\frac{x}{l}\right), & & \displaystyle\end{eqnarray}$$

where $l$ is arbitrary and $\text{d}T_{e}/\text{d}{\it\xi}={\it\alpha}_{0}/l$ , ${\it\alpha}_{e}$ being equal to ${\it\alpha}_{0}$ . This case may be regarded as the limiting case of (5.52) as ${\it\beta}$ tends to zero with ${\it\beta}x_{\infty }$ fixed to be the constant $l$ . Both distributions appear to be plausible if $x_{\infty }$ in (5.52) were located outside of a domain concerned.

For such distributions, (5.48) is integrated by making use of the mean of (5.51) to yield

(5.54) $$\begin{eqnarray}\displaystyle \widetilde{\frac{\partial p^{\prime }}{\partial {\it\xi}}}+\widetilde{\frac{p^{\prime }}{p_{0}}\frac{\partial p^{\prime }}{\partial {\it\xi}}} & = & \displaystyle -\frac{T_{e}}{2{\it\gamma}\mathscr{R}p_{0}}\left[X\frac{\partial }{\partial {\it\xi}}\widetilde{\left(\frac{\partial p^{\prime }}{\partial {\it\xi}}\right)^{2}}+\frac{(1+{\it\beta})X-n{\it\gamma}Y/m}{T_{e}}\frac{\text{d}T_{e}}{\text{d}{\it\xi}}\widetilde{\left(\frac{\partial p^{\prime }}{\partial {\it\xi}}\right)^{2}}\right]\nonumber\\ \displaystyle & & \displaystyle -\,\frac{1}{6p_{0}^{2}}\frac{\partial \widetilde{p^{\prime 3}}}{\partial {\it\xi}}+\text{const.},\end{eqnarray}$$

where const. is an arbitrary constant to be determined. Reverting ${\it\xi}$ in (5.54) to $x$ , it follows that

(5.55) $$\begin{eqnarray}\displaystyle \widetilde{\frac{\partial p^{\prime }}{\partial x}}+\widetilde{\frac{p^{\prime }}{p_{0}}\frac{\partial p^{\prime }}{\partial x}} & = & \displaystyle -\frac{H^{2}}{{\it\nu}_{e}p_{0}}\left\{\frac{mX}{n{\it\gamma}}\widetilde{\frac{\partial p^{\prime }}{\partial t}\frac{\partial p^{\prime }}{\partial x}}+\frac{1}{2}\left[\frac{m}{n}\frac{(1+{\it\beta})}{{\it\gamma}}X-Y\right]\frac{{\it\alpha}_{e}}{T_{e}}\frac{\text{d}T_{e}}{\text{d}x}\widetilde{\left(\frac{\partial p^{\prime }}{\partial x}\right)^{2}}\right\}\nonumber\\ \displaystyle & & \displaystyle -\,\frac{1}{6p_{0}^{2}}\frac{\partial \widetilde{p^{\prime 3}}}{\partial x}+\text{const.}\times \frac{T_{e}}{{\it\alpha}_{e}},\end{eqnarray}$$

where (5.43) and (5.46) have been used. We remark that this relation is valid only when $({\it\alpha}_{e}/T_{e})\,\text{d}T_{e}/\text{d}x$ is constant, which includes the case without a temperature gradient. When (5.55) is introduced into (5.17), (5.23) and (5.37), respectively, for the heat, mass and energy fluxes, these mean values are expressed in terms of the products of the spatial and/or temporal pressure gradients. In addition, the presence of the cubic term in $p^{\prime }$ should be noted. For the mean heat flux, however, it yields a term proportional to $(HT_{e}^{-1}\,\text{d}T_{e}/\text{d}x)^{2}$ , which is negligible by the narrow-tube approximation. Because $X$ and $Y$ in (5.55) are associated with the finite effect of span length, this effect appears only through (5.55).

The constant involved in (5.55) should be determined by a boundary or a matching condition. If the channel or the pore is closed by a flat plate, then the mean of the axial velocity vanishes over the cross-section. This will determine the constant. If the channel or the pore is open, the constant should be determined by matching conditions at both ends.

For reference, the numerical values of $X$ and $Y$ are given for the case of ${\it\gamma}=1.4$ , $Pr=0.72$ and ${\it\beta}=0.5$ . Letting the factor $[m(1+{\it\beta})X/n{\it\gamma}-Y]/2$ in (5.55) be $Z$ ,  $X=1.4$ , $Y=0.89$ and $Z=-0.19$ for the case of the channel, while $X=1.6$ , $Y=0.37$ and $Z=-0.076$ for the case of the pore.