3.1.2 Summary of key equations

As discussed previously, in this section we provide a summary of the key results and refer the reader to appendix A for the full derivation. Proceeding as outlined in § 3.1.1, we substitute (3.2) into (2.16). This gives the required boundary conditions for ${\it\phi}$ at the walls in terms of $D_{\pm }$ for symmetric uniform heating:

(3.3) $$\begin{eqnarray}{\it\phi}|_{y=\pm 1/2}={\it\gamma}\left(D_{\pm }+v_{i}^{2}-{\textstyle \frac{3}{2}}\right).\end{eqnarray}$$

Enforcing no penetration at each wall yields an expression for ${\it\phi}$ ; see (A 2). Substituting (A 2) into the moment equations in (2.12) then yields coupled integral equations for three macroscopic flow quantities, ${\it\sigma}$ , $\bar{v}_{y}$ and ${\it\tau}$ , and two constants $D_{\pm }$ , in terms of the normal coordinate $y$ ; see (A 3)–(A 5), where the Abramowitz functions (Abramowitz & Stegun Reference Abramowitz and Stegun1965) are defined as

(3.4) $$\begin{eqnarray}J_{n}({\it\eta})=\int _{0}^{\infty }t^{n}\exp \left(-\frac{{\it\eta}}{t}-t^{2}\right)\text{d}t,\end{eqnarray}$$

and the sign function $\text{sgn}(y-y_{0})$ extracts the sign of the real argument $y-y_{0}$ .

We now determine expressions for the constants $D_{\pm }$ using the procedure in § 3.1.1. First, the integral equation for $\bar{v}_{y}$ is evaluated at each wall, i.e. at $y=\pm 1/2$ , and the no-penetration condition from (2.16d ) is enforced. This yields the simultaneous equations for $D_{\pm }$ that are summarised in (A 6a ) and (A 6b ). The simultaneous equations (A 6a ) and (A 6b ) are solved by using the symmetry relation from (A 7) and (A 8). This yields the required unique solution for  $D_{\pm }$ :

(3.5) $$\begin{eqnarray}D_{+}\equiv D=D_{-},\end{eqnarray}$$

where $D$ is defined as:

(3.6) $$\begin{eqnarray}\displaystyle D & = & \displaystyle \frac{2}{1-2J_{1}(a)}\left[-\frac{1}{2}\left(J_{1}(a)-\frac{1}{2}\right)+J_{3}(a)-\frac{1}{2}\right.\nonumber\\ \displaystyle & & \displaystyle +\,\frac{1}{k{\it\gamma}}\int _{-1/2}^{1/2}\left\{{\it\sigma}\left[-\frac{2\text{i}{\it\theta}}{1-\text{i}{\it\theta}}J_{2}\left(a\left[\frac{1}{2}-y_{0}\right]\right)+J_{0}\left(a\left[\frac{1}{2}-y_{0}\right]\right)\right]\right.\nonumber\\ \displaystyle & & \displaystyle +\left.\!\left.{\it\tau}\left[J_{2}\left(a\left[\frac{1}{2}-y_{0}\right]\right)-\frac{1}{2}J_{0}\left(a\left[\frac{1}{2}-y_{0}\right]\right)\right]\right\}\text{d}y_{0}\right].\end{eqnarray}$$

The required system of three integral equations for the hydrodynamic quantities ${\it\sigma}$ , $\bar{v}_{y}$ and ${\it\tau}$ are thus obtained; see (A 3)–(A 5), (3.5) and (3.6). These equations will be solved numerically to provide benchmark results in which to assess the validity of the asymptotic theories in Nassios & Sader (Reference Nassios and Sader2012, Reference Nassios and Sader2013) and Takata et al. (Reference Takata, Aoki, Hattori and Hadjiconstantinou2012) – the numerical solution is detailed in § 3.3 whereas the asymptotic results are presented in § 4.

Importantly, these equations are valid for all $k$ and ${\it\theta}$ . The equation for the mean velocity $\bar{v}_{y}$ can be formally decoupled from the equations for the density ${\it\sigma}$ and temperature ${\it\tau}$ to reduce computational time in the numerical solution. This is achieved by integrating terms involving $\bar{v}_{y}$ in (A 3) and (A 5) by parts, and substituting for (A 8) and (3.5). Formally, we therefore consider the following integral equations for the gas density and temperature:

(3.7a ) $$\begin{eqnarray}\displaystyle \displaystyle \frac{\sqrt{{\rm\pi}}}{{\it\gamma}}{\it\sigma} & = & \displaystyle \left(D-\frac{1}{2}\right)\left[J_{0}\left(a\left[y+\frac{1}{2}\right]\right)+J_{0}\left(a\left[\frac{1}{2}-y\right]\right)\right]+J_{2}\left(a\left[y+\frac{1}{2}\right]\right)\nonumber\\ \displaystyle & & \displaystyle \displaystyle +\;J_{2}\left(a\left[\frac{1}{2}-y\right]\right)+\frac{1}{k{\it\gamma}}\int _{-1/2}^{1/2}\left\{{\it\sigma}\left[-\frac{2\text{i}{\it\theta}}{1-\text{i}{\it\theta}}J_{1}(a|y-y_{0}|)+J_{-1}(a|y-y_{0}|)\right]\right.\nonumber\\ \displaystyle & & \displaystyle +\left.{\it\tau}\left[J_{1}(a|y-y_{0}|)-\frac{1}{2}J_{-1}(a|y-y_{0}|)\right]\right\}\text{d}y_{0},\end{eqnarray}$$

(3.7b ) $$\begin{eqnarray}\displaystyle \frac{3\sqrt{{\rm\pi}}}{2{\it\gamma}}{\it\tau} & = & \displaystyle D\left[J_{2}\left(a\left[y+\frac{1}{2}\right]\right)+J_{2}\left(a\left[\frac{1}{2}-y\right]\right)-\frac{1}{2}J_{0}\left(a\left[y+\frac{1}{2}\right]\right)\right.\nonumber\\ \displaystyle & & \displaystyle -\left.\frac{1}{2}J_{0}\left(a\left[\frac{1}{2}-y\right]\right)\right]+J_{4}\left(a\left[y+\frac{1}{2}\right]\right)+J_{4}\left(a\left[\frac{1}{2}-y\right]\right)\nonumber\\ \displaystyle & & \displaystyle -\,J_{2}\left(a\left[y+\frac{1}{2}\right]\right)-J_{2}\left(a\left[\frac{1}{2}-y\right]\right)+\frac{5}{4}J_{0}\left(a\left[y+\frac{1}{2}\right]\right)\nonumber\\ \displaystyle & & \displaystyle +\,\frac{5}{4}J_{0}\left(a\left[\frac{1}{2}-y\right]\right)+\frac{1}{k{\it\gamma}}\int _{-1/2}^{1/2}\left\{{\it\sigma}\left[\frac{2\text{i}{\it\theta}}{1-\text{i}{\it\theta}}J_{3}(a|y-y_{0}|)\right.\right.\nonumber\\ \displaystyle & & \displaystyle +\left.\!\left(\frac{\text{i}{\it\theta}}{1-\text{i}{\it\theta}}+1\right)J_{1}(a|y-y_{0}|)-\frac{1}{2}J_{-1}(a|y-y_{0}|)\right]\nonumber\\ \displaystyle & & \displaystyle +\left.{\it\tau}\left[J_{3}(a|y-y_{0}|)-J_{1}(a|y-y_{0}|)+\frac{5}{4}J_{-1}(a|y-y_{0}|)\right]\vphantom{\left[\left(\frac{\text{i}{\it\theta}}{\text{i}{\it\theta}}\right)\right]}\right\}\text{d}y_{0},\end{eqnarray}$$

$D$

The complete set of integral equations for the symmetric heating problem defined by (A 4), (3.6) and (3.7) is numerically solved in § 3.3.

4.1.1 Symmetric wall temperatures

First, consider the leading-order equations in table 1(a). The first equation yields the hydrostatic pressure

(4.5) $$\begin{eqnarray}P_{H}^{(0)}=C_{\mathit{sym}},\end{eqnarray}$$

where $C_{\mathit{sym}}$ is a constant to be determined. From the second equation in table 1(a), the flow is incompressible at leading order and satisfies the no-slip boundary condition; see table 2(a). This gives zero mean wall-normal velocity, because the wall is stationary,

(4.6) $$\begin{eqnarray}\bar{v}_{H|y}^{(0)}=0.\end{eqnarray}$$

Therefore, no flow occurs in the continuum limit, i.e. for $k=0$ .

At leading order, the temperature of the gas matches the wall temperatures given by (4.4a ) with ${\it\upsilon}=+1$ ; see table 2(a). The third equation in table 1(a) then yields the temperature in the gas at leading order. This is defined in terms of the hydrostatic pressure from (4.5):

(4.7) $$\begin{eqnarray}{\it\tau}_{H}^{(0)}=\left({\it\gamma}-\frac{2C_{\mathit{sym}}}{5}\right)\frac{\cosh ({\it\delta}y)}{\cosh \left({\displaystyle \frac{{\it\delta}}{2}}\right)}+\frac{2C_{\mathit{sym}}}{5}.\end{eqnarray}$$

Substituting (4.5) and (4.7) into the continuity equation in the fourth line of table 1(a) yields

(4.8) $$\begin{eqnarray}\frac{\partial \bar{v}_{H|y}^{(1)}}{\partial y}=\frac{{\it\delta}^{2}}{2}\left({\it\gamma}-\frac{2C_{\mathit{sym}}}{5}\right)\frac{\cosh ({\it\delta}y)}{\cosh \left({\displaystyle \frac{{\it\delta}}{2}}\right)}-\frac{3{\it\delta}^{2}}{10}C_{\mathit{sym}}.\end{eqnarray}$$

Equation (4.8) is a first-order differential equation for $\bar{v}_{H|y}^{(1)}$ , which must satisfy the no-penetration condition at both the upper and lower walls. The constant $C_{\mathit{sym}}$ therefore ensures the necessary degree of freedom exists in (4.8) to simultaneously satisfy no penetration at both walls. This establishes that mass conservation is the dominant physical mechanism driving the flow, which generates a hydrostatic pressure in the gas at leading order. This pressure is present across the full range of inertia, i.e. for all ${\it\beta}$ , which ensures the impermeability condition at the walls is satisfied.

In the symmetric case, the unique solution for $\bar{v}_{H|y}^{(1)}$ and $C_{\mathit{sym}}$ from (4.8) is

(4.9a ) $$\begin{eqnarray}\displaystyle & \displaystyle C_{\mathit{sym}}={\it\gamma}\frac{10\tanh \left({\displaystyle \frac{{\it\delta}}{2}}\right)}{3{\it\delta}+4\tanh \left({\displaystyle \frac{{\it\delta}}{2}}\right)}=P_{H}^{(0)}, & \displaystyle\end{eqnarray}$$

(4.9b ) $$\begin{eqnarray}\displaystyle & \displaystyle G_{\mathit{sym}}=0, & \displaystyle\end{eqnarray}$$

(4.9c ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\tau}_{H}^{(0)}={\it\gamma}\frac{3{\it\delta}\cosh ({\it\delta}y)+4\sinh \left({\displaystyle \frac{{\it\delta}}{2}}\right)}{3{\it\delta}\cosh \left({\displaystyle \frac{{\it\delta}}{2}}\right)}, & \displaystyle\end{eqnarray}$$

(4.9d ) $$\begin{eqnarray}\displaystyle & \displaystyle \bar{v}_{H|y}^{(1)}={\it\gamma}\frac{3{\it\delta}^{2}\left(\sinh ({\it\delta}y)-2y\sinh \left({\displaystyle \frac{{\it\delta}}{2}}\right)\right)}{6{\it\delta}\cosh \left({\displaystyle \frac{{\it\delta}}{2}}\right)+8\sinh \left({\displaystyle \frac{{\it\delta}}{2}}\right)}. & \displaystyle\end{eqnarray}$$

A similar procedure can be applied to solve the first order ( $n=1$ ) bulk flow quantities, using the equations in table 1(b). For brevity, we omit the derivation and summarise the unique solutions to $P_{H}^{(1)}$ , ${\it\tau}_{H}^{(1)}$ , and $\bar{v}_{H|y}^{(2)}$ ,

(4.10a ) $$\begin{eqnarray}\displaystyle & \displaystyle P_{H}^{(1)}=-{\it\gamma}\frac{30{\it\delta}^{2}d_{1}\tanh ^{2}\left({\displaystyle \frac{{\it\delta}}{2}}\right)}{\left(3{\it\delta}+4\tanh \left({\displaystyle \frac{{\it\delta}}{2}}\right)\right)^{2}}, & \displaystyle\end{eqnarray}$$

(4.10b ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\tau}_{H}^{(1)}=-{\it\gamma}\frac{3{\it\delta}^{2}d_{1}\sinh \left({\displaystyle \frac{{\it\delta}}{2}}\right)\left(3{\it\delta}\cosh ({\it\delta}y)+4\sinh \left({\displaystyle \frac{{\it\delta}}{2}}\right)\right)}{\left(3{\it\delta}\cosh ({\it\delta}y)+4\sinh \left({\displaystyle \frac{{\it\delta}}{2}}\right)\right)^{2}}, & \displaystyle\end{eqnarray}$$

(4.10c ) $$\begin{eqnarray}\displaystyle & \displaystyle \bar{v}_{H|y}^{(2)}={\it\gamma}\frac{9{\it\delta}^{4}d_{1}\sinh \left({\displaystyle \frac{{\it\delta}}{2}}\right)\left(2y\sinh \left({\displaystyle \frac{{\it\delta}}{2}}\right)-\sinh ({\it\delta}y)\right)}{2\left(3{\it\delta}\cosh ({\it\delta}y)+4\sinh \left({\displaystyle \frac{{\it\delta}}{2}}\right)\right)^{2}}. & \displaystyle\end{eqnarray}$$

$O(k)$

$n=2$

$\bar{v}_{H|y}^{(2)}$

4.1.2 Comparison with Manela & Hadjiconstantinou (Reference Manela and Hadjiconstantinou2010)

As discussed, the symmetric heating problem was previously studied by Manela & Hadjiconstantinou (Reference Manela and Hadjiconstantinou2010) using the linearised Navier–Stokes–Fourier equation. Interestingly, the $k$ -expansion of their solution (not shown here) yields algebraic expressions for the flow up to $O(k^{2})$ , which are identical to those reported in (4.9) and (4.10). However, the results in (4.9) and (4.10) are derived using the hydrodynamic equations in table 1, which differ from the linearised Navier–Stokes–Fourier equation. This is shown in (4.11), where we give the momentum conservation equations for the symmetric uniform heating problem, at both first and second order in $k$ , as formally derived from the Boltzmann–BGK equation,

(4.11a ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\delta}^{2}\bar{v}_{H|y}^{(1)}=-\frac{\partial P_{H}^{(2)}}{\partial y}+\frac{4}{3}\frac{\partial ^{2}\bar{v}_{H|y}^{(1)}}{\partial y^{2}}-\frac{2{\it\delta}^{2}}{3}\frac{\partial {\it\tau}_{H}^{(0)}}{\partial y}, & \displaystyle\end{eqnarray}$$

(4.11b ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\delta}^{2}\bar{v}_{H|y}^{(2)}=-\frac{\partial P_{H}^{(3)}}{\partial x_{y}}+\frac{4}{3}\frac{\partial ^{2}\bar{v}_{H|y}^{(2)}}{\partial y^{2}}-\frac{2{\it\delta}^{2}}{3}\frac{\partial {\it\tau}_{H}^{(1)}}{\partial y}. & \displaystyle\end{eqnarray}$$

$O(k^{2})$

$O(k)$

$O(k^{2})$

$O(k)$

${\it\beta}$

4.1.3 Antisymmetric wall temperatures

We follow an identical approach to the one employed in § 4.1.1 to study the flow caused by antisymmetric heating; see (4.4), with ${\it\upsilon}=\pm 1$ . The leading order ( $n=0$ ) pressure, mean wall-normal velocity and temperature of the gas have the unique solutions:

(4.12a ) $$\begin{eqnarray}\displaystyle & \displaystyle P_{H}^{(0)}=C_{\mathit{asym}}, & \displaystyle\end{eqnarray}$$

(4.12b ) $$\begin{eqnarray}\displaystyle & \displaystyle \bar{v}_{H|y}^{(0)}=0, & \displaystyle\end{eqnarray}$$

(4.12c ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\tau}_{H}^{(0)}=\left({\it\gamma}-\frac{2C_{\mathit{asym}}}{5}\right)\frac{\sinh ({\it\delta}y)}{\sinh \left({\displaystyle \frac{{\it\delta}}{2}}\right)}+\frac{2C_{\mathit{asym}}}{5}. & \displaystyle\end{eqnarray}$$

$k$

$C_{\mathit{asym}}$

(4.13) $$\begin{eqnarray}\bar{v}_{H|y}^{(1)}=\frac{{\it\delta}}{2}\left({\it\gamma}-\frac{2C_{\mathit{asym}}}{5}\right)\frac{\cosh ({\it\delta}y)}{\cosh \left({\displaystyle \frac{{\it\delta}}{2}}\right)}+\frac{{\it\delta}^{2}C_{\mathit{asym}}y}{5}+G_{\mathit{asym}},\end{eqnarray}$$

where $G_{\mathit{asym}}$ is a constant of integration. Enforcing no penetration at both the upper and lower walls gives the unique solution for $\bar{v}_{y}^{(1)}$ ,

(4.14a ) $$\begin{eqnarray}\displaystyle & \displaystyle C_{\mathit{asym}}=0=P_{H}^{(0)}, & \displaystyle\end{eqnarray}$$

(4.14b ) $$\begin{eqnarray}\displaystyle & \displaystyle G_{\mathit{asym}}=-{\it\gamma}\frac{{\it\delta}}{2\tanh \left({\displaystyle \frac{{\it\delta}}{2}}\right)}, & \displaystyle\end{eqnarray}$$

(4.14c ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\tau}_{H}^{(0)}={\it\gamma}\frac{\sinh ({\it\delta}y)}{\sinh \left({\displaystyle \frac{{\it\delta}}{2}}\right)}, & \displaystyle\end{eqnarray}$$

(4.14d ) $$\begin{eqnarray}\displaystyle & \displaystyle \bar{v}_{H|y}^{(1)}={\it\gamma}\frac{{\it\delta}\left(\cosh ({\it\delta}y)-\cosh \left({\displaystyle \frac{{\it\delta}}{2}}\right)\right)}{2\sinh \left({\displaystyle \frac{{\it\delta}}{2}}\right)}. & \displaystyle\end{eqnarray}$$

$P_{H}^{(0)}=C_{\mathit{asym}}$

$n=0$

Closure of this system to first order in $k$ yields the following expressions:

(4.15a ) $$\begin{eqnarray}\displaystyle & \displaystyle P_{H}^{(1)}=0, & \displaystyle\end{eqnarray}$$

(4.15b ) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\tau}_{H}^{(1)}=-{\it\gamma}\frac{{\it\delta}d_{1}\sinh ({\it\delta}y)}{\tanh \left({\displaystyle \frac{{\it\delta}}{2}}\right)\sinh \left({\displaystyle \frac{{\it\delta}}{2}}\right)}, & \displaystyle\end{eqnarray}$$

(4.15c ) $$\begin{eqnarray}\displaystyle & \displaystyle \bar{v}_{H|y}^{(2)}={\it\gamma}\frac{{\it\delta}^{2}d_{1}\left(\cosh \left({\displaystyle \frac{{\it\delta}}{2}}\right)-\cosh ({\it\delta}y)\right)}{2\tanh \left({\displaystyle \frac{{\it\delta}}{2}}\right)\sinh \left({\displaystyle \frac{{\it\delta}}{2}}\right)}. & \displaystyle\end{eqnarray}$$

$n=1$

$k$

4.2.1 Solution for an isolated wall

The solution of an isolated wall is obtained by substituting (4.18), (4.19) and (4.21) into general formulae for the collisionless layer quantities presented in Nassios & Sader (Reference Nassios and Sader2013). This yields expressions for the density ( ${\it\sigma}_{C}^{(0)}$ and ${\it\sigma}_{C}^{(1)}$ ), mean wall-normal velocity ( $\bar{v}_{C|y}^{(0)}$ and $\bar{v}_{C|y}^{(1)}$ ) and the temperature ( ${\it\tau}_{C}^{(0)}$ and ${\it\tau}_{C}^{(1)}$ ) of the gas in terms of ${\it\tau}_{b}^{(0)}$ , which are summarised in table 5. Importantly, these formulae are identical to the corresponding expressions for oscillatory thermal creep, i.e. the flows in each case are identical functions of ${\it\tau}_{b}^{(0)}$ . This is a direct result of the explicit solvability property of the flow in this limit. The key point of difference between these two problems is the first-order mean tangential velocity, i.e. $\bar{v}_{H|x}^{(1)}$ , which is non-zero in the oscillatory thermal creep problem. As required, this thermal creep flow disappears in the uniform heating case, due to the absence of a temperature gradient along the wall, i.e. $\partial {\it\tau}_{b}^{(0)}/\partial x$ is zero; see (4.18). As before, the argument ${\it\eta}_{C}$ in table 5 is the wall-normal coordinate $y$ from figure 2, rescaled by the acoustic length scale $L_{c}=v_{mp}(T_{0})/{\it\omega}$ defined in (1.4).

4.2.2 Symmetric and antisymmetric cases

The solution for the two-wall problem depicted in figure 1 can be constructed from a translation and linear superposition of the isolated wall solutions (see figure 2 and table 5), as we now discuss. The boundary temperature field for the symmetric and antisymmetric two-wall problem is reported in (2.17); the ${\it\theta}^{-1}$ -expansion of (2.17) is

(4.23) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}{\it\tau}_{\pm b}^{(0)}={\it\upsilon}{\it\gamma},\quad {\it\upsilon}=\pm 1,\\ {\it\tau}_{\pm b}^{(1)}=0.\end{array}\right\}\end{eqnarray}$$

Next, we rescale the wall-normal coordinate ${\it\eta}_{C}$ by the geometric length scale, which is the distance between the walls $d$ ; see figure 1. The new scaled normal coordinate is denoted as $y$ , and is related to ${\it\eta}_{C}$ by

(4.24) $$\begin{eqnarray}{\it\eta}_{C}=\frac{y}{\mathit{St}},\end{eqnarray}$$

where the $\mathit{St}={\it\omega}L/v_{mp}(T_{0})$ ; see (1.5) and (2.6).

The temperature of the upper wall ${\it\tau}_{+b}^{(0)}$ is identical for both the symmetric and antisymmetric problems; see (4.23). To account for the flow at the upper wall in each case, the isolated wall solutions in table 5 are translated such that the wall located at ${\it\eta}_{C}=0$ is positioned at $y=1/2$ , while the wall temperature distribution ${\it\tau}_{b}^{(0)}$ is set in accord with the above discussion. A similar procedure can be applied at the lower wall ( $y=-1/2$ ), where ${\it\tau}_{-b}^{(0)}={\it\gamma}$ in the symmetric problem while ${\it\tau}_{-b}^{(0)}=-{\it\gamma}$ in the antisymmetric case. Using ${\it\alpha}$ to denote any of the macroscopic flow quantities ${\it\sigma}_{C}^{(n)}$ , $\bar{v}_{C|y}^{(n)}$ , ${\it\tau}_{C}^{(n)}$ and $P_{C}^{(n)}$ , the general solution for symmetric uniform heating of two walls is

(4.25) $$\begin{eqnarray}{\it\alpha}_{\mathit{sym}}(y)=\frac{1}{{\it\gamma}}\left[{\it\alpha}\left(\frac{{\textstyle \frac{1}{2}}-y}{\mathit{St}}\right)+{\it\alpha}\left(\frac{{\textstyle \frac{1}{2}}+y}{\mathit{St}}\right)\right],\end{eqnarray}$$

while the corresponding expression for antisymmetric heating is

(4.26) $$\begin{eqnarray}{\it\alpha}_{\mathit{asym}}(y,A)=\frac{1}{{\it\gamma}}\left[{\it\alpha}\left(\frac{{\textstyle \frac{1}{2}}-y}{\mathit{St}}\right)-{\it\alpha}\left(\frac{{\textstyle \frac{1}{2}}+y}{\mathit{St}}\right)\right].\end{eqnarray}$$

The two-wall solutions can be constructed by substituting the isolated wall expressions from table 5 into (4.25) and (4.26); for brevity, we omit the lengthy expressions that result.

4.2.3 Symmetric uniform heating by Manela & Hadjiconstantinou (Reference Manela and Hadjiconstantinou2008)

In addition to the (near continuum) low-frequency limit, Manela & Hadjiconstantinou (Reference Manela and Hadjiconstantinou2008) also studied the symmetric two-wall uniform heating problem for a free molecular gas, i.e. $k^{-1}\rightarrow 0$ . The collisionless Boltzmann equation was therefore considered, subject to the linearised diffuse reflection condition at both walls:

(4.27) $$\begin{eqnarray}-\text{i}\,\mathit{St}\,{\it\phi}+v_{y}\frac{\partial {\it\phi}}{\partial y}=0,\end{eqnarray}$$

where $\mathit{St}$ is the Strouhal number. The particle velocity $v_{y}$ has been scaled by the most probable speed at the equilibrium temperature of the gas $v_{mp}(T_{0})$ defined in (2.6), while the wall-normal coordinate $y$ has been scaled by the channel width $d$ . Integral equations for the macroscopic flow quantities were derived from (4.27), defined in terms of the Strouhal number and the reflected particle densities ${\it\sigma}_{\pm b}$ at the upper and lower walls. This was achieved using similar methods to those outlined in § 3.1.

The method of steepest descent was then applied to study the resulting integral equations for the density ${\it\sigma}$ , mean wall-normal velocity $\bar{v}_{y}$ and temperature ${\it\tau}$ derived from (4.27) for large Strouhal number, i.e. $\mathit{St}^{-1}\rightarrow 0$ . This yielded the closed form expressions for the reflected particle densities at both walls,

(4.28) $$\begin{eqnarray}{\it\sigma}_{\pm b}=-\frac{{\it\tau}_{\pm b}}{2},\end{eqnarray}$$

where ${\it\tau}_{\pm b}={\it\gamma}$ because the wall temperatures were assumed to be symmetric and ${\it\gamma}\equiv A/T_{0}$ is the normalised amplitude of the wall temperature. The reflected particle densities ${\it\sigma}_{\pm b}$ therefore decouple, which gives explicit formulae for the density, mean wall-normal velocity and temperature of the gas. Importantly, these formulae are identical to those constructed using (4.25) and the isolated wall solutions in table 5. Interestingly, despite the assumption of a free molecular gas, good agreement was observed between LVDSMC simulations of the flow and the analytical formulae for large Strouhal numbers and small Knudsen numbers, i.e. $\mathit{St}\gg 1$ and $k\ll 1$ .

We now discuss these observations in the context of the solution presented here, derived using the formalism in Nassios & Sader (Reference Nassios and Sader2013). Given that the frequency ratio ${\it\theta}$ can be expressed in terms of the Strouhal number (1.5), the limit $\mathit{St}^{-1}\rightarrow 0$ is equivalent to the high-frequency flow regime, i.e. where ${\it\theta}^{-1}\rightarrow 0$ . When ${\it\theta}\gg 1$ , gas particle advection parallel to a uniformly heated wall does not affect the flow, because the wall temperature is independent of the tangential coordinates $x$ and  $z$ .

As discussed in Nassios & Sader (Reference Nassios and Sader2013), the scaled Knudsen number $k$ determines the strength of gas particle advection parallel to a solid wall. Since gas particles do not interact in the leading-order flow (i.e. ${\it\theta}^{-1}\rightarrow 0$ ) generated by oscillatory uniform heating of two plane walls at high frequency for all $k$ , any effect of $k$ is eliminated. Consequently, (4.27) can be applied to study this flow.

However, collisional effects become important at higher order, and cannot be studied using the collisionless equation in (4.27). In this case, the asymptotic formulae derived in Nassios & Sader (Reference Nassios and Sader2013) and employed in this section provide a general framework with which to study flows for ${\it\theta}\gg 1$ and $k\ll 1$ up to $O({\it\theta}^{-1})$ . These higher-order asymptotic results account for the effect of collisions and advection along a solid wall, which are not captured by (4.27).

5.1.1 Symmetric wall temperatures

Since the gas flow is driven by temperature oscillations at the walls, we initially examine the temperature field as a function of the inertial parameter  ${\it\beta}$ .

Temperature field. The real and imaginary components of the leading-order scaled temperature ${\it\tau}_{H}^{(0)}/{\it\gamma}$ from (4.9) are shown in figure 3, as a function of ${\it\beta}$ . Due to the no-slip condition inherent in the leading-order solution, the real and imaginary components of ${\it\tau}_{H}^{(0)}/{\it\gamma}$ at each wall are independent of ${\it\beta}$ , and equal to 1 and 0, respectively.

Figure 3.

Scaled leading-order temperature of the symmetric problem: ${\it\tau}_{H}^{(0)}/{\it\gamma}$ . (a) Real component; (b) imaginary component. Results given for ${\it\beta}=1$ (small dashed), ${\it\beta}=10$ (medium dashed), ${\it\beta}=30$ (large dashed), ${\it\beta}=100$ (dot-dashed), ${\it\beta}=1000$ (dotted) and ${\it\beta}=10\,000$ (solid).

For zero inertia, i.e. ${\it\beta}=0$ , the expected steady solution is recovered, with constant and purely real gas temperature between the walls,

(5.1) $$\begin{eqnarray}{\it\tau}_{H|\mathit{sym}}^{(0)}={\it\gamma}.\end{eqnarray}$$

For non-zero but small inertia, i.e. ${\it\beta}\ll 1$ , a non-zero imaginary component arises, indicating a lag in the gas temperature response. The imaginary component increases in magnitude for small ${\it\beta}$ , attaining a peak value at $y=0$ for ${\it\beta}=13.6657\ldots \,$ before inflecting; see figure 3(b). For large inertia, i.e. ${\it\beta}\gg 1$ , thermal boundary layers form in both real and imaginary components near the walls; thermal gradients within these layers are large. Away from the walls, the temperature ${\it\tau}_{H}^{(0)}$ displays $1/\sqrt{{\it\beta}}$ decay throughout the bulk of the channel due to diffusive heat transport from the walls. This is evident from the small and large- ${\it\beta}$ asymptotic behaviour away from any walls:

(5.2) $$\begin{eqnarray}{\it\tau}_{H}^{(0)}={\it\gamma}\left\{\begin{array}{@{}ll@{}}\displaystyle 1+\frac{3\text{i}{\it\beta}}{40}(1-4y^{2})-\frac{{\it\beta}^{2}}{3200}(80y^{4}-88y^{2}+17) & \\ \displaystyle \quad -\,\frac{\text{i}{\it\beta}^{3}}{384\,000}(141-764y^{2}+880y^{4}-320y^{6}), & {\it\beta}\ll 1\\ \displaystyle \frac{4}{3\sqrt{2{\it\beta}}}(1+\text{i})-\frac{16\text{i}}{9{\it\beta}}, & {\it\beta}\gg 1.\end{array}\right.\end{eqnarray}$$

Thus, the temperature field exhibits spatial variations at finite inertia, which in principle can induce pressure and density variations in the gas – this coupling is now examined.

Pressure and density fields. The leading-order pressure, $P_{H}^{(0)}$ , i.e. which corresponds to the true pressure at zero frequency, is hydrostatic and non-zero; see (4.9). Operation at non-zero ${\it\beta}$ produces a deviation from this hydrostatic state, with the real and imaginary components of $P_{H}^{(0)}/{\it\gamma}$ for ${\it\beta}\in [0,1000]$ given in figure 4.

For zero inertia, ${\it\beta}=0$ , the pressure is identical to the temperature field ${\it\tau}_{H}^{(0)}/{\it\gamma}$ throughout the channel; see figure 4, where the real component is equal to $1$ and the imaginary component is zero. A lag in the pressure of the gas in response to the oscillatory wall temperatures arises for small ${\it\beta}$ . Interestingly, the imaginary component of $P_{H}^{(0)}/{\it\gamma}$ is non-monotonic in ${\it\beta}$ , whereas the real component is a monotonically decreasing function of ${\it\beta}$ . At ${\it\beta}=15.6470\ldots \,$ , the imaginary component attains its maximum value, and subsequently begins to decrease. For large ${\it\beta}$ , both real and imaginary components of the pressure again exhibit $1/\sqrt{{\it\beta}}$ decay, in line with the temperature field in (5.2) that is driving the pressure; the small- and large- ${\it\beta}$ asymptotics for the pressure are given by

(5.3) $$\begin{eqnarray}P_{H}^{(0)}={\it\gamma}\left\{\begin{array}{@{}ll@{}}\displaystyle 1+\frac{\text{i}{\it\beta}}{20}-\frac{{\it\beta}^{2}}{300}-\frac{23\text{i}{\it\beta}^{3}}{100\,800}, & {\it\beta}\ll 1\\ \displaystyle \frac{10}{3\sqrt{2{\it\beta}}}(1+\text{i})-\frac{40\text{i}}{9{\it\beta}}, & {\it\beta}\gg 1.\end{array}\right.\end{eqnarray}$$

The gas density is then determined from the ideal gas law in (4.8),

(5.4) $$\begin{eqnarray}{\it\sigma}_{H}^{(0)}=P_{H}^{(0)}-{\it\tau}_{H}^{(0)}.\end{eqnarray}$$

The small- and large- ${\it\beta}$ asymptotics of ${\it\sigma}_{H}^{(0)}$ (away from the walls) are:

(5.5) $$\begin{eqnarray}{\it\sigma}_{H}^{(0)}={\it\gamma}\left\{\begin{array}{@{}ll@{}}\displaystyle \frac{\text{i}{\it\beta}}{40}(12y^{2}-1)-\frac{{\it\beta}^{2}}{9600}(19-264y^{2}+240y^{4}) & \\ \displaystyle \quad -\,\frac{\text{i}{\it\beta}^{3}}{8\,064\,000}(-1121+16044y^{2}-18480y^{4}+6720y^{6}) & {\it\beta}\ll 1\\ \displaystyle \frac{10}{3\sqrt{2{\it\beta}}}(1+\text{i})-\frac{40\text{i}}{9{\it\beta}} & {\it\beta}\gg 1.\end{array}\right.\end{eqnarray}$$

Figure 4.

Scaled leading-order hydrostatic pressure of the symmetric problem: $P_{H}^{(0)}/{\it\gamma}$ . (a) Real component; (b) imaginary component.

Plots of the real and imaginary components of ${\it\sigma}_{H}^{(0)}/{\it\gamma}$ are given in figure 5. For zero inertia, ${\it\sigma}_{H}^{(0)}=0$ throughout the channel; see the asymptotic formula for ${\it\beta}\ll 1$ in (5.5). For small ${\it\beta}$ , the imaginary (out-of-phase) component increases in magnitude, and is non-zero at the walls; the real component monotonically decreases towards the walls, for all ${\it\beta}$ . At the critical value of ${\it\beta}=15.6471\ldots \,$ , the imaginary component of ${\it\sigma}_{H}^{(0)}/{\it\gamma}$ attains its maximum value at the walls; at this ${\it\beta}$ the imaginary component of $P_{H}^{(0)}/{\it\gamma}$ is maximised. For ${\it\beta}\gg 1$ , the density exhibits $1/\sqrt{{\it\beta}}$ decay away from the walls, as expected; see (5.5). At the walls, the real component of ${\it\sigma}_{H}^{(0)}/{\it\gamma}$ asymptotes to 1 with increasing ${\it\beta}$ , because $P_{H}^{(0)}/{\it\gamma}$ decays to 0; see figures 4(a) and 5(b).

With the temperature, pressure and density fields determined, we now turn our attention to their effect on the velocity field (our primary goal).

Figure 5.

Scaled leading-order density of the symmetric problem: ${\it\sigma}_{H}^{(0)}/{\it\gamma}$ . (a) Real component; (b) imaginary component. Results given for ${\it\beta}=1$ (small dashed), ${\it\beta}=15.6471$ (medium dashed), ${\it\beta}=30$ (large dashed), ${\it\beta}=100$ (dot-dashed), ${\it\beta}=1000$ (dotted) and ${\it\beta}=10\,000$ (solid).

Mean wall-normal velocity. We now study the first-order ( $n=1$ ) component of the mean flow, $\bar{v}_{H|y}^{(1)}$ (see (4.9)), because the $n=0$ term is zero – the component $\bar{v}_{H|y}^{(1)}$ thus specifies the leading-order behaviour of the velocity field. To facilitate discussion, we calculate an average spatial velocity in the gas. The velocity is antisymmetric in $y$ , and we therefore calculate the average wall-normal velocity $u_{y}^{(1)}$ in the upper half-channel, $0\leqslant y\leqslant 1/2$ :

(5.6) $$\begin{eqnarray}u_{y}^{(1)}=2\int _{0}^{1/2}\bar{v}_{H|y}^{(1)}\,\text{d}y=-\frac{3{\it\delta}{\it\gamma}}{4}\frac{4-4\cosh \left({\displaystyle \frac{{\it\delta}}{2}}\right)+{\it\delta}\sinh \left({\displaystyle \frac{{\it\delta}}{2}}\right)}{3{\it\delta}\cosh \left({\displaystyle \frac{{\it\delta}}{2}}\right)+4\sinh \left({\displaystyle \frac{{\it\delta}}{2}}\right)}.\end{eqnarray}$$

The asymptotic behaviour of this average wall-normal velocity, for small and large ${\it\beta}$ (away from the walls), is

(5.7) $$\begin{eqnarray}u_{y}^{(1)}={\it\gamma}\left\{\begin{array}{@{}ll@{}}\displaystyle \frac{{\it\beta}^{2}}{640}+\frac{3\text{i}{\it\beta}^{3}}{25\,600}, & {\it\beta}\ll 1\\ \displaystyle \frac{1}{4}\sqrt{\frac{{\it\beta}}{2}}(1-\text{i})+\frac{4}{3}, & {\it\beta}\gg 1.\end{array}\right.\end{eqnarray}$$

We consider four separate limits: (i) zero inertia, ${\it\beta}=0$ ; (ii) small inertia, ${\it\beta}\ll 1$ ; (iii) intermediate inertia, ${\it\beta}\sim O(1)$ ; and (iv) large inertia, ${\it\beta}\gg 1$ .

Zero inertia ( ${\it\beta}=0$ ). The flow is steady and incompressible; see the continuity equation in the last line of table 1(a). As such, the mean wall-normal velocity, $\bar{v}_{H|y}^{(1)}$ , is zero throughout the gas.

Small inertia ( ${\it\beta}\ll 1$ ). For small inertia, the leading-order density ${\it\sigma}_{C}^{(0)}$ is of $O({\it\beta})$ and varies quadratically in the normal coordinate, $y$ ; see the ${\it\beta}\ll 1$ asymptotic in (5.5). The gas density is therefore out of phase with the wall temperature oscillations. By the continuity equation (mass conservation), this drives a compressible flow that is both (i) in-phase with the wall temperature oscillations, and (ii) antisymmetric in the wall-normal coordinate $y$ ; see figure 6(a) for ${\it\beta}=10^{-4}$ , and the ${\it\beta}\ll 1$ asymptotic in (5.7). As ${\it\beta}$ increases, an additional out-of-phase correction to $\bar{v}_{H|y}^{(1)}$ arises due to the $O({\it\beta}^{2})$ term in the density; see figure 6(b) for ${\it\beta}=10^{-4}$ and (5.7).

Figure 6.

Scaled first-order mean wall-normal velocity of symmetric problem: $\bar{v}_{H|y}^{(1)}/({\it\beta}^{l}{\it\gamma})$ , where $l$ is chosen to match dominant asymptotic behaviour in ${\it\beta}$ . Real, ( $l=2$ ), imaginary ( $l=3$ ). (a) Real component; (b) imaginary component. Results given for ${\it\beta}=10^{-4}$ (small dashed), ${\it\beta}=10^{-2}$ (small dashed), ${\it\beta}=10$ (large dashed), ${\it\beta}=15.647$ (dot-dashed), ${\it\beta}=30$ (dotted) and ${\it\beta}=50.4757$ (solid). Note that results for ${\it\beta}=10^{-4}$ and ${\it\beta}=10^{-2}$ are indistinguishable on the graph.

Intermediate inertia ( ${\it\beta}\sim O(1)$ ). For intermediate inertia, the imaginary component of ${\it\sigma}_{H}^{(0)}$ is positive at the walls, $y=\pm 1/2$ . In contrast, the density throughout the bulk of the channel is otherwise negative. As discussed, for ${\it\beta}=15.6471\ldots \,$ the imaginary component of the gas density attains its maximum value at the walls, and begins to decrease with further increases in ${\it\beta}$ . This causes a similar inflection in the real component of the average velocity in the upper half-channel, $u_{y}^{(1)}$ , where the maximum in the real part of the average normal velocity, $u_{y}^{(1)}$ , in (5.6) occurs for

(5.8) $$\begin{eqnarray}{\it\beta}={\it\beta}_{c}=22.5002\ldots .\end{eqnarray}$$

For ${\it\beta}>{\it\beta}_{c}$ , the real and imaginary components of $u_{y}^{(1)}$ are monotonically decreasing functions of ${\it\beta}$ . This causes a flow reversal in the real component of $\bar{v}_{H|y}^{(1)}$ , which occurs at ${\it\beta}=50.4757\ldots ;$ the onset of this reversal is illustrated by plots of the real component of $\bar{v}_{H|y}^{(1)}$ from ${\it\beta}=10$ to ${\it\beta}=50.4757$ , which are given in figure 6(a). No reversal occurs in the imaginary component of $\bar{v}_{H|y}^{(1)}$ , because it is a monotonically decreasing function of ${\it\beta}$ , for all  ${\it\beta}$ .

Large inertia ( ${\it\beta}\gg 1$ ). In the limit of large inertia, the temperature and density fields are confined to thin boundary layers near the walls, and the hydrostatic pressure decays to zero; see the ${\it\beta}\gg 1$ asymptotics in (5.2), (5.3) and (5.5). Away from the walls, the real and imaginary components of ${\it\sigma}_{H}^{(0)}$ are therefore independent of $y$ ; see figure 5. From the continuity equation, a suitable choice of scale for $\bar{v}_{H|y}^{(1)}$ then yields a linear velocity profile away from the walls that is evident in figure 7; see ${\it\beta}=1000$ and ${\it\beta}=10\,000$ . This contrasts with the near-wall velocity profile, which exhibits strong gradients.

Figure 7.

Scaled first-order mean wall-normal velocity of symmetric problem: $\bar{v}_{H|y}^{(1)}/(\sqrt{{\it\beta}}{\it\gamma})$ . (a) Real component; (b) imaginary component. Results given for ${\it\beta}=10$ (medium dashed), ${\it\beta}=50.4757$ (large dashed), ${\it\beta}=100$ (dot-dashed) and ${\it\beta}=1000$ (dotted) and ${\it\beta}=10\,000$ (solid).

Summary. This analysis establishes the overriding physical mechanism driving gas flow in this problem: finite density variations at continuum order, $O(1)$ , induced by temperature and pressure variations of the same order (via the ideal gas law), and mass conservation. Note that this non-continuum $O(k)$ velocity field is driven by continuum-order effects in the temperature and pressure, as dictated by the asymptotic theory. Momentum transport does not play a role in driving this non-continuum flow.

5.1.2 Antisymmetric wall temperatures

We now provide an analogous (and abbreviated) discussion for the problem where the wall temperatures are antisymmetric, i.e. they are equal in magnitude but opposite in sign.

Temperature field. Plots of the real and imaginary components of the scaled temperature ${\it\tau}_{H}^{(0)}/{\it\gamma}$ from (4.9) are given in figure 8. As for the symmetric case, a steady (purely real) temperature profile is recovered for zero inertia, i.e. ${\it\beta}=0$ ,

(5.9) $$\begin{eqnarray}{\it\tau}_{H|\mathit{asym}}^{(0)}=2{\it\gamma}y.\end{eqnarray}$$

A phase lag in the gas temperature field occurs as ${\it\beta}$ increases, and thermal boundary layers form near the walls for large inertia, i.e. ${\it\beta}\gg 1$ ; see figure 8 for ${\it\beta}=1000$ and ${\it\beta}=10\,000$ .

Figure 8.

Scaled leading-order temperature of antisymmetric problem: ${\it\tau}_{H}^{(0)}/{\it\gamma}$ . (a) Real component; (b) imaginary component; (c) magnitude. Results given for ${\it\beta}=1$ (small dashed), ${\it\beta}=10$ (medium dashed), ${\it\beta}=30$ (large dashed), ${\it\beta}=100$ (dot-dashed), ${\it\beta}=1000$ (dotted) and ${\it\beta}=10\,000$ (solid).

Density field. As discussed in § 4.1.3, the hydrostatic pressure at leading order ( $n=0$ ) is zero for the antisymmetric case, which contrasts with the symmetric problem; see (4.14). From the ideal gas law in (4.8), the gas density is thus:

(5.10) $$\begin{eqnarray}{\it\sigma}_{H}^{(0)}=-{\it\tau}_{H}^{(0)}.\end{eqnarray}$$

Substituting (5.10) into the continuity equation in the final line of table 1, we thus find that gradients in the mean wall-normal velocity are caused solely by the temperature field of the gas; this contrasts strongly with the symmetric case, where gas density is controlled by both the temperature and pressure.

We now use this observation to study the first-order mean wall-normal velocity, $\bar{v}_{H|y}^{(1)}$ , which is the leading-order mean flow (as for the symmetric problem).

Mean wall-normal velocity. We again calculate the average velocity in the upper half-channel,

(5.11) $$\begin{eqnarray}u_{y}^{(1)}=2\int _{0}^{1/2}\bar{v}_{H|y}^{(1)}\,\text{d}y={\it\gamma}\left[1-\frac{{\it\delta}}{2}\coth \left(\frac{{\it\delta}}{2}\right)\right].\end{eqnarray}$$

The corresponding small- and large- ${\it\beta}$ asymptotic results are

(5.12) $$\begin{eqnarray}\displaystyle u_{y}^{(1)}={\it\gamma}\left\{\begin{array}{@{}ll@{}}\displaystyle \frac{\text{i}{\it\beta}}{12}-\frac{{\it\beta}^{2}}{720}-\frac{\text{i}{\it\beta}^{3}}{30\,240}, & {\it\beta}\ll 1\\ \displaystyle 1+\sqrt{\frac{{\it\beta}}{2}}(\text{i}-1), & {\it\beta}\gg 1.\end{array}\right.\end{eqnarray}$$

These formulae are again used in conjunction with the temperature field, ${\it\tau}_{H}^{(0)}$ , to study $\bar{v}_{H|y}^{(1)}$ across the full range of inertia,  ${\it\beta}$ .

Zero inertia ( ${\it\beta}=0$ ). Again, no flow is generated in the steady limit.

Small inertia ( ${\it\beta}\ll 1$ ). For small and non-zero inertia, the steady and purely real component for the temperature ${\it\tau}_{H|\mathit{asym}}^{(0)}$ drives a compressible (out-of-phase) mean flow via mass conservation; see asymptotics for ${\it\beta}\ll 1$ in (5.12), and plots of the scaled mean wall-normal flow profile in the upper half-channel, $0\leqslant y\leqslant 1/2$ , given in figure 9. The real component of the temperature field, ${\it\tau}_{H}^{(0)}$ , is monotonically increasing in $y$ for small ${\it\beta}$ ; see figure 8, with ${\it\beta}=1$ .

Figure 9.

Scaled first-order mean wall-normal velocity of antisymmetric problem: $\bar{v}_{H|y}^{(1)}/({\it\beta}^{l}{\it\gamma})$ , where $l$ is chosen to match the dominant asymptotic behaviour in ${\it\beta}$ . (a) Real ( $l=2$ ); (b) imaginary ( $l=1$ ). Results given for ${\it\beta}=10^{-4}$ (small dashed), ${\it\beta}=10$ (large dashed), ${\it\beta}=30$ (dot-dashed) and ${\it\beta}=50$ (solid).

Figure 10.

Scaled first-order mean wall-normal velocity of antisymmetric problem: $\bar{v}_{H|y}^{(1)}/(\sqrt{{\it\beta}}{\it\gamma})$ . (a) Real component; (b) imaginary component. Results given for ${\it\beta}=10$ (medium dashed), ${\it\beta}=50$ (large dashed), ${\it\beta}=100$ (dot-dashed) and ${\it\beta}=1000$ (dotted) and ${\it\beta}=10\,000$ (solid).

From the continuity equation in table 1(a), and (5.10), the imaginary component of $\bar{v}_{H|y}^{(1)}$ is therefore monotonically decreasing in  $y$ . Together with the no-penetration condition at the walls, this causes the inverted parabolic flow profile in figure 9(b) for ${\it\beta}=10^{-4}$ .

The real component of $\bar{v}_{H|y}^{(1)}$ is driven by the $O({\it\beta})$ correction to ${\it\tau}_{H}^{(0)}$ for small inertia, which is zero at (i) the centre of the channel and (ii) the upper wall. Once more, from the continuity equation and (5.10), this explains the inflected flow profile in the real component of $\bar{v}_{H|y}^{(1)}$ for ${\it\beta}=10^{-4}$ ; see figure 9(a). The magnitude of both the real and imaginary component of $\bar{v}_{H|y}^{(1)}$ are also increasing functions of ${\it\beta}$ at small inertia; this is evident from plots of the scaled profiles in figure 9 for ${\it\beta}=10^{-4}$ and ${\it\beta}=10^{-2}$ .

Intermediate inertia ( ${\it\beta}\sim O(1)$ ). As ${\it\beta}$ is increased further, the magnitude of $\bar{v}_{H|y}^{(1)}$ continues to increase. Both components of the mean wall-normal velocity $\bar{v}_{H|y}^{(1)}$ therefore display identical behaviour to that established for small inertia; see figures 9 and 10 for ${\it\beta}=10$ , ${\it\beta}=30$ and ${\it\beta}=50$ .

High inertia ( ${\it\beta}\gg 1$ ). Contrasting behaviour to the symmetric problem is observed at high inertia. As for the symmetric case, perturbations to the temperature, ${\it\tau}_{H}^{(0)}$ , are confined to boundary layers near the walls, with exponential decay exhibited in the (scaled) normal coordinate, $y$ , away from the walls; see figure 8 for ${\it\beta}=1000$ and ${\it\beta}=10\,000$ . However, $P_{H}^{(0)}=0$ here; hence, by continuity and (5.10), flow throughout the bulk of the channel (away from any walls) is divergence free (incompressible), and the mean wall-normal velocity outside the boundary layers is thus independent of  $y$ ; see figure 10. Within the boundary layers near the walls, gradients in the temperature field are large. This drives similarly large gradients in the scaled mean wall-normal velocity near the walls; see plots of the scaled mean wall-normal velocity in figure 10 for ${\it\beta}=1000$ and ${\it\beta}=10\,000$ .

Figure 11.

Scaled temperature, ${\it\tau}_{C}^{(0)}/{\it\tau}_{b}^{(0)}$ and ${\it\tau}_{C}^{(1)}/{\it\tau}_{b}^{(0)}$ : (a) Real; (b) imaginary. Results given for the leading-order solution ( $n=0$ , solid) and first-order correction ( $n=1$ , dashed). Numerical data for the imaginary components in figures 11–15 are affected by round-off error (noise) due to integrable singularities in the integrands. This could not be suppressed with available computational resources and was smoothed using a polynomial regression fit of order 50; the coefficients were determined using ordinary least squares. This does not affect the shape of the presented curves however suppresses the numerical noise.

Summary. This analysis shows that the mechanisms driving gas flow in the antisymmetric problem differ to those of the symmetric problem: pressure no longer has an effect here, with density variations being driven purely by temperature variations and mass conservation. Momentum transport again does not play a role.

5.3.1 Low oscillation frequency, ${\it\theta}\ll 1$

Figure 16 presents a comparison of the asymptotic and direct numerical solutions for the symmetric problem in the low-frequency regime. Due to symmetry, results for the top half of the channel only are presented; density and temperature fields are symmetric about $y=0$ while the normal velocity is antisymmetric about this position.

The low-frequency asymptotic solution is found to be in excellent agreement with the numerical solution for ${\it\theta}=0.0125$ (or ${\it\beta}=10$ ); we remind the reader that ${\it\beta}$ is linked to the frequency ratio ${\it\theta}$ and scaled Knudsen number $k$ in the low-frequency limit by (4.1a ). As may be expected, this agreement weakens with increasing ${\it\theta}$ as the low frequency (quasi-steady) assumption ${\it\theta}\ll 1$ is violated. However, even at ${\it\theta}=0.0625$ ( ${\it\beta}=50$ ), the asymptotic solution compares well with the numerical solution and provides a good approximation. Flow characteristics of the symmetric heating problem are clearly captured. As ${\it\theta}$ increases, the imaginary component of the wall density attains a maximum and decreases for values of ${\it\theta}$ higher than ${\approx}0.025$ . This causes the real part of the normal velocity to undergo a flow reversal, as discussed in § 5.1.1. The imaginary part of the normal velocity increases monotonically with ${\it\theta}$ due to the effects of increased inertia.

Figure 16.

Asymptotic solutions for symmetric problem versus numerical results for low-frequency flow when $k=0.05$ . Scaled temperature: (a) real; (d) imaginary. Scaled density: (b) real; (e) imaginary. Scaled mean wall-normal velocity: (c) real; (f) imaginary. All solutions are scaled by ${\it\gamma}\equiv A/T_{0}$ . Asymptotic solutions: solid. Numerical solutions: dashed. Comparisons provided for ${\it\beta}=10$ (orange), ${\it\beta}=30$ (green), and ${\it\beta}=50$ (blue).

Figure 17 shows complementary flow profiles for the antisymmetric problem, with $k=0.05$ and ${\it\theta}=0.0125,0.0375,0.0625$ ( ${\it\beta}=10,30,50$ ). Correspondingly, the density and temperature are antisymmetric about $y=0$ while the normal velocity is symmetric about this position. As before, the low-frequency asymptotic and full numerical solutions are in excellent agreement for ${\it\theta}=0.0125$ , while divergence appears as ${\it\theta}$ increases. The low-frequency asymptotic solution predicts the antisymmetric uniform heating problem flow well for values of ${\it\theta}$ less than ${\approx}0.0625$ .

Figure 17.

Asymptotic solutions for antisymmetric problem versus numerical results for low-frequency flow when $k=0.05$ . Scaled temperature: (a) real; (d) imaginary. Scaled density: (b) real; (e) imaginary. Scaled mean wall-normal velocity: (c) real; (f) imaginary. All solutions are scaled by ${\it\gamma}\equiv A/T_{0}$ . Asymptotic solutions: solid. Numerical solutions: dashed. Comparisons provided for ${\it\beta}=10$ (orange), ${\it\beta}=30$ (green) and ${\it\beta}=50$ (blue).

5.3.2 High oscillation frequency, ${\it\theta}\gg 1$

Corresponding results in the high-frequency limit are given in figure 18, which shows a comparison between the asymptotic formulae and the direct numerical solution for the symmetric unsteady heating problem; results for $k=0.1$ and ${\it\theta}=7$ are shown. Both the leading order (free molecular) and first-order asymptotic solutions are presented for the density, temperature and normal velocity fields. Only the bottom half of the channel is shown due to symmetry; density and temperature profiles are symmetric about $y=0$ while the normal velocity profile is antisymmetric. We only consider the symmetric unsteady heating problem here; similar trends are observed for the antisymmetric problem.

These results show that the leading-order asymptotic solution, i.e. at $O(1)$ (free molecular flow), captures the density, temperature and velocity near the wall. However, discrepancies appear between the asymptotic and numerical solutions away from the wall. This is due to use of a finite frequency ratio, ${\it\theta}=7$ , which leads to collisional interactions between gas particles that becomes increasingly important away from the wall. Including the first-order correction results in significantly better agreement with the direct numerical solution, showing that this higher-order term correctly captures the effects of interparticle interactions. Despite this solution being accurate only up to $O({\it\theta}^{-1})$ , it provides an excellent approximation for large but finite ${\it\theta}$ and small  $k$ . Note that interaction between the two walls is negligible, because the density, temperature and velocity profiles all decay sufficiently between the walls (at $x=0$ ) for ${\it\theta}=7$ . Strikingly, the temperature field exhibits an unusual kink near the wall at $x\approx 0.45$ , which is accurately captured by the asymptotic solution – this provides further validation for its robustness.

Figure 18.

Asymptotic solutions versus numerical results for high-frequency flow where ${\it\theta}=7$ and $Kn=0.1$ ( $k=0.0886$ ). Images at the lower wall for symmetric problem. Scaled temperature: (a) real; (d) imaginary. Scaled density: (b) real; (e) imaginary. Scaled mean wall-normal velocity: (c) real; (f) imaginary. All solutions are scaled by ${\it\gamma}\equiv A/T_{0}$ . Free molecular solution: red dotted line. First-order asymptotic solution: green solid line. Numerical solutions: dashed.

Summary. The results in figures 17 and 18 show that the asymptotic formulae in § 4, which are derived using the general theories in Nassios & Sader (Reference Nassios and Sader2012, Reference Nassios and Sader2013) and Takata et al. (Reference Takata, Aoki, Hattori and Hadjiconstantinou2012), provide excellent quantitative agreement with direct numerical solutions, in their respective regimes.