2.1. The conformal transformation

For periodic corrugations of arbitrary shape and amplitude, a conformal transformation can be constructed to map the flow domain onto a uniform strip (Yu & Howard Reference Yu and Howard2012). This is adapted here. Let the undisturbed channel wall position be written as $y_{s}=kh_{0}\left[1+{\it\epsilon}_{b}f(x)\right]$ , where $f(x)$ is the sawtooth profile, ${\it\epsilon}_{b}$ is a dimensionless amplitude parameter. We shall require $f(x)$ to have zero average in $x$ , so that $h_{0}$ is the mean half channel width in the absence of motion. When $f(x)$ is so normalized that the peak-to-trough height is unity, ${\it\epsilon}_{b}h_{0}$ is the sawtooth height. We consider ${\it\epsilon}_{b}=O(1)$ . Following Yu & Howard (Reference Yu and Howard2012), the mapping between the physical plane $(x,y)$ and the mapped plane $({\it\xi},{\it\eta})$ , is given by the transformation functions

where $kh$ , and the coefficients $b_{j}$ and $c_{j}$ are determined by the channel wall profile. The Cauchy–Riemann conditions, $\partial x/\partial {\it\xi}=\partial y/\partial {\it\eta}$ and $\partial x/\partial {\it\eta}=-\partial y/\partial {\it\xi}$ , can be readily verified. The centreline $y=0$ is mapped into ${\it\eta}=0$ . The periodicity of the problem is preserved, as the map is periodic in $x$ and in ${\it\xi}$ with the same period ${\rm\pi}$ . By requiring $y=y_{s}(x)$ to be mapped into a straight line ${\it\eta}=kh$ , we get

with

Averaging (2.9) over a spatial period, recalling that $f(x)$ has zero average in $x$ and using (2.10) to relate $\text{d}x$ and $\text{d}{\it\xi}$ , we obtain

Solving (2.9)–(2.11), we determine $kh$ , $b_{j}$ and $c_{j}$ for the given profile $f(x)$ . Since $y$ is antisymmetric and $x$ is symmetric in ${\it\eta}$ , the opposite channel wall at $y=-y_{s}$ is mapped into ${\it\eta}=-kh$ . Under the transformation, the centreline remains to be a straight line at ${\it\eta}=0$ , while the corrugated channel walls are stretched to become flat surfaces at ${\it\eta}=\pm kh$ . It is thus clear that $kh$ is the dimensionless half channel width in the mapped plane, in the absence of wall movement. With the vibrations, the instantaneous wall position $y(x,t)=y_{s}(x)+{\it\alpha}{\it\beta}\cos t$ is mapped into a curve ${\it\eta}={\it\eta}_{sw}({\it\xi},t)$ , i.e. given $\left[x,y(x,t)\right]$ , the inverse of the map (2.7) and (2.8) gives $\left[{\it\xi},{\it\eta}_{sw}({\it\xi},t)\right]$ . For relatively small amplitude of oscillations, this curve should be fairly close to ${\it\eta}=kh$ .

The vorticity equation (2.3) is transformed into

where

and the Jacobian of transformation is

Since

the boundary conditions at the channel wall are written from (2.6) as

From (2.6), the boundary conditions at the centreline ${\it\eta}=0$ are

2.2. The sawtooth wall profile

An iterative algorithm for calculating the map is given in Yu & Howard (Reference Yu and Howard2012) which can determine the coefficients $b_{j}$ and $c_{j}$ , and $kh$ with fairly good accuracy and efficiency, even for non-smooth functions such as square wave and sawtooth profiles. In this study, we shall smooth the ‘conventional’ sawtooth profile, avoiding the sharp corners and jumps thus removing the discontinuity from the function and its derivatives. This is to avoid the complications in the perturbation analysis, as follows, rather than any concerns on the computation of the map. The procedure of obtaining a smooth sawtooth profile that is exact to a prescribed map is as follows.

In one period $-{\rm\pi}/2\leqslant x\leqslant {\rm\pi}/2$ , an antisymmetric function for the ‘conventional’ sawtooth profile is written as

where $q$ controls the skewness of the sawtooth. When $q={\rm\pi}/2$ , $f_{1}(x)$ gives a symmetric sawtooth profile. When $q=0$ , $f_{1}(x)$ becomes discontinuous at $x=0$ , having a jump of magnitude 1. The Fourier series is

Using the first few terms of (2.19), we get a preliminary smooth profile $\widetilde{f}_{1}(x)$ for which we shall construct the conformal map satisfying (2.9)–(2.11). If we wish to use $\widetilde{f}_{1}(x)$ as the actual wall profile, we would have to keep as many terms of $b_{j}$ and $c_{j}$ as necessary, depending on the accuracy specified. We can also turn the problem around: we truncate the map just constructed to the first few terms, treating it as the new map and subsequently determining the shape of wall corrugations under this new map, i.e. explicitly calculating $\left[x({\it\xi},kh),f(x)\right]$ . In this way, we obtain a sawtooth profile $f(x)$ for which we have an exact map that is expressed in a closed form with finite terms. The channel width $kh_{0}$ in the physical plane, that corresponds to the chosen map and newly determined profile, is then evaluated from (2.11) using the appropriate terms. An example is shown in figure 2, comparing the sawtooth profiles at different stages of approximation. The parameter measuring the skewness $q=0.2{\rm\pi}$ , meaning that the section of steep slope is approximately 20 % of the wavelength. The smooth sawtooth profile, given by the map consisting of the first three terms of $b_{j}$ and $c_{j}$ , is fairly similar to the conventional sawtooth geometry given by (2.18), except for its broadened troughs. In this example, the mean half channel width is $kh_{0}=0.3130$ ; with ${\it\epsilon}_{b}=1.0$ , the narrowest channel width at a sawtooth peak is approximately $y=0.17$ from the centreline, and the maximum width at a sawtooth trough is approximately $y=0.44$ .

Figure 2.

Sawtooth profiles at $y_{s}=kh_{0}\left[1+{\it\epsilon}_{b}f(x)\right]$ : ——, conventional sawtooth $f(x)=f_{1}(x)$ given by (2.18) with $q=0.2{\rm\pi}$ , ${\it\epsilon}_{b}=1.0$ , $kh_{0}=0.3142$ ( ${\it\lambda}_{s}=10h_{0}$ ); ⋯⋯, smooth profile $f(x)=\widetilde{f}_{1}(x)$ given by the first five terms of (2.19), for which $b_{j}$ and $c_{j}$ are computed; –  –  –, $f(x)$ given by the map which consists of the first three terms of $b_{j}$ and $c_{j}$ , ${\it\epsilon}_{b}=1.0$ , $kh_{0}=0.3130$ ; — ⋅ —, $f(x)$ plotted in terms of ${\it\xi}$ .

3.1. The leading order solution

To the lowest order approximation, the fluid flow is primarily driven by the wall velocity component normal to the corrugation surface, cf. (3.13a ), as a result of the sawtooth slope being of $O({\it\beta})$ . The solution is similar to that of an ordinary Stokes boundary layer, except for a parametric dependence on ${\it\xi}$ , representing the effects of wall sawteeth. The solution is written as ${\it\psi}_{0}=\widehat{{\it\psi}}_{0}({\it\xi},\widetilde{{\it\eta}})\text{e}^{-\text{i}t}+\text{c.c.}$ , where

$A\equiv (1-\text{i}){\it\sigma}^{-1}$ , and $C({\it\xi})$ is to be determined. It can be readily checked that (3.19) satisfies (3.11), (3.12) and (3.13b ). To satisfy (3.13a ), $\partial \widehat{{\it\psi}}_{0}/\partial {\it\xi}=(1/2)\text{i}x_{{\it\xi}}|_{w}$ at $\widetilde{{\it\eta}}={\it\gamma}$ , which upon integrating in ${\it\xi}$ leads to

where $x_{w}({\it\xi})\equiv x({\it\xi},kh)$ using (2.7). The constant of integration $C_{0}$ is in general complex. Since $x_{w}({\it\xi})$ measures the distance in $x$ following the corrugated wall, $C({\it\xi})$ is overall proportional to $x$ . As the fluid is pushed in and out of the channel due to wall oscillations, the intensity of the flow field is expected to increase outwards from the channel centre towards the ends. This is similar to the boundary-layer solution when the channel walls are flat plates.

Figure 3.

Contours of stream functions in the half plane $y>0$ for an asymmetrical sawtooth wall profile: (a) the leading order flow ${\it\psi}_{0}$ at $t={\rm\pi}/2$ , ${\rm\Delta}{\it\psi}_{0}=0.5$ , $-8.2228\leqslant {\it\psi}_{0}\leqslant 7.4852$ ; (b) ${\it\psi}_{0}$ at $t={\rm\pi}$ near the reversal of the flow, ${\rm\Delta}{\it\psi}_{0}=0.05$ , $-0.5268\leqslant {\it\psi}_{0}\leqslant 0.1813$ ; (c) the induced steady streaming ${\it\alpha}{\it\psi}_{10}$ , ${\it\alpha}{\rm\Delta}{\it\psi}_{10}=0.05$ , $-0.1250\leqslant {\it\psi}_{10}\leqslant 1.0496$ . Dashed lines indicate the negative contour values. Dimensionless parameters: oscillation amplitude ${\it\alpha}=0.2$ and frequency ${\it\sigma}=0.6538$ , sawtooth wavelength ${\it\beta}=0.3130$ and height ${\it\epsilon}_{b}=1.0$ .

The arbitrary constant $C_{0}$ can be determined by imposing lateral boundary conditions at the channel ends. Consider a channel of finite length, whose two ends, $x=x_{0}$ and $x=x_{1}$ , are submerged in reservoirs of the same fluid. Suppose the reservoirs are so large that no significant fluid motions are excited there, and have the same fluid levels so that there is zero imposed pressure difference. Thus, the dynamic pressure $p$ at the two ends must vanish, i.e.  $p=0$ at $x=x_{0},x_{1}$ , or equivalently

The condition of zero mean pressure drop has been similarly used in some previous studies on peristaltic transport (Yin & Fung Reference Yin and Fung1971; Takagi & Balmforth Reference Takagi and Balmforth2011). The pressure gradient along channel is obtained from the momentum equation. Since the Stokes boundary layer follows the sawtooth profile, the pressure gradient $\partial p/\partial x$ varies both in $x$ and $y$ even at the leading order approximation. On the other hand, lines $y=\text{constant}=\bar{y}$ are in general curves in the $({\it\xi},{\it\eta})$ plane. Thus, to carry out the integration in (3.21), one must invert the map (2.7) and (2.8), finding the curve in ( ${\it\xi},{\it\eta}$ ) that corresponds to $y=\bar{y}$ . This is in general not convenient for an arbitrary choice of $\bar{y}$ . However, the centreline $y=0$ corresponds to ${\it\eta}=0$ , along which $\text{d}x=x_{{\it\xi}}|_{{\it\eta}=0}\,\text{d}{\it\xi}$ . This greatly simplifies the work to deal with the pressure condition (3.21).

Upon integrating (3.21) along ${\it\eta}=0$ , we obtain the condition for determining the constant $C_{0}$ , i.e. 

where ${\it\xi}_{0}(0)$ and ${\it\xi}_{1}(0)$ correspond, respectively, to $x_{0}$ and $x_{1}$ at ${\it\eta}=0$ . For details, a reader is referred to appendix B. As an example, contours of stream function ${\it\psi}_{0}$ are shown in figure 3(a,b) at different phases of an oscillation cycle, for the sawtooth profile (dashed line) in figure 2. The increase of flow intensity towards the two ends of the channel is particularly noted. Near the time of flow reversal, vortices are formed on the steeper sides of the teeth. Close to the channel ends, the vortices are strong, rotating in opposite senses.

3.2. The rectified steady flow

At $O({\it\alpha})$ , the stream function ${\it\psi}_{1}={\it\psi}_{10}+(\widehat{{\it\psi}}_{11}\text{e}^{-\text{i}t}+\widehat{{\it\psi}}_{12}\text{e}^{-2\text{i}t}+\text{c.c.})$ , in view of (3.14) and (3.17). The zeroth and second time harmonics, ${\it\psi}_{10}$ and $\widehat{{\it\psi}}_{12}$ , are induced by the nonlinear convective inertia of the vorticity, as well as its interaction with the wall movement. Owing to the sawtooth slope, the $y$ velocity of wall motion has a component tangential to the corrugated surface. This is the source for $\widehat{{\it\psi}}_{11}$ . The focus here is ${\it\psi}_{10}$ , the induced steady streaming. From (3.14) to (3.18), the equations for ${\it\psi}_{10}$ are written, as follows:

where

arises from the time average of nonlinear inertia, and

Here $\ast$ denotes the complex conjugate and $C^{\prime }=\text{d}C/\text{d}{\it\xi}$ . Owing to the viscosity, the fluid velocity near the wall has a phase shift with respect to the wall velocity. This leads to the correlation between $\partial ^{2}{\it\psi}_{0}/\partial \widetilde{{\it\eta}}^{\,2}$ and the wall displacement ${\rm\Delta}{\it\eta}$ , which is the physical meaning of $B({\it\xi})$ . Derivations of (3.26) and (3.27) are detailed in appendix C.

The solution, that satisfies (3.23) and (3.24), is

where

and $f_{5}$ and $f_{6}$ are to be determined by satisfying the wall boundary conditions. It follows from (3.25b ) that ${\it\psi}_{10}=Q$ at $\widetilde{{\it\eta}}={\it\gamma}$ , where $Q$ is an arbitrary constant, indicating that the undisturbed channel wall position is effectively a streamline in the sense of time average. Thus, from (3.25a,b ) we get where

The physical interpretation of $Q$ is as follows. The time-averaged volume flow rate through a cross-section of the channel is a constant, independent of $x$ , as the fluid is incompressible. Denoting the time average by overlines, the net volume flow rate is

noting the boundary conditions (2.5a ) and (2.6a ). Thus, to the lowest order approximation $2{\it\alpha}Q$ represents the net volume flow rate averaged over one cycle, at which the fluid is pumped from one end to another through the channel. By satisfying the imposed condition of zero pressure drop between the channel ends, we obtain the condition for determining $Q$ .

where

Algebraic details can be found in appendix B. Once $Q$ is determined, $f_{5}$ and $f_{6}$ follow from (3.30) and (3.31), completing the solution ${\it\psi}_{10}$ . The flow pattern of the induced steady streaming ${\it\alpha}{\it\psi}_{10}$ is shown in figure 3(c). The overall steady flow in the positive $x$ direction is observed. Steady circulations of size smaller than sawtooth wavelength are seen to attach to the teeth, with their strengths increasing as we move toward the channel ends. The net transport of fluid in the $+x$ direction is at a rate ${\it\alpha}Q=0.3037$ (over half channel width).

Figure 4.

Time averaged volume flow rate ${\it\alpha}Q$ varying with phase angle ${\it\theta}$ which indicates the left end position: (a) symmetrical sawtooth with $q=0.5{\rm\pi}$ . (b) Asymmetrical sawtooth with $q=0.2{\rm\pi}$ . Parameters: oscillation amplitude ${\it\alpha}=0.3$ , sawtooth wavelength ${\it\beta}=0.3130$ and height ${\it\epsilon}_{b}=1.0$ , and channel length $N{\rm\pi}$ , $N=10$ . Oscillation frequencies: ▵, ${\it\sigma}=0.6524$ ; ○, ${\it\sigma}=0.8008$ ; ▫, ${\it\sigma}=1.1325$ ; ●, ${\it\sigma}=1.6016$ .

4.1. Effects of flow conditions at channel ends

If the left end is taken at $x_{0}+{\it\theta}$ (and the right end then at $N{\rm\pi}+x_{0}+{\it\theta}$ ), where ${\it\theta}$ measures the phase of a sawtooth wave, by varying ${\it\theta}$ over a range of ${\rm\pi}$ we cover all of the possibilities. Figure 4(a) shows the net pumping rate ${\it\alpha}Q$ varying with ${\it\theta}$ for a symmetrical sawtooth $q={\rm\pi}/2$ . When the ends are located on a peak ( ${\it\theta}=-{\rm\pi}/4$ ) or trough ( ${\it\theta}=-3{\rm\pi}/4$ , ${\rm\pi}/4$ ) of the symmetrical sawtooth, the left–right symmetry is restored and ${\it\alpha}Q$ is identically zero, as expected; but for other locations, ${\it\alpha}Q\neq 0$ . Here the net non-zero pumping can only be caused by the different geometries at the two channel ends. Figure 4(b) shows the case for an asymmetrical sawtooth with $q=0.2{\rm\pi}$ . While these are similar to those in figure 4(a), note that the net pumping is mostly positive; slight negative pumping only occurs for large ${\it\sigma}$ (low-frequency oscillation). This is generally found when asymmetrical sawteeth are oriented as in figure 2. As the oscillation frequency increases ( ${\it\sigma}$ decreases), the fluid velocity becomes greater, and so the nonlinearity is stronger. Effects of channel ends and sawtooth asymmetry are therefore both enhanced. In the case of symmetrical sawtooth, changing $N$ does not alter the graphs in figure 4(a). In the case of asymmetrical sawtooth, however, ${\it\alpha}Q$ changes in magnitude for different $N$ ; specifically, the positive pumping increases and the negative pumping decreases (in magnitude) with $N$ , indicating the cumulative effects of sawteeth on breaking the left–right symmetry of the system. With $N=1$ or 2 asymmetrical sawteeth, the negative minimum and positive maximum of the curve ${\it\alpha}Q$ versus ${\it\theta}$ are comparable in magnitude, with the latter being slightly greater.

The flow intensity increases towards the ends, overall proportional to the distance away from the channel centre, cf. § 3.1. As the channel length is increased, the flow velocity and vortex activity become stronger, producing a greater difference between the flow conditions at the two ends, if a difference in the end geometries is present. On the other hand, the total wall friction is also greater for longer channels. In the absence of sawtooth asymmetry, a state of rectified steady flow is established which provides an overall viscous friction necessary to balance the effects of channel ends such that the mean pressure drop is zero (as imposed). The volume flow rate of this steady state, a manifestation of the spatially averaged velocity, is invariant to the length $N$ , given the end geometries and oscillation conditions. As a loose comparison, this is analogous to the problem of a thin layer of viscous fluid on a slope, in which the gravity–friction balanced steady flow is independent of the length of the slope. The cumulative effects of sawtooth asymmetry are anticipated, and discussed in the section that follows.

The flow patterns of the induced streaming ${\it\psi}_{10}$ are shown in figure 5 for the case of symmetrical sawtooth, using different end positions. For ${\it\theta}=-{\rm\pi}/4$ in figure 5(b), both ends are at a peak, and the flow is left–right antisymmetric, having zero spatial average. The end positions corresponding to the negative minimum and positive maximum pumping rate in figure 4(a) are exactly the mirror reflection of each other, and so are the flow patterns, cf. figure 5(a,c). It is found that the effects of entrance and flow conditions in general induce a net pumping directed towards the end that is closer to a sawtooth peak. Because the sawtooth height is large, the local channel width at a sawtooth peak is much smaller than that at a trough. Therefore, a small change in the phase ${\it\theta}$ near a sawtooth peak results in a larger relative change in the local channel width than a similar change of ${\it\theta}$ near a trough. This perhaps is the reason that the location of a peak is more sensitive in determining the direction of net pumping caused by the end flow conditions. It also explains the sharp variation in ${\it\alpha}Q$ between the positions ${\it\theta}$ corresponding to the negative minimum and positive maximum ${\it\alpha}Q$ .

Figure 5.

Contours of the stream function ${\it\alpha}{\it\psi}_{10}$ for a symmetrical sawtooth wall profile. The left end position: (a) ${\it\theta}=-1.0996$ , $-0.5933\leqslant {\it\alpha}{\it\psi}_{10}\leqslant 0.1958$ and ${\it\alpha}Q=-0.2355$ ; (b) ${\it\theta}=-0.7854$ (i.e.  $-{\rm\pi}/4$ ), $-0.3773\leqslant {\it\alpha}{\it\psi}_{10}\leqslant 0.3773$ and ${\it\alpha}Q=0.0$ ; (c) ${\it\theta}=-0.4712$ , $-0.1958\leqslant {\it\alpha}{\it\psi}_{10}\leqslant 0.5933$ and ${\it\alpha}Q=0.2355$ . The contour increment ${\it\alpha}{\rm\Delta}{\it\psi}_{10}=0.05$ . Parameters: oscillation amplitude ${\it\alpha}=0.3$ and frequency ${\it\sigma}=0.8008$ , sawtooth wavelength ${\it\beta}=0.3130$ and height ${\it\epsilon}_{b}=1.0$ , and $N=5$ for the channel length.

4.2. Net pumping rate

In the discussion that follows, the channel ends have been placed at or near the positions corresponding to the maximum ${\it\alpha}Q$ in figure 4(b). The entrance and exit geometries at the channel ends are the same as indicated in figure figure 3.

Given a sawtooth profile and oscillation amplitude, the net pumping rate ${\it\alpha}Q$ increases rapidly as ${\it\sigma}$ decreases (i.e. as the oscillation frequency increases or the fluid viscosity decreases), as shown in figure 6(a). The rate of increase is approximately exponential. Holding ${\it\sigma}$ fixed, the pumping rate increases linearly with the oscillation amplitude ${\it\alpha}$ , cf. figure 6(b), with the slope of the linear lines being $Q$ and increasing with $N$ . From (3.35), $Q$ is a function of channel length $N$ , frequency ${\it\sigma}$ and sawtooth geometry. Noting that $a$ appears in the normalization for ${\it\psi}$ , cf. (2.2), the dimensional pumping rate increases quadratically with oscillation amplitude. It is clear from figure 6 that the channel length, in terms of the number $N$ of sawteeth, has strong influence on the net pumping, in particular for oscillations at high frequencies and large amplitudes.

Figure 6.

Effects of oscillation on the net pumping rate ${\it\alpha}Q$ (for half channel width) for an asymmetry sawtooth with $q=0.2{\rm\pi}$ , wavelength ${\it\beta}=0.3130$ and height ${\it\epsilon}_{b}=1.0$ :(a) ${\it\alpha}Q$ versus oscillation frequency ${\it\sigma}$ for amplitude ${\it\alpha}=0.3$ ; (b) ${\it\alpha}Q$ versus ${\it\alpha}$ for ${\it\sigma}=0.9247$ . For channel lengths: ○, $N=15$ ; ▫, $N=10$ ; ▵, $N=5$ .

Figure 7.

Effects of sawtooth geometry on the net pumping rate ${\it\alpha}Q$ (for half channel width) for oscillation amplitude ${\it\alpha}=0.3$ and frequency ${\it\sigma}=0.8953$ . (a) Plots of ${\it\alpha}Q$ versus $N$ . Sawtooth height ${\it\epsilon}_{b}=1.0$ , and wavelengths: ●, ${\it\beta}=0.2090$ ( ${\it\lambda}_{s}\simeq 15h_{0}$ ); ▵, ${\it\beta}=0.2506$ ( ${\it\lambda}_{s}\simeq 12.5h_{0}$ ); ○, ${\it\beta}=0.3130$ ( ${\it\lambda}_{s}\simeq 10h_{0}$ ); ▫, ${\it\beta}=0.4166$ ( ${\it\lambda}_{s}\simeq 7.5h_{0}$ ). (b) Plots of ${\it\alpha}Q$ versus ${\it\epsilon}_{b}$ for ${\it\beta}\simeq 0.31$ : ○, $N=15$ ; ▫, $N=10$ . The skewness parameter for the sawtooth $q=0.2{\rm\pi}$ .

The cumulative effect of sawteeth is illustrated in figure 7(a), in which ${\it\alpha}Q$ increases with $N$ , fixing all other parameters. The curves match with polynomial fitting of order two. When we reduce ${\it\beta}$ , holding ${\it\epsilon}_{b}$ fixed, we effectively make the sawtooth more gently sloped while keeping the same sawtooth height relative to the channel width. As a result, ${\it\alpha}Q$ decreases. For sufficiently small $N$ , the effects of sawtooth asymmetry on the flow cannot be fully developed during any half cycle of an oscillation due to the short channel length, and are reversed during the other half of the cycle as the flow directions are reversed. Thus, the sawtooth asymmetry is insufficient to produce a net transport of fluid when there are only a few sawteeth in the channel. In such a case, the net pumping rate ${\it\alpha}Q$ is dominantly influenced by the entrance and exit flow conditions at the channel ends. This expectation is confirmed by the result in figure 7(a) that ${\it\alpha}Q$ becomes independent of ${\it\beta}$ as $N$ is reduced while keeping the end positions fixed with respect to the phase of the sawtooth wave. Given an oscillation condition, ${\it\alpha}Q$ increases with the sawtooth height ${\it\epsilon}_{b}$ , following a polynomial fitting of order four within the range of ${\it\epsilon}_{b}$ explored, cf. figure 7(b). We note that ${\it\beta}$ varies slightly from $0.3141$ to $0.3120$ , as we fix the sawtooth wavelength while increasing the height ${\it\epsilon}_{b}$ from 0.4 to 1.2. This is due to the fact that we choose the map and adapt the wall geometry subsequently.

To fix the idea, let us take for example the sawtooth wavelength ${\it\lambda}_{s}=5$  mm. For the sawtooth profile (dashed curve) in figure 2, ${\it\beta}=0.3130$ and ${\it\epsilon}_{b}=1.0$ correspond to a half channel width $h_{0}=0.4982$  mm and sawtooth height $h_{s}=0.4650$ mm. For water, ${\it\nu}=1\ \text{mm}^{2}\ \text{s}^{-1}$ . From figure 6(a), ${\it\alpha}Q=0.6863$ for ${\it\alpha}=0.3$ , ${\it\sigma}=0.8953$ , $N=15$ . Thus, in a channel of length 75 mm, when the sawtooth channel walls oscillate at an amplitude $a=0.1495$  mm and frequency $f=1.60\ \text{Hz}$ , the fluid is pumped in the $+x$ direction at a rate $2\times ({\it\alpha}Q)a{\it\omega}k^{-1}=197.0\ {\rm\mu}\text{L}\ \text{min}^{-1}\text{mm}^{-1}$ wall width. If the sawtooth wavelength is increased to ${\it\lambda}_{s}=10$  mm, subsequently the half channel width $h_{0}=0.9964$  mm and sawtooth height $h_{s}=0.930$  mm. With the oscillation amplitude of 0.2990 mm but frequency of 0.4 Hz (keeping ${\it\sigma}=0.8953$ ), the same pumping rate can be achieved in a channel of length 150 mm. Similarly, we can also reinterpret the case by taking ${\it\lambda}_{s}=2.5$  mm and reducing all other lengths by a factor of two but increasing the frequency to $f=6.4$  Hz.

In view of the discussion in § 4.1, the effects of entrance and exit flow conditions have minimum contribution to the net directional pumping when both ends are placed at a sawtooth peak or trough. On the other hand, to maximize the net pumping in the positive $x$ direction for the sawteeth oriented as indicated in figure 2, it is in one’s interest to keep a peak closer to the right end from inside the channel. Starting with the end positions as indicated in figure 3 which give the maximum positive ${\it\alpha}Q$ seen in figure 4(b), if the right end is moved outward (corresponding to increasing ${\it\theta}$ ), the pumping rate tends to be reduced but still positive; however, if the right end is moved inward (corresponding to decreasing ${\it\theta}$ ), ${\it\alpha}Q$ decreases rapidly and could even become negative depending on ${\it\sigma}$ , as the sawtooth peak previously close to the right end is disappearing from the channel and a sawtooth peak appears at the left end. The effects of ends is less important for long channels in which the pumping rate and direction are primarily determined by the effects of sawtooth asymmetry. However, very long channels may not be of practical interest considering the power that is needed to vibrate the channel walls against incompressible fluids.

At the time of writing, the experimental results referred to Thiria & Zhang (Reference Thiria and Zhang2010) have not been published. While the pumping direction is not explicitly noted in the report, a reference in Dupeux et al. (Reference Dupeux, Le Merrer, Lagubeau, Clanet, Hardt and Quéré2011) indicates that the net transport of fluid in Thiria & Zhang’s experiment is in the $+x$ direction if the sawteeth are oriented as in figure 2. A private communication with Professor R. Camassa seems to confirm the author’s interpretation of the pumping direction in the experiment.

4.3. Power input and rate of viscous dissipation

To the lowest order approximation, the rate ${\dot{W}}$ at which the work is delivered to the channel wall from outside means is

averaged over the period of oscillation. Here ${\it\tau}_{ny}$ is the $y$ component of the normal stress at the wall. When the channel walls are flat plates, ${\it\tau}_{ny}={\it\tau}_{yy}=0$ . Since there is zero mean pressure drop between the channel ends, the rate at which the useful energy is stored in the fluid is identically zero. Thus, the rate of work delivered to the channel wall must be balanced entirely by the rate of viscous dissipation ${\dot{D}}$ in the fluid,

This situation, ${\dot{W}}={\dot{D}}$ , similarly occurs in Takagi & Balmforth (Reference Takagi and Balmforth2011), in which pumping of viscous fluid is induced by steady peristaltic waves propagating down an elastic tube. In those cases, the mechanical efficiency based on the rate of useful energy stored in the fluid (flow rate $\times$ pressure drop against pumping) as in Shapiro, Jaffrin & Weinberg (Reference Shapiro, Jaffrin and Weinberg1969), is zero. Takagi & Balmforth defined an alternative pumping efficiency which measures the fluid transport per unit input power. (This definition does not have the appropriate dimensions for efficiency.) The concern, of course, is a different one from that in fluid machinery. Here the question is how much energy is needed to achieve a desired transport of fluid, rather than the energy efficiency of the transport mechanism.

The calculation of ${\dot{D}}$ , or ${\dot{W}}$ can be carried out using the solution in § 3.1, but it is unavoidably tedious, attainable only by numerical evaluations because of the wall geometry. In an averaged sense, it is reasonable to expect that ${\dot{W}}$ could be characterized by ${\dot{W}}_{f}$ for flat channel walls with same oscillation frequency and amplitude, but with a channel width $kh$ seen in the mapped plane to account for the effects of sawtooth shape and height. For simple purposes and an order of magnitude estimate, this may be useful. The analytical expression for ${\dot{W}}_{f}$ is derived in appendix D, for the benefits of readers who may be interested.