2.1 Experimental set-up

In the experiment, we used a rectangular Plexiglas tank divided into three cells by two optical glass plates (figure 1a). The middle cell was the working cell, with the dimensions $L_{x}=2.2~\text{cm}$ in length, $L_{y}=5.0~\text{cm}$ in width and $L_{z}=10.0~\text{cm}$ in height. Deionized water filled the working cell to a depth of $h=6.5~\text{cm}$. Two types of meniscus symmetry were created by selecting the wettability conditions of the glass plates (see figure 1b). The clean optical glass surface was hydrophilic, with a contact angle close to $45^{\circ }$. A hydrophobic boundary was made by applying a coating of hydrophobic nano-$\text{SiO}_{2}$ film (CHEMNANO^®, NC317) to an optical glass surface, creating a contact angle close to $135^{\circ }$. A working cell with two hydrophilic lateral boundaries creates the symmetric setting, while choosing one boundary hydrophobic and the other hydrophilic creates an antisymmetric system. We chose experiments that had contact angles closest to the ideally symmetric and antisymmetric cases. In the chosen symmetric case, the left contact angle is $\unicode[STIX]{x1D703}_{l}=44.01^{\circ }$ ($43.71^{\circ }$, $48.02^{\circ }$) and the right contact angle is $\unicode[STIX]{x1D703}_{r}=48.57^{\circ }$ ($46.18^{\circ }$, $50.75^{\circ }$). In the chosen antisymmetric case, the left contact angle is $\unicode[STIX]{x1D703}_{l}=142.66^{\circ }$ ($137.41^{\circ }$, $146.92^{\circ }$) and the right contact angle is $\unicode[STIX]{x1D703}_{r}=45.84^{\circ }$ ($43.65^{\circ }$, $47.86^{\circ }$). The numbers in parentheses are the 95 % confidence intervals. The contact angle of water with the remaining Plexiglas boundary was close to $90^{\circ }$.

The fluid tank was installed on an electrodynamic shaker (Labworks^® ET-139) that vibrated sinusoidally in the $z$ direction with an excess gravitational acceleration varying from $0.02g_{0}$ to $0.36g_{0}$ and a frequency of $f_{0}=8~\text{Hz}$. The control signal produced from a wavefunction generator was amplified using a linear power amplifier to drive the shaker. An accelerometer mounted on the base plate of the fluid tank (not shown in figure 1) measured the instantaneous acceleration, which was fed back to the wavefunction generator to regulate the control signal. The closed-loop control system reduced error and ensured the stability of the shaker oscillation.

We measured the fluid velocity field in the working cell using a particle imaging velocimetry (PIV) system. Neutrally buoyant hollow glass microspheres of diameter $10~\unicode[STIX]{x03BC}\text{m}$ (DANTEC^®, HGS-10) were seeded in the flow. Because of the effect of surface tension, some of the seeding particles were suspended on the free surface. We used these particles to track the surface deformation (see § 2.2). We found that these surface particles also played a role in enhancing surface damping and surface-deformation-induced streaming. The PIV particles in the system were illuminated from the right using a continuous, diode-pumped solid-state (DPSS) laser, creating a 1 mm thick light sheet that passed through the $y=2.5~\text{cm}$ plane. PIV images that cover a region from the surface to a fluid depth of approximately 3 cm were captured by a high-speed camera (Photron UX100) placed perpendicular to the laser sheet, at a frame rate of 80 Hz, achieving a time resolution of 10 frames per oscillation period. The surface oscillation caused by the primary flow is estimated by measuring the surface profile deformation in one period. Two-dimensional velocity maps for the secondary flow were obtained by cross-correlating two images with the same phase in two consecutive periods using LaVision^® PIV software. Each velocity vector was calculated from interrogation windows spanning $32~\text{pixel}\times 32~\text{pixel}$, with 50 % overlap of neighbouring sub-windows. Each vector covers a region of $16~\text{pixel}\times 16~\text{pixel}$ and there are $80\times 64$ velocity vectors per frame, achieving a spatial resolution of 0.34 mm (one order in magnitude smaller than the capillary length scale).

The laser light sheet may heat up the fluid on the right side of the working cell and give rise to an overturning circulation. The speed of this overturning flow could be up to the order of $O(10^{-3}~\text{cm}~\text{s}^{-1})$, comparable to the speed of the streaming flow we study. To avoid this laser-induced convection, the two lateral cells were filled with water to reduce the thermal heating of the fluid in the working cell. With this thermal protection, the speed of the overturning flow was reduced to $O(10^{-5}~\text{cm}~\text{s}^{-1})$ and was negligible.

2.2 Observation of the surface deformation

The time-varying acceleration of the fluid tank in the laboratory frame is $a(t)=a_{0}\sin \unicode[STIX]{x1D711}(t)$, where $a_{0}$ is the forcing amplitude and $\unicode[STIX]{x1D711}(t)=2\unicode[STIX]{x03C0}f_{0}t$ is the oscillation phase. From Newton’s third law, the effective gravity felt by the fluid in the tank is

$$\begin{eqnarray}g(t)=-[g_{0}+a_{0}\sin \unicode[STIX]{x1D711}(t)],\end{eqnarray}$$

in which $g_{0}$ is the gravitational acceleration constant. When $\unicode[STIX]{x1D711}=\unicode[STIX]{x03C0}/2$, the fluid tank is at its highest position with maximum super-gravity; when $\unicode[STIX]{x1D711}=3\unicode[STIX]{x03C0}/2$, the tank is at its lowest position with maximum sub-gravity. To force the system without exciting the Faraday instability, we select $f_{0}=8.0~\text{Hz}$ and $0.02\leqslant a_{0}/g_{0}\leqslant 0.36$ (see appendix B). Thus the periodic deformation of the liquid–air interface excites the cavity modes, giving rise to standing meniscus waves. We observe that such a surface deformation mode is strongly dependent on the geometry of the static meniscus. To quantify the relation between this surface deformation mode and the shape of the static meniscus, we measure the surface profile in different oscillation phases from the PIV images. It is found that a large proportion of the liquid–air interface is clearly visible due to seeding particles accumulated at the interface, which are illuminated by the laser sheet (figure 2a,c). The free surface is tracked using light intensity: particles are densely populated on the free surface so the light intensity is maximal there. Using this method, we extract the surface profiles $\unicode[STIX]{x1D701}(x)$ at four oscillatory phases, $\unicode[STIX]{x1D711}=0$, $\unicode[STIX]{x03C0}/2$, $\unicode[STIX]{x03C0}$ and $3\unicode[STIX]{x03C0}/2$, as shown in figure 2(b) and 2(d).

Figure 2.

Images and surface profiles of meniscus shape deformation at $\unicode[STIX]{x1D711}=0$, $\unicode[STIX]{x03C0}/2$, $\unicode[STIX]{x03C0}$ and $3\unicode[STIX]{x03C0}/2$, with $f_{0}=8~\text{Hz}$ and $a_{0}=0.2g_{0}$. (a) A PIV image of the vibrating surface in the symmetric configuration. (b) The extracted surface profiles corresponding to (a). (c) A PIV image of the vibrating surface in the antisymmetric configuration. (d) The extracted surface profiles of symmetric scenario corresponding to (c). The insets in (b) and (d) show the horizontally averaged surface deformation amplitude $\langle \unicode[STIX]{x1D701}\rangle$ as a function of $a_{0}/g_{0}$. The symbols in the insets are experimental measurements and the solid line is a linear regression.

The meniscus oscillates at the forcing frequency $f_{0}$. The response of meniscus wave amplitude $\unicode[STIX]{x1D701}(x)$ to the forcing acceleration amplitude $a_{0}/g_{0}$ is estimated by the horizontally averaged surface deformation amplitude $\langle \unicode[STIX]{x1D701}\rangle$, which is defined by

$$\begin{eqnarray}\langle \unicode[STIX]{x1D701}\rangle =\overline{|\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D711}=\unicode[STIX]{x03C0}/2}-\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D711}=3\unicode[STIX]{x03C0}/2}|},\end{eqnarray}$$

where $\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D711}=\unicode[STIX]{x03C0}/2}$ and $\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D711}=3\unicode[STIX]{x03C0}/2}$ are the surface profiles captured at $\unicode[STIX]{x1D711}=\unicode[STIX]{x03C0}/2$ and $\unicode[STIX]{x1D711}=3\unicode[STIX]{x03C0}/2$, respectively; the overline denotes an average in the $x$ direction. The mean amplitude of the meniscus oscillation $\langle \unicode[STIX]{x1D701}\rangle$ grows linearly with the forcing amplitude (see the insets figure 2b,d). The scale for the speed of the primary flow is $\unicode[STIX]{x1D714}\langle \unicode[STIX]{x1D701}\rangle$, which ranges from $0.5~\text{cm}~\text{s}^{-1}$ to $5~\text{cm}~\text{s}^{-1}$. The Reynolds number for the primary flow, $\mathit{Re}=\unicode[STIX]{x1D714}\langle \unicode[STIX]{x1D701}\rangle L/\unicode[STIX]{x1D708}$, therefore ranges from 110 to 1100, and we can infer that the effect of viscosity on the primary oscillation is weak. In the symmetric case, the excited mode is symmetric and has two nodes. At maximum super-gravity ($\unicode[STIX]{x1D711}=\unicode[STIX]{x03C0}/2$) the liquid surface bulges upwards in the middle of the cell, analogous to the phenomenon of the liquid jet observed by Antkowiak et al. (Reference Antkowiak, Bremond, Le Dizès and Villermaux2007). At maximum sub-gravity ($\unicode[STIX]{x1D711}=3\unicode[STIX]{x03C0}/2$), the meniscus bends downwards to its greatest extent. Between these two states, the fluid flows periodically between the lateral boundary and the middle of the cell. In the antisymmetric case, an oscillatory mode with one node at the centre ($x=L/2$) is instead excited. When the system is at maximum sub-gravity ($\unicode[STIX]{x1D711}=3\unicode[STIX]{x03C0}/2$), the fluid surface forms an ‘S’ shape, with the convex part on the hydrophobic side ($x<L/2$) and the concave part on the hydrophilic side ($x>L/2$). When the system is at maximum super-gravity ($\unicode[STIX]{x1D711}=\unicode[STIX]{x03C0}/2$), the fluid surface approaches a straight slope. Between these two states, the fluid sloshes between the two lateral boundaries. No observable surface oscillation is found in the $y$ direction. Representing the mode number as $(k,l)$, where $k$ and $l$ are positive integers denoting the mode number on $x$ and $y$ axes, respectively, the symmetric and antisymmetric configurations give rise to mode $(2,0)$ and $(1,0)$ waves, respectively.

2.3 Observation of the streaming motion

The streaming circulation is observed when the PIV images are broadcast stroboscopically. The streaming circulation is one order of magnitude weaker than the instantaneous primary flow. Once the circulation forms, it persists as long as the forcing oscillation continues. During the experiment, we collected the PIV images one minute after the forcing was turned on, when a steady streaming circulation had been established. The flow velocity and the vertical scale of the circulations, for both wettability configurations, appear to increase if the oscillation amplitude $a_{0}$ increases. The streaming circulation has distinct structures in symmetric and antisymmetric cases. The streaming circulation patterns are taken when $\unicode[STIX]{x1D711}\approx \unicode[STIX]{x03C0}$, when the free surface is close to its equilibrium state.

In the symmetric case (figure 3), the secondary circulation has a stable four-vortex structure, with clockwise (anticlockwise) rotation near the right (left) lateral boundary. When $a_{0}\sim 0.1g_{0}$, the central vortices are not clearly visible in the PIV image. As the forcing gets stronger, the central vortices become apparent with approximately circular streamlines. At the strongest forcing amplitude, the boundary vortices are well developed, whereas the central vortices become suppressed and asymmetric.

Figure 3.

Velocity fields of the streaming flows overlying the raw PIV image in the symmetric case for three values of $a_{0}$. For each case, the driving frequency is $f_{0}=8~\text{Hz}$. The liquid–air interface has been highlighted manually. The streaming flows in (a), (b) and (c) are computed at the oscillation phases $\unicode[STIX]{x1D711}=9/10\unicode[STIX]{x03C0}$, $\unicode[STIX]{x1D711}=13/10\unicode[STIX]{x03C0}$ and $\unicode[STIX]{x1D711}=13/10\unicode[STIX]{x03C0}$, respectively.

In the antisymmetric case, the secondary flow field has a dipole circulation structure with a weaker anticlockwise circulation near the hydrophobic side and a stronger clockwise circulation near the hydrophilic side (figure 4). When the forcing is weak, $a_{0}\sim 0.1g_{0}$, the anticlockwise circulation is only partially visible in the PIV image, but a strong leftward velocity is found below the free surface. As the forcing amplitude increases, the anticlockwise circulation becomes more dominant and downward motion becomes more prominent near the hydrophobic side.

Figure 4.

Velocity fields of the streaming flows overlying the raw PIV image in the antisymmetric case for four values of $a_{0}$. For each case, the driving frequency is $f_{0}=8~\text{Hz}$. The liquid–air interface has been highlighted manually. The streaming flows in (a), (b), (c) and (d) are computed at the oscillation phases $\unicode[STIX]{x1D711}=5/6\unicode[STIX]{x03C0}$, $\unicode[STIX]{x1D711}=9/10\unicode[STIX]{x03C0}$, $\unicode[STIX]{x1D711}=9/10\unicode[STIX]{x03C0}$ and $11/10\unicode[STIX]{x03C0}$, respectively.

Figure 5.

Response of the (a,b) r.m.s. speed, $v_{rms}$, and (c,d) symmetry parameter, $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{0}$, to changes in forcing amplitude $a_{0}/g_{0}$ in (a,c) the laboratory experiments and (b,d) the simulations. The fitted trends in (a) have $\unicode[STIX]{x1D6FD}=2.02$ and $\unicode[STIX]{x1D6FD}=1.97$ for symmetric and antisymmetric cases, respectively, while the fitted trends in (b) both have $\unicode[STIX]{x1D6FD}=2.0$. The symmetry parameter $\unicode[STIX]{x1D6FE}$ is defined in (2.3) and normalized by $\unicode[STIX]{x1D6FE}_{0}=\unicode[STIX]{x1D6FE}|_{a_{0}/g_{0}=0.1}$. The insets in (d) show schematic plots illustrating the streaming velocity distribution under the free surface.

To quantify the dynamical behaviour of the secondary flow, we compute the r.m.s. fluid speed in the observed section:

(2.1)$$\begin{eqnarray}\displaystyle v_{rms}=\frac{1}{A}\sqrt{\iint _{A}(u^{2}+w^{2})\,\text{d}x\,\text{d}z}. & & \displaystyle\end{eqnarray}$$

Here $A$ is the total cross-sectional area of the observed fluid bulk, and $u$ and $w$ are the velocity components in the $x$ and $z$ directions, respectively. As shown in figure 5(a), the kinetic energy density for fully established secondary flow structure $v_{rms}$ scales like

(2.2)$$\begin{eqnarray}\displaystyle v_{rms}\sim (a_{0}/g_{0})^{\unicode[STIX]{x1D6FD}}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6FD}\approx 2$ for both symmetric and antisymmetric cases. This indicates that the secondary circulation is caused by the quadratic nonlinearity of the velocity field, which is consistent with the direct numerical simulations of Carrión et al. (Reference Carrión, Herrada, Montanero and Vega2017). Note that the flow velocity near the boundary is not resolvable by PIV, so the integration in (2.1) excludes the region within 0.034 cm of the lateral boundaries. Further, the camera does not capture the whole water bulk, so the integration covers the region $z>-1.79~\text{cm}$ in the symmetric case and $z>-1.65~\text{cm}$ in the antisymmetric case. To avoid edge effects from the bottom, we limit the maximum forcing amplitude so that the streaming velocity at the bottom edge of the observation window is never more than 10 % of the characteristic secondary flow speed in the bulk.

The spatial symmetry of the secondary circulation pattern is quantified by the symmetry parameter

(2.3)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FE}=\frac{\displaystyle \iint _{x<L/2}\,\text{d}x\,\text{d}z\,|\unicode[STIX]{x1D6FA}(x,z)|/A_{x<L/2}}{\displaystyle \iint _{x>L/2}\,\text{d}x\,\text{d}z\,|\unicode[STIX]{x1D6FA}(x,z)|/A_{x>L/2}}, & & \displaystyle\end{eqnarray}$$

which denotes the ratio of mean vorticity magnitude of the left region over that of the right region of the working cell. In (2.3), $|\unicode[STIX]{x1D6FA}(x,z)|$ is the magnitude of the vorticity, and $A_{x>L/2}$ and $A_{x<L/2}$ are the liquid bulk areas at the right or left side, respectively, of $x=1.1~\text{cm}$ below the static meniscus. Figure 5(b,d) shows the variation of $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{0}$ against $a_{0}/g_{0}$, where $\unicode[STIX]{x1D6FE}_{0}=\unicode[STIX]{x1D6FE}|_{a_{0}/g_{0}=0.1}$ is a reference $\unicode[STIX]{x1D6FE}$ value. The variation of $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{0}$ with forcing amplitude is different in the symmetric and antisymmetric cases. In the symmetric case, the secondary flows remains highly symmetric, with no obvious trend in $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FE}_{0}$. In contrast, in the antisymmetric case, the symmetry parameter $\unicode[STIX]{x1D6FE}$ decreases rapidly as the forcing amplitude increases, then saturates at $\unicode[STIX]{x1D6FE}\approx 0.5\unicode[STIX]{x1D6FE}_{0}$ for large forcing amplitudes.

3.1 Meniscus wave

The meniscus wave is solved by linearization of the ideal fluid equations with a free surface (e.g. Batchelor Reference Batchelor2000). The nonlinear fluid equations are

(3.1)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D719}=0, & \displaystyle\end{eqnarray}$$

(3.2)$$\begin{eqnarray}\displaystyle & \displaystyle \left.\left[\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D719}+\frac{1}{2}(\unicode[STIX]{x1D735}\unicode[STIX]{x1D719})^{2}\right]\right|_{z=\unicode[STIX]{x1D701}}+[g_{0}+a_{0}\sin (\unicode[STIX]{x1D714}t)]\unicode[STIX]{x1D701}=\frac{\unicode[STIX]{x1D70E}}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}_{xx}^{2}\unicode[STIX]{x1D701}}{[1+(\unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D701})^{2}]^{3/2}}, & \displaystyle\end{eqnarray}$$

(3.3)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D701}+\unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D719}|_{z=\unicode[STIX]{x1D701}}\unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D701}=\unicode[STIX]{x2202}_{z}\unicode[STIX]{x1D719}|_{z=\unicode[STIX]{x1D701}}, & \displaystyle\end{eqnarray}$$

(3.4)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}|_{z\rightarrow -\infty }=0, & \displaystyle\end{eqnarray}$$

(3.5)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D719}|_{x=0,L}=0, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D701}=\unicode[STIX]{x1D701}(x,t)$ is the surface elevation and $\unicode[STIX]{x1D719}=\unicode[STIX]{x1D719}(x,z,t)$ is the velocity potential field. The parameter notation is the same as in the previous section. The cell width is $L=2.2~\text{cm}$, $g_{0}=980~\text{cm}~\text{s}^{-2}$ is the gravitational acceleration, $\unicode[STIX]{x1D70C}=1~\text{g}~\text{cm}^{-3}$ is the density of water, $\unicode[STIX]{x1D70E}=72~\text{dyne}~\text{cm}^{-1}$ is the surface tension of water, $a_{0}$ is the forcing acceleration, and $\unicode[STIX]{x1D714}=2\unicode[STIX]{x03C0}f_{0}$ is the forcing angular frequency.

The linearization is done in the following way. We use the capillary length $l_{c}=\sqrt{\unicode[STIX]{x1D70E}/(\unicode[STIX]{x1D70C}g_{0})}=0.27~\text{cm}$ as the characteristic length scale in the $x$ and $z$ directions. The scale of the surface displacement due to the forced oscillation is $a_{0}/\unicode[STIX]{x1D714}^{2}\approx 0.008{-}0.13~\text{cm}$, so the ratio between these two length scales is a small dimensionless number, $\unicode[STIX]{x1D716}=a_{0}/(l_{c}\unicode[STIX]{x1D714}^{2})\approx 0.03{-}0.5$, that scales the linear system response. In the experiment, the dimensionless forcing strength $a_{0}/g_{0}$ has the magnitude range of $0.02{-}0.36$, which is $O(\unicode[STIX]{x1D716})$. The mean static surface slope $\unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D701}_{0}$ approaches zero at $x=1.1~\text{cm}$ and ${\approx}1$ at both boundaries. Nevertheless, to make progress, we assume that $\unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D701}_{0}\sim O(\unicode[STIX]{x1D716})$ everywhere, while acknowledging that this will lead to errors near the boundaries. After scaling the velocity potential, $\unicode[STIX]{x1D719}$, by $g_{0}a_{0}/\unicode[STIX]{x1D714}^{3}$, the solution of the ideal fluid equations (3.1)–(3.5) can be expanded in an asymptotic series in powers of $\unicode[STIX]{x1D716}$:

(3.6)$$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D701}=\unicode[STIX]{x1D701}_{0}+\unicode[STIX]{x1D701}_{1}, & \displaystyle\end{eqnarray}$$

(3.7)$$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D719}=\unicode[STIX]{x1D719}_{0}+\unicode[STIX]{x1D719}_{1}. & \displaystyle\end{eqnarray}$$

The quantities with the subscript 0 are $O(1)$, which represents a situation when the forcing is neglected and depicts the static balance in the fluid bulk. Thus, both $\unicode[STIX]{x1D719}_{0}$ and $\unicode[STIX]{x1D701}_{0}$ are independent of time. The quantities with subscript 1, that is, $O(\unicode[STIX]{x1D716})$, represent the linear wave forced by the vertical oscillation. As the depth is three times larger than the cell width, we apply an infinite-depth bottom boundary condition.

3.1.1 Zeroth-order solution

At $O(1)$, equations (3.1) and (3.2) become the Young–Laplace equation:

(3.8)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{xx}^{2}\unicode[STIX]{x1D701}_{0}=\frac{\unicode[STIX]{x1D70C}g_{0}}{\unicode[STIX]{x1D70E}}(\unicode[STIX]{x1D701}_{0}-z^{\ast }). & & \displaystyle\end{eqnarray}$$

Since the left plate and right plate completely prevent liquid mass exchange, the experiment is done with total fluid volume conserved. The reference elevation,

(3.9)$$\begin{eqnarray}z^{\ast }=-\frac{\unicode[STIX]{x1D70E}}{\unicode[STIX]{x1D70C}g_{0}L}(\unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D701}_{0}|_{x=L}-\unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D701}_{0}|_{x=0})=-\frac{\unicode[STIX]{x1D70E}}{\unicode[STIX]{x1D70C}g_{0}L}[\tan (\unicode[STIX]{x1D703}_{r})-\tan (\unicode[STIX]{x03C0}/2-\unicode[STIX]{x1D703}_{l})],\end{eqnarray}$$

is determined by integrating equation (3.8) in $x$, applying the lateral boundary conditions, and requiring that the horizontal integral of $\unicode[STIX]{x1D701}_{0}$ vanish. The quantities $\unicode[STIX]{x1D703}_{l}$ and $\unicode[STIX]{x1D703}_{r}$ are the contact angles at the left and right boundaries, respectively. Following the experimental set-up, in the symmetric case $\unicode[STIX]{x1D703}_{l}=\unicode[STIX]{x1D703}_{r}=45^{\circ }$, and in the antisymmetric case $\unicode[STIX]{x1D703}_{l}=135^{\circ }$ and $\unicode[STIX]{x1D703}_{r}=45^{\circ }$. The corresponding boundary conditions for (3.8) are written as

(3.10)$$\begin{eqnarray}\displaystyle & \unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D701}_{0}|_{x=0}=-s, & \displaystyle\end{eqnarray}$$

(3.11)$$\begin{eqnarray}\displaystyle & \unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D701}_{0}|_{x=L}=1. & \displaystyle\end{eqnarray}$$

Here, $s$ controls the symmetry of the meniscus; it is 1 in the symmetric case and $-1$ in the antisymmetric case. These boundary conditions and the above zeroth-order equation are satisfied by

(3.12)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D701}_{0}=\frac{1+s\cosh (mL)}{m\sinh (mL)}\cosh (mx)-\frac{s}{m}\sinh (mx)+z^{\ast }, & & \displaystyle\end{eqnarray}$$

where $m=l_{c}^{-1}$. In figure 7, the solutions are compared to the static surface profile measurements for both symmetric and antisymmetric cases – the theoretical result captures the shape of the free surface well.

Figure 6.

The zeroth-order meniscus solution $\unicode[STIX]{x1D701}_{0}$ (solid line) compared to the experimental measurement (dotted line) for (a) symmetric and (b) antisymmetric cases. The $+$ and $-$ signs represent the surplus and deficit of dynamical pressure $p_{d}$, respectively, that arises when the acceleration is changed by $a_{0}\unicode[STIX]{x1D714}\unicode[STIX]{x1D6FF}t$, where $\unicode[STIX]{x1D6FF}t$ is a small time increment. The $p_{d}$ distribution outside (inside) of the bracket is for the situation when $g$ is increased (decreased).

3.1.2 First-order solution

At $O(\unicode[STIX]{x1D716})$, the governing equations and boundary conditions become

(3.13)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D719}_{1}=0, & \displaystyle\end{eqnarray}$$

(3.14)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D719}_{1}|_{z=0}+g_{0}\unicode[STIX]{x1D701}_{1}-\frac{\unicode[STIX]{x1D70E}}{\unicode[STIX]{x1D70C}}\unicode[STIX]{x2202}_{xx}^{2}\unicode[STIX]{x1D701}_{1}=-a_{0}\unicode[STIX]{x1D701}_{0}\sin (\unicode[STIX]{x1D714}t), & \displaystyle\end{eqnarray}$$

(3.15)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{z}\unicode[STIX]{x1D719}_{1}|_{z=0}-\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D701}_{1}=0, & \displaystyle\end{eqnarray}$$

(3.16)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}_{1}|_{z=-\infty }=0, & \displaystyle\end{eqnarray}$$

(3.17)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D719}_{1}|_{x=0,L}=0. & \displaystyle\end{eqnarray}$$

The first-order surface pressure balance (3.14) suggests a physical interpretation for the mechanism of the meniscus wave. Suppose that the system is initially at rest, so that $\unicode[STIX]{x1D701}_{1}=0$, and the vertical acceleration increases by an amount $\unicode[STIX]{x1D6FF}g=a_{0}\unicode[STIX]{x1D714}\unicode[STIX]{x1D6FF}t$ over the background value $g_{0}$. At this instant, the dynamic pressure is

(3.18)$$\begin{eqnarray}p_{d}=-\unicode[STIX]{x1D70C}\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D719}_{1}|_{z=0}=\unicode[STIX]{x1D70C}\unicode[STIX]{x1D6FF}g\unicode[STIX]{x1D701}_{0},\end{eqnarray}$$

so that there is an excess of dynamic pressure where $\unicode[STIX]{x1D701}_{0}>0$ and a deficit where $\unicode[STIX]{x1D701}_{0}<0$. In the symmetric case, this dynamic pressure perturbation will cause the free surface to fall in the centre of the cell and rise on the sides, leading to a wave with two nodes (figure 6a). For the antisymmetric case, the same process will cause the free surface to fall and rise on the left and right, respectively, sides of the cell, so the meniscus wave will have a single node (figure 6b). A decrease in the vertical acceleration will lead to the opposite pattern of dynamic pressure perturbations, resulting in the opposite surface deflection.

Figure 7.

The first-order meniscus solutions $\hat{\unicode[STIX]{x1D701}}_{1}$ (solid line) compared to the experimental measurements (dotted line) when $a_{0}=0.2g_{0}$ and $f_{0}=8~\text{Hz}$ for (a) the symmetric and (b) the antisymmetric cases. The experimental $\hat{\unicode[STIX]{x1D701}}_{1}$ are produced by subtracting the static surface profiles from the oscillating surface profiles captured during the slack phase. The insets show the difference between the theoretical first-order slope and the experimental first-order slope, $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x2202}_{x}\hat{\unicode[STIX]{x1D701}}_{1})$.

The solution that satisfies (3.13)–(3.17) takes the form

(3.19)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}_{1}=\hat{\unicode[STIX]{x1D719}}_{1}\cos (\unicode[STIX]{x1D714}t)=\mathop{\sum }_{n=1}^{\infty }A_{n}\text{e}^{k_{n}z}\cos (k_{n}x)\cos (\unicode[STIX]{x1D714}t), & \displaystyle\end{eqnarray}$$

(3.20)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D701}_{1}=\hat{\unicode[STIX]{x1D701}}_{1}\sin (\unicode[STIX]{x1D714}t)=\mathop{\sum }_{n=1}^{\infty }\frac{A_{n}k_{n}}{\unicode[STIX]{x1D714}}\cos (k_{n}x)\sin (\unicode[STIX]{x1D714}t), & \displaystyle\end{eqnarray}$$

where the eigen-wavenumber $k_{n}=n\unicode[STIX]{x03C0}/L$ and coefficients $A_{n}$ are given by

(3.21)$$\begin{eqnarray}\displaystyle A_{n}=\frac{2\unicode[STIX]{x1D714}a_{0}}{\unicode[STIX]{x1D714}^{2}-\unicode[STIX]{x1D714}_{n}^{2}}\frac{(-1)^{n}+s}{(m^{2}+k_{n}^{2})L}, & & \displaystyle\end{eqnarray}$$

where

(3.22)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}_{n}^{2}=g_{0}k_{n}(1+k_{n}^{2}l_{c}^{2}) & & \displaystyle\end{eqnarray}$$

is the (squared) natural frequency of a capillary–gravity wave with wavenumber $k_{n}$. Note that these wave modes (3.19) and (3.20) are not parametric resonance modes (Faraday modes), but ordinary resonance modes, since they have the same frequency as the forcing. The wave energy is concentrated around those $n$ for which $\unicode[STIX]{x1D714}_{n}$ is close to $\unicode[STIX]{x1D714}$. If $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{n}$, for a given $n$, the capillary–gravity wave can travel integer multiples of the cell length during one forcing cycle. For a particular wave mode $n$ and oscillation frequency $\unicode[STIX]{x1D714}$, the mode amplitude $A_{n}$ is also controlled by the distance between the boundary plates $L$. When the resonance condition $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{n}(L)$ is satisfied by tuning $L$, the mode amplitude $A_{n}\rightarrow \infty$. The mode amplitude $A_{n}$ goes to zero like $L^{-1}$ as $L\rightarrow \infty$ and like $L^{4}$ as $L\rightarrow 0$. The forcing frequency $f_{0}=8~\text{Hz}$ is approximately at the midpoint between two flat-surface resonant frequencies, $f_{1}=6.38~\text{Hz}$ and $f_{2}=10.65~\text{Hz}$, for modes $n=1$ and $n=2$, respectively. If the equilibrium surface were flat, the amplitudes $A_{1}$ and $A_{2}$ would be comparable to each other. However, the (anti)symmetry of the equilibrium surface selects excitations with the corresponding (anti)symmetry: $A_{1}=0$ in the symmetric case and $A_{2}=0$ in the antisymmetric case (see (3.21)). The amplitude $A_{n}$ decays like $O(n^{-5})$, so the series converges quickly. In figure 7, the infinite series is terminated at $n=200$ to produce the estimate of the meniscus wave amplitude $\hat{\unicode[STIX]{x1D701}}_{1}$ for both symmetric and antisymmetric cases.

Figure 8.

The streaming flow velocity along the free surface at $\unicode[STIX]{x1D711}=\unicode[STIX]{x03C0}$, when $a_{0}=0.2g_{0}$ and $f_{0}=8~\text{Hz}$ for (a) the symmetric and (b) the antisymmetric cases. The solid line is the result (3.26) truncated at $m+n=200$, while the dashed line contains contributions from the gravest mode only.

In figure 7, the experimental wave amplitudes are 67 % and 42 % larger than the theoretical values for the symmetric and antisymmetric cases, respectively. The linearized theory is inadequate to produce a more accurate result, because our basic scaling approximation $\unicode[STIX]{x2202}_{x}(\unicode[STIX]{x1D701}_{0}+\unicode[STIX]{x1D701}_{1})\rightarrow 0$ is not satisfied since $\unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D701}_{0}\sim 1$ near the contact angle. The insets in figure 7 show the differences between the theoretical and experimental first-order slopes, $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x2202}_{x}\hat{\unicode[STIX]{x1D701}}_{1})$. The experimental slope is determined by fitting a fifth-order polynomial to the measured $\hat{\unicode[STIX]{x1D701}}_{1}$ profile. We can see that $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x2202}_{x}\hat{\unicode[STIX]{x1D701}}_{1})$ is large near the contact line and zero near the centre of the cell.

The deviation between the experiment and theory may also due to the contact-line dynamics. In the theory, the contact line is assumed to be free-slipping with the liquid body. In the experimental measurement shown in figure 8, the amplitude of the oscillation near the contact line is larger (smaller) than predicted in the symmetric (antisymmetric) case. These interesting behaviours may contribute to the difference between the predicted and observed meniscus wave shapes. Nevertheless, a precise analytical model of the meniscus wave is not the main purpose of this work. In the next section, we will show that the meniscus wave solution obtained here is sufficient to explain the streaming circulation pattern formation.

3.2 Streaming circulation

In this section, we explain the secondary circulation pattern shown in figures 3 and 4. Firstly, we note that the surface appears to play a very important role in streaming circulation generation, since the number of secondary vortices doubles as the dominant mode number of the meniscus wave flow doubles. Moreover, the nodes of the meniscus wave are above upwelling flow, while the antinodes cap downwelling flows. The spatial relationships between the meniscus wave and the streaming flow indicate that the streaming flow is generated by the meniscus wave itself. This streaming flow affects the bulk through viscous momentum transfer, so as to generate a secondary circulation pattern.

The fluid surface was not frequently cleaned during the experiments, which leads to the build-up of surface contamination. The free surface is additionally contaminated by a large number of seeding particles which form a membrane (see figure 2). Under these conditions, the free surface acts like a no-slip boundary (Henderson & Segur Reference Henderson and Segur2013; Moisy et al. Reference Moisy, Bouvard and Herreman2018). We therefore treat the free surface as a vibrating solid when applying the streaming theory of Nicolás & Vega (Reference Nicolás and Vega2003).

3.2.1 Streaming under the free surface

Let the primary-flow-induced deflection normal to the unperturbed boundary, $f$, the primary-flow oscillation along the boundary, $u_{\Vert }$, and the tangential boundary-oscillation velocity, $v^{w}$, be written as

$$\begin{eqnarray}\displaystyle & f=F\text{e}^{\text{i}\unicode[STIX]{x1D714}t}+\text{c.c.}, & \displaystyle \nonumber\\ \displaystyle & u_{\Vert }=U\text{e}^{\text{i}\unicode[STIX]{x1D714}t}+\text{c.c.}, & \displaystyle \nonumber\\ \displaystyle & v^{w}=U^{w}\text{e}^{\text{i}\unicode[STIX]{x1D714}t}+\text{c.c.}, & \displaystyle \nonumber\end{eqnarray}$$

where $F$ is the normal oscillation amplitude and $U^{w}$ is the tangential velocity amplitude of the boundary. With the present notation, the streaming velocity along a vibrating solid boundary in two dimensions is given by equations (6.2) of Nicolás & Vega (Reference Nicolás and Vega2003):

(3.23)$$\begin{eqnarray}\displaystyle u_{\bot }^{s}=\unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}}(F^{\ast }U)+\text{c.c.}, & & \displaystyle\end{eqnarray}$$

(3.24)$$\begin{eqnarray}\displaystyle u_{\Vert }^{s} & = & \displaystyle \unicode[STIX]{x1D705}F^{\ast }(2U-3U^{w})-\text{i}\unicode[STIX]{x1D714}F^{\ast }\unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}}F-\frac{3}{2\unicode[STIX]{x1D714}}[(U^{\ast }-U^{w\ast })\unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}}(U-U^{w})]\nonumber\\ \displaystyle & & \displaystyle -\,\frac{\text{i}}{\unicode[STIX]{x1D714}}[(3U-5U^{w})\unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}}U^{\ast }]-\frac{1}{\unicode[STIX]{x1D714}}(3U-U^{w})\unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}}U^{w\ast }+\text{c.c.},\end{eqnarray}$$

where $\unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}}$ is the derivative along the steady meniscus, $u_{\bot }^{s}$ and $u_{\Vert }^{s}$ are the streaming velocity normal and parallel to the steady meniscus, respectively, $\unicode[STIX]{x1D705}\approx \unicode[STIX]{x2202}_{xx}^{2}\unicode[STIX]{x1D701}_{0}$ is the curvature of the unperturbed boundary, and $U$ is the oscillation amplitude of the primary flow tangential to the steady free surface. According to our approximation $\unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D701}_{0}\rightarrow 0$, the tangential sliding of the unperturbed surface is negligible and we have $f\approx \unicode[STIX]{x1D701}_{1}$ and $v^{w}\approx 0$.

The results (3.23) and (3.24) are independent of the oscillation phase, since the result is evaluated at the unperturbed boundary (equivalent to $\unicode[STIX]{x1D711}=\unicode[STIX]{x03C0}$ in our case) and averaged over one period. For a standing wave, the normal deformation amplitude, $f$, and primary-flow velocity amplitude parallel to the free surface, $u_{\Vert }$, are given by

$$\begin{eqnarray}\displaystyle & \displaystyle f\approx \hat{\unicode[STIX]{x1D701}}_{1}\sin (\unicode[STIX]{x1D714}t)=\frac{\hat{\unicode[STIX]{x1D701}_{1}}}{2\text{i}}(\text{e}^{\text{i}\unicode[STIX]{x1D714}t}-\text{e}^{-\text{i}\unicode[STIX]{x1D714}t}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle u_{\Vert }=u_{\Vert }\cos (\unicode[STIX]{x1D714}t)=\frac{\hat{u} _{\Vert }}{2}(\text{e}^{\text{i}\unicode[STIX]{x1D714}t}+\text{e}^{-\text{i}\unicode[STIX]{x1D714}t}), & \displaystyle \nonumber\end{eqnarray}$$

where $\hat{\unicode[STIX]{x1D701}}_{1}$ and $\hat{u} _{\Vert }$ are real functions, so that $F$ and $U$ are given by

$$\begin{eqnarray}\displaystyle F=\frac{\hat{\unicode[STIX]{x1D701}_{1}}}{2i},\quad U=\frac{\hat{u} _{\Vert }}{2}. & & \displaystyle \nonumber\end{eqnarray}$$

Substitution of $F$, $U$ and $U^{w}$ into (3.23) and (3.24) gives

(3.25)$$\begin{eqnarray}\displaystyle \boldsymbol{u}_{s}|_{z=\unicode[STIX]{x1D701}_{0}}=-\frac{3}{4\unicode[STIX]{x1D714}}\hat{u} _{\Vert }\unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}}\hat{u} _{\Vert }\hat{\boldsymbol{t}}, & & \displaystyle\end{eqnarray}$$

where $\hat{\boldsymbol{t}}$ is the tangential unit vector along the meniscus. Note that, since the surface elevation is $90^{\circ }$ out of phase with the flow field, the streaming flow normal to the meniscus surface $u_{\bot }^{s}=0$. Moreover, the tangential streaming flow components proportional to the curvature $\unicode[STIX]{x1D705}$ and surface deformation $F$ do not appear in the final result, which is of the same form as for an undeformable solid surface (Nicolás & Vega Reference Nicolás and Vega2003). Although the term ${\sim}\unicode[STIX]{x1D705}FU$ disappears in our system, its physical meaning is of interest. When there is a local curvature on the unperturbed free surface, the perpendicular deformation shrinks the curve element length in the curvature direction. Thus the boundary viscous stress does more (less) work at the phases when the surface curve element stretches (shrinks). For progressive waves, when the velocity oscillation along the surface is in phase with the normal deformation, the damping asymmetry within one period produces a net Eulerian flow. For standing waves, the deceleration of the velocity along the surface in both directions cancels each other to produce a zero periodical mean.

The tangential velocity at the meniscus when $\unicode[STIX]{x1D711}=\unicode[STIX]{x03C0}$, i.e. $u_{\Vert }$, is obtained by linear extrapolation from $z=0$ (see appendix B). By substituting the expression for $u_{\Vert }$ into (3.23), we find

(3.26)

where and  are the horizontal and vertical unit vectors, respectively. From this result, we can see that the surface profile $\unicode[STIX]{x1D701}_{0}$ modifies the streaming velocity distribution. To be consistent with the scaling assumptions of the primary flow solution, we have neglected terms that are $O(\unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D701}_{0})$, which eliminates the vertical velocity along the meniscus.

The horizontal velocity along the meniscus, $u_{s}|_{z=\unicode[STIX]{x1D701}_{0}}$, is shown in figure 8. In both cases, the gravest mode gives the largest contribution to the streaming velocity. With the contribution from the higher-order waves, the magnitude decreases in the symmetric case and increases in the antisymmetric case. The streaming velocity field is completely antisymmetric in the symmetric case, but lacks any definite symmetry in the antisymmetric case. This asymmetry can be understood by considering the fact that the velocity of deep-water waves decreases exponentially from the mean water depth (i.e. from $z=0$). Since the steady liquid surface is higher on the hydrophilic side than on the hydrophobic side, the streaming velocity induced by the meniscus wave is also higher on the hydrophilic side. From (3.23), the wave-induced secondary flow inherits this strength distribution (see the schematic inset in figure 5), leading to an asymmetric streaming velocity distribution.

3.2.2 Streaming near the lateral boundary

The (flat) lateral solid boundaries are a limiting case of the deformable solid surface boundary where $\unicode[STIX]{x1D6FE}=0$ and $F=0$. The previous results, equations (3.23) and (3.24), yield

(3.27)$$\begin{eqnarray}\displaystyle \boldsymbol{u}_{s}|_{x=0,L}=-\frac{3}{4\unicode[STIX]{x1D714}}{\hat{w}}_{1}\unicode[STIX]{x2202}_{z}{\hat{w}}_{1}|_{x=0,L}\hat{\boldsymbol{t}}, & & \displaystyle\end{eqnarray}$$

where $\hat{w_{1}}|_{x=0,L}=\unicode[STIX]{x2202}_{z}\unicode[STIX]{x1D719}_{1}|_{x=0,L}$ is the vertical velocity amplitude at the lateral boundary. Substituting the primary flow solution (3.19) into (3.27) and applying linear extrapolation (see appendix B), we find that

(3.28)$$\begin{eqnarray}w_{s}|_{x=0,L}=\left\{\begin{array}{@{}ll@{}}-{\displaystyle \frac{3}{4\unicode[STIX]{x1D714}}}\displaystyle \mathop{\sum }_{n,m}A_{n}A_{m}k_{n}k_{m}^{2}(1+zk_{n})(1+zk_{m})\quad & z\geqslant 0,\\ -{\displaystyle \frac{3}{4\unicode[STIX]{x1D714}}}\displaystyle \mathop{\sum }_{n,m}A_{n}A_{m}k_{n}k_{m}^{2}\text{e}^{(k_{n}+k_{m})z}\quad & z<0.\end{array}\right.\end{eqnarray}$$

This expression is valid for both symmetric and antisymmetric cases.

Figure 9.

The streaming flow velocity distribution along the lateral boundary when $a_{0}=0.2g_{0}$, $f_{0}=8~\text{Hz}$ for (a) the symmetric and (b) the antisymmetric cases. The solid line is the result (3.27) with the sum terminated at $m+n=200$, while the dashed line contains contributions from the gravest mode only. The upper limit of the plot is the position of the liquid surface at hydrophilic boundary. The dash-dotted line in (b) shows the liquid surface position at the hydrophobic boundary.

The vertical velocity distribution near the lateral boundary is shown in figure 9. The contribution from the gravest mode is similar in both the symmetric and antisymmetric cases. The two cases become distinct once higher-order contributions are included. The streaming velocity becomes stronger, but remains monotonic and negative in the symmetric case. In the antisymmetric case, the higher-order contributions weaken the streaming velocity and create a profile that is positive above $z=0.013~\text{cm}$ and has a negative extremum approximately 0.2 cm below the surface. This effect is due to phase modulation within the harmonic spectrum, $A_{n}$. All the $A_{n}$ are positive in the symmetric case, so the components add constructively. In the antisymmetric case, $A_{1}$ is negative and the other $A_{n}$ are positive, so interactions between the gravest and higher-order modes interfere destructively with the contributions from the wave self-interactions. Thus, in contrast to the monochromatic wave situation discussed by Périnet et al. (Reference Périnet, Gutiérrez, Urra, Mujica and Gordillo2017), the lateral boundary streaming velocity profile is affected by the relative phases of the component waves. In this experiment, the relative phases are controlled by the symmetry of static meniscus. Note that the streaming velocity is left–right symmetric in the symmetric case, but asymmetric in the antisymmetric case because the free surface on the hydrophobic side is lower than that on the hydrophilic side (see figure 9b).

3.2.3 Numerical simulation of streaming circulation

Following Périnet et al. (Reference Périnet, Gutiérrez, Urra, Mujica and Gordillo2017), we obtain the secondary flow in the bulk by using the steady streaming velocity profile as boundary conditions for the laminar flow solver in COMSOL Multiphysics^® using a fixed (i.e. not deforming) domain at $\unicode[STIX]{x1D711}=\unicode[STIX]{x03C0}$. Taking the characteristic velocity $U$ to be the r.m.s. velocity of the surface streaming velocity distribution, and the characteristic length to be the cell width $L$, the maximum Reynolds number is 24 for streaming circulation. We simulate the steady secondary circulation at seven different oscillation amplitudes with the driving frequency $f_{0}=8~\text{Hz}$.

Figure 10.

The secondary flow velocity beneath the oscillating meniscus for the symmetric case from (a) the numerical simulation and (b) the experiment. The forcing amplitude is $a_{0}=0.2g_{0}$ and the driving frequency is $f_{0}=8~\text{Hz}$. The phase chosen for simulation is $\unicode[STIX]{x1D711}=\unicode[STIX]{x03C0}$. The centre of the anticlockwise (clockwise) circulation is denoted by $E_{acw}$ ($E_{cw}$). The point where the velocity perpendicular to the line connecting $E_{acw}$ and $E_{cw}$ is maximal, denoting a transition between anticlockwise and clockwise vortices, is given by $T$.

Figure 11.

The secondary flow velocity beneath the oscillating meniscus for the antisymmetric case from (a) the numerical simulation and (b) the experiment. The forcing amplitude is $a_{0}=0.2g_{0}$ and the driving frequency is $f_{0}=8~\text{Hz}$. The phase chosen for simulation is $\unicode[STIX]{x1D711}=\unicode[STIX]{x03C0}$. The centre of the anticlockwise (clockwise) circulation is denoted by $E_{acw}$ ($E_{cw}$). The point where the velocity perpendicular to the line connecting $E_{acw}$ and $E_{cw}$ is maximal, denoting a transition between anticlockwise and clockwise vortices, is given by $T$.

The steady secondary circulation induced by the oscillation at $a_{0}=0.2g_{0}$ is shown in figures 10 and 11. A nonlinear transformation is applied to the magnitude of the flow vectors to better visualize their direction. The numerical simulation successfully reproduces the four-vortex circulation structure in the symmetric case and the dipole circulation structure in the antisymmetric case. The scalings for the r.m.s. speed, $v_{rms}$, and the symmetry parameter, $\unicode[STIX]{x1D6FE}$, from the numerical simulations are shown in figure 5. The numerical solutions follow the same scaling law, $v_{rms}\sim (a_{0}/g_{0})^{2}$, as the experiments, but the value of the r.m.s. speed is approximately one order of magnitude smaller (figure 5a). The error in the first-order solution (see figure 7) accounts for approximately 15 % of the total error in both symmetric and antisymmetric cases; the remaining error must be due to other effects.

In order to determine the source of this additional error, we quantify the rotation speed of the eddies to gain additional insight into the streaming circulation structure and the energy distribution in the bulk. The rotation speed of each vortex is measured along a section connecting the centre of a anticlockwise circulation, $E_{acw}$, and the adjacent clockwise circulation, $E_{cw}$. The two vortices are separated by the point $T$ where the velocity perpendicular to $E_{acw}E_{cw}$ is maximal. The radii and speed of the vortices are then defined by

(3.29a,b)$$\begin{eqnarray}r_{acw}=\unicode[STIX]{x0394}(E_{acw}T),\quad v_{acw}=\frac{1}{r_{acw}}\int _{E_{acw}\rightarrow T}v_{n}\,\text{d}l,\end{eqnarray}$$

(3.30a,b)$$\begin{eqnarray}r_{cw}=\unicode[STIX]{x0394}(E_{cw}T),\quad v_{cw}=\frac{1}{r_{cw}}\int _{T\rightarrow E_{cw}}v_{n}\,\text{d}l,\end{eqnarray}$$

where $\unicode[STIX]{x0394}(E_{acw}T)$ and $\unicode[STIX]{x0394}(E_{cw}T)$ are the lengths of the lines connecting $E_{acw}$ and $E_{cw}$, respectively, to $T$, and $v_{n}$ is the velocity perpendicular to $E_{acw}E_{cw}$. The resulting speed and radii for the different cases are shown in table 1.

Table 1.

Radii and speed for vortices from the simulations and experiments.

We find that $r_{cw}$ is always smaller than $r_{acw}$, while $v_{cw}$ is always larger than $v_{acw}$. This means that the vortices near the hydrophilic boundary are smaller and relatively faster in both the symmetric and antisymmetric cases. The speed of the vortices is of the order $10^{-3}{-}10^{-2}~\text{cm}~\text{s}^{-1}$ for both the experiments and simulations. Consistent with the result from the $v_{rms}$ calculation, the experiment gives a result approximately five or six times larger than the simulation. The difference between simulation and experiment is slightly larger for the clockwise vortex than for the anticlockwise vortex. This suggests that our model fails to completely capture the physics of the hydrophilic boundary. The impact of three-dimensional (3-D) effects on the circulation is estimated in appendix A by calculating the compression rate. We find a significant secondary flow convergence in the $y$ direction near the hydrophilic boundary (figure 12). The convergence near the hydrophilic boundary induces a downwelling jet on the observational plane and makes a significant contribution to the secondary circulation near the hydrophilic boundary. This effect explains the significant symmetry loss in the antisymmetric case and may account for the remaining discrepancy between the simulations and experiments.

The numerical experiments were repeated with the surface streaming condition replaced by a free-slip boundary. The four-vortex structure is lost in the symmetric case and the kinetic energy drops by an order of magnitude in both cases. These results demonstrate that the free surface is a significant source of momentum for the secondary circulation and is important for determining its qualitative structure.

Figure 12.

Divergence of the 2-D streaming flow at $a_{0}/g_{0}=0.2$ and $f_{0}=8~\text{Hz}$ for (a) the symmetric and (b) the antisymmetric cases. Negative contours are dashed. The phase $\unicode[STIX]{x1D711}=9/10\unicode[STIX]{x03C0}$ ($\unicode[STIX]{x1D711}=13/10\unicode[STIX]{x03C0}$) for the symmetric (antisymmetric) case.