3.1. SWs confined near the end walls

We find that SWs can be excited in the close vicinity of the end walls of the container depending on the amplitude and frequency of the imposed horizontal forcing, as illustrated in figure 1. The differential inertial forcing of the two fluid layers drives synchronous (harmonic) propagating waves which decay away from the end walls due to viscous dissipation (Perez-Gracia et al. Reference Perez-Gracia, Porter, Varas and Vega2014; Sánchez et al. Reference Sánchez, Fernández, Tinao and Porter2019a). Figure 3(b) shows side-view snapshots of this harmonic wave at the right-hand side wall for the HT135-SO50 fluid pair with $f=35$ Hz and $A=1.30$ mm, which is below the threshold of onset of SWs. The four snapshots were taken every quarter of a period of oscillation $T$, with corresponding displacements of the container shown in figure 3(a). The images indicate that fluid near the wall rises and recedes harmonically with the horizontal oscillation of the container, resulting in the local vertical displacement of the interface (Sánchez et al. Reference Sánchez, Yasnou, Gaponenko, Mialdun, Porter and Shevtsova2019b). We estimate that the harmonic wave front, which undergoes the largest vertical forcing and, thus, where the SWs are first observed, extends across approximately $3l_{ca}$. The interface adjacent to the right hand-side wall is displaced downwards while the container moves to the right, because the denser fluid in the lower layer flows away from the end wall due to inertia. The minimum height of the interface is reached when the container reaches its far-right position ($+0.25T$). The subsequent change in the direction of motion of the container leads to the development of an opposite counterflow where the lower layer now flows towards the end wall. This, in turn, drives the upwards displacement of the interface at the end wall ($+0.5T$) up to a maximum height reached at the maximum container displacement ($+0.75T$). Note that the upwards displacement of the interface is larger than its downwards displacement because of the reduced viscosity of the lower layer. We observed this harmonic wave which drives interface deformation at the end walls for all values of the forcing acceleration, $a=A\omega ^2$, and fluid pairs investigated. However, the distance from the end walls over which interfacial deformation was observed, which we refer to as the harmonic wave field, increased from the minimum value of $3l_{ca}$ as the mean viscosity in the system was reduced. This is because the spread of the wave field is set by the relative magnitude of viscous and capillary effects (Perez-Gracia et al. Reference Perez-Gracia, Porter, Varas and Vega2014). In figure 3(b), the height of the harmonic wave front is uniform across the width of the channel ($W\approx 75 l_{ca}$). However, an increase in the amplitude of forcing to $A=1.45$ mm destabilises this flat front into a standing wave pattern in the $y$-direction as shown in figure 3(c). Figure 3(d) shows the evolution of the standing wave over two periods of oscillation in angled top view for $f=40$ Hz and $A=1.30$ mm for HT135-SO50. The two cycles of oscillation shown in figure 3(d) are superposed in figure 3(e) so that each snapshot now combines images one period apart, with the interface from the second cycle highlighted with a red dotted line; see, e.g. the images at the times $t^*$ and $t^*+T$ in figure 3(e). We find a phase shift of 180$^\circ$ between the superposed wave patterns, which indicates that the standing wave oscillates at half of the forcing frequency (i.e. $\omega = 2 \omega _0$) and is therefore subharmonic (see also the Supplementary Material).

Figure 3.

(a) Schematic diagrams showing the $x$-position of the centre of the container over one period (top) and the ROI (bottom). (a–d) Snapshot images of the ROI in different views (see figure 2b) for the HT135-SO50 fluid pair, with the $x$ displacement of the end wall (vertical yellow line) indicating the instantaneous displacement of the vessel. (b) Harmonic wave front of uniform height along $y$, visualised in angled side view ($\,f=35$ Hz, $A=1.30$ mm). (c) Development of SWs on the harmonic wave front in angled side view ($\,f=35$ Hz, $A=1.45$ mm). (d) Experimental snapshots of the SWs visualised in angled top view ($\,f=40$ Hz, $A=1.30$ mm). (e) Superimposed 180$^\circ$ out-of-phase images for patterns observed in (d).

The distance over which the harmonic wave field decays away from the end wall determined the spread of the SWs in the $x$-direction. Figure 4 compares SWs excited in experiments with the fluid pairs HT135-SO10 (figure 4a) and HT135-SO50 (figure 4b). The upper and lower rows show experiments performed with $f=25$ and 40 Hz, respectively, and in each figure the amplitudes of forcing increase from left to right. The yellow arrows highlight SWs which extend beyond the near-end-wall region where the harmonic wave front oscillates vertically (${\sim }3l_{ca}$). SWs formed in the HT135-SO10 fluid pair with lower mean viscosity, $\nu _{m}= (\rho _{u} \nu _{u} + \rho _{l} \nu _{l})/(\rho _{u} + \rho _{l})$, tend to extend beyond the near-wall region regardless of the forcing frequency; in contrast, SWs formed in the HT135-SO50 fluid pair with higher mean viscosity localise in the close vicinity of the end wall for the higher forcing frequency. As shown by Bechhoefer et al. (Reference Bechhoefer, Ego, Manneville and Johnson1995) and Puthenveettil & Hopfinger (Reference Puthenveettil and Hopfinger2009), the harmonic wave decays on a characteristic damping length scale, $\sigma /(\nu _{m} \Delta \rho \omega )$. This decay length increases with a reduction in either $\nu _{m}$ or $\omega$, which facilitates the spread of SWs away from the end wall through wave interaction. The harmonic forcing is only vertical in the near-end-wall region, beyond which the harmonic wave front is subject to a combination of vertical and horizontal forcing, which renders the SWs spatially inhomogeneous (Porter et al. Reference Porter, Tinao, Laverón-Simavilla and Lopez2012; Tinao et al. Reference Tinao, Porter, Laverón-Simavilla and Fernández2014). For sufficiently low values of $\nu _{m}$ or $\omega$, wave interaction upon a small increase in the forcing amplitude beyond the onset of the SWs can lead to irregular patterns, as illustrated in figure 4. Hence, the nature of the SWs observed experimentally differs depending on vibrational parameters and mean viscosity. However, at onset, they always arise next to the end wall where vertical forcing is strongest.

Figure 4.

SW patterns near the end walls for fluid pairs (a) HT135-SO10 and (b) HT135-SO50, where images are captured in angled top view (see figure 2b) with the container at its leftmost position (blue circle as in figure 3), for different forcing frequencies: $f=25$ Hz (upper row) and $f=40$ Hz (lower row). In (a), the forcing amplitude $A$ increased from 1.5 mm (1.30$A_c$) to 1.7 mm (1.48$A_c$) at $f=25$ Hz, and from 0.85 mm (1.06$A_c$) to 1.0 mm (1.25$A_c$) at $f=40$ Hz. In (b), $A$ increased from 1.9 mm (1.00$A_c$) to 2.0 mm (1.05$A_c$) at $f=25$ Hz and from 1.3 mm (1.00$A_c$) to 1.4 mm (1.08$A_c$) at $f=40$ Hz. The coordinate system shown was defined in figure 3(a). The yellow line indicates the position of the end wall and the yellow arrow points to the appearance of SWs outside the near-end-wall region (${\sim }3l_{ca}$, as indicated by the red boxes).

3.2. Effect of fluid properties and vibrational parameters

Figure 5(a) shows the threshold forcing acceleration $a_{c}=A_{c}\omega ^2$ at which SWs appear as a function of the forcing frequency for the five fluid pairs introduced in § 2. These onset measurements were taken by imposing the frequency of forcing and gradually increasing the amplitude of forcing in increments of 0.01 mm until the first appearance of the instability at the threshold amplitude $A_{c}$, taking care to let transients decay for at least 600 cycles of oscillation after each increment. We find that for each fluid pair, the threshold acceleration increases monotonically with increasing forcing frequency. Remarkably, the onset curves are stacked in order of increasing mean kinematic viscosity, $\nu _{m}$, with the lowest curve corresponding to the lowest value of $\nu _{m}$. Note that the viscosity ratio $\nu _{u}/\nu _{l}$, which governs the onset of the frozen wave instability (Talib et al. Reference Talib, Jalikop and Juel2007), does not increase monotonically with $\nu _{m}$ for the experiments shown in figure 5(a). This is because of the red data which has the lowest viscosity ratio, due to the tenfold increase of the lower layer viscosity in these experiments compared with the other data shown with black symbols. In fact, the results of figure 5(a) bear no resemblance to the onset of the frozen wave instability, whose threshold acceleration decreases monotonically as a function of frequency, and whose onset curves are stacked in order of decreasing $\nu _{u}/\nu _{l}$. Figure 5(a) also indicates that the datasets do not remain similar as $\nu _{m}$ is increased, because of increasingly different threshold acceleration values at high frequency. This means that a rescaling in terms of only the mean viscosity is not sufficient to collapse the experimental data onto a master curve, and we refer to § 3.3 for a more detailed scaling analysis.

Figure 5.

Critical acceleration (a) and critical wavenumber (b) as a function of forcing frequency indicating the onset of SWs in the experiment for five fluid pairs with different mean kinematic viscosity, $\nu _{m}$, represented by different symbols. Experimental measurements in (b) are compared with the Faraday wave dispersion relation (lines) for each fluid pair. The red line sits slightly below the black lines because the density of HT270 is approximately 5 % larger than that of HT135 (see table 1).

The dispersion relation of the SWs is shown in figure 5(b), where the wavenumber ($k_{c}$) measured at onset is plotted as a function of frequency for the five fluid pairs investigated. The symbols used to indicate different values of $\nu _{m}$ are the same as in figure 5(a). Each data point represents the average of several separate experiments, and the error bar indicates the standard deviation of this value. The lines show the theoretical prediction for Faraday waves at the interface between two infinite-depth fluid layers, which is given by the gravity–capillary wave dispersion relation, $(1/2\omega )^2={\omega _0}^2=(\sigma k^3+\Delta \rho gk)/(\rho _{l}+\rho _{u})$, where $\omega _0$ is the natural angular frequency (Kumar & Tuckerman Reference Kumar and Tuckerman1994; Rajchenbach & Clamond Reference Rajchenbach and Clamond2015). The experimental dependence of wavelength on frequency is captured satisfactorily by this dispersion relation. The agreement is within error bars for the lowest values of $\nu _{m}$, but for $\nu _{m}>20.1$ mm$^2$ s$^{-1}$, the experimental data sit marginally below the theoretical lines. This discrepancy is associated with the increase in the rate of viscous dissipation with forcing frequency and mean viscosity, which has been highlighted previously in studies of Faraday waves (Edwards & Fauve Reference Edwards and Fauve1994; Bechhoefer et al. Reference Bechhoefer, Ego, Manneville and Johnson1995).

3.3. Scaling analysis and physical interpretation

Measured wavelengths of the order of the capillary length, and the increase of the critical acceleration for the onset of SWs with increasing $\nu _{m}$ discussed in § 3.2, suggest that the onset of SWs is governed by the relative magnitude of capillary and viscous forces. Hence, we follow Goodridge et al. (Reference Goodridge, Shi, Hentschel and Lathrop1997) in defining a capillary–viscous length, $l_{cv}=\nu _{m}^{2}/(\sigma /\Delta \rho )$, and a capillary–viscous frequency, $\varOmega _{cv}=(\sigma /\Delta \rho )^2/(\nu _{m}^3)$, and use them to scale the critical forcing acceleration $a_{c}^*= a_{c}/(l_{cv} \varOmega _{cv}^2)$ and the forcing frequency, $\omega ^* = \omega /\varOmega _{cv}$. Figure 6(a) shows that the scaled critical acceleration data previously shown in figure 5(a) approximately collapse onto a master curve proportional to ${\omega ^*}^{1.46\pm 0.01}$, which we obtained by linear regression of the experimental data (blue dashed line). The scaled critical wavenumber, $k_{c}^*=k_{c} l_{cv}$, is plotted as a function of $\omega ^*$ in the inset of figure 6(a), where the solid lines correspond to the capillary–gravity wave dispersion relation shown in figure 5(b). The scaled dispersion relation is shifted to larger frequencies as $\nu _{m}$ is increased. The upper parts of each of these curves which correspond to sufficiently large values of $\omega ^*$ approximately collapse onto a single line indicating capillary–viscous wavenumber selection. We find that each set of experimental data aligns onto the appropriate dispersion relation curve close to this line, which further supports our choice of scaling.

Figure 6.

(a) Log–log plot of non-dimensional acceleration vs non-dimensional frequency for different fluid pairs. A least-squares fit yields $a_{c}^* \sim \omega ^{*1.46 \pm 0.01}$ (dashed line) and the solid line indicates $a_{c}^*\omega ^{*3/2}$ predicted by (3.4) (solid line). Inset: Non-dimensional wavenumber vs $\omega ^*$. The solid lines indicate the Faraday wave dispersion relation on the capillary–viscous scale. (b) Experimental critical acceleration $a_{c}$ plotted against the prediction of (3.4).

To predict the critical value of scaled acceleration for the onset of SWs within the limit of weak viscous effects (Kumar Reference Kumar2000), we use the Mathieu equation with linear viscous damping as a simple model describing a parametrically forced interface separating infinite-depth layers (Kumar & Tuckerman Reference Kumar and Tuckerman1994; Porter et al. Reference Porter, Tinao, Laverón-Simavilla and Rodríguez2013). The dimensional critical acceleration is given by

(3.1)\begin{equation} a_{c}=2\gamma(\rho_{l}+\rho_{u})\omega/(\Delta \rho k_{c}). \end{equation}

Following Landau & Lifshitz (Reference Landau and Lifshitz1987), the damping rate is defined as $\gamma =\bar {\dot {E}}/(2\bar {E})$, where $\bar {\dot {E}}$ and $\bar {E}$ denote the time-averaged rate of viscous dissipation and the time-averaged mechanical energy, respectively. Kumar & Tuckerman (Reference Kumar and Tuckerman1994) showed that for two fluid layers of the same depth, $d$, the time-averaged rate of dissipation and total mechanical energy can be estimated through volume integrals as $\bar {\dot {E}}=-2(\rho _{l}\nu _{l}\int (\boldsymbol {\nabla }\boldsymbol {\cdot }\bar {\boldsymbol{u}}_{l})^2\,{\rm d}V+\rho _{u}\nu _{u}\int (\boldsymbol {\nabla }\boldsymbol {\cdot }\bar {\boldsymbol{u}}_{u})^2\,{\rm d}V)$ and $\bar {E}=\rho _{l}\int \bar {\boldsymbol{u}}_{l}^2\,{\rm d}V+\rho _{u}\int \bar {\boldsymbol{u}}_{u}^2\,{\rm d}V$, where $\bar {\boldsymbol{u}}_{l}$, $\bar {\boldsymbol{u}}_{u}$ are the time-averaged fluid velocities that can be characterised by $u=A(\omega /2)$ (Bechhoefer et al. Reference Bechhoefer, Ego, Manneville and Johnson1995). Using the onset wavelength of the Faraday wave pattern, $\lambda _{c}=2{\rm \pi} /k_{c}$ as the length scale of damping in the limit of deep fluid layers where $kd \gg 1$, the damping rate is evaluated in a volume $\lambda _c^2 W$ as

(3.2)\begin{equation} \gamma \sim \nu_{m} \frac{(u/ \lambda_{c})^2 (W \lambda_{c}^2)}{(u)^2 (W \lambda_{c}^2)}=\nu_{m} k_{c}^2. \end{equation}

As shown in § 3.1, the SWs are adjacent to the end walls and, thus, the damping rate over $\lambda _{c}$ is dominated by the Stokes boundary layers of thickness $\delta \equiv (2\nu _{m}/\omega )^{1/2}$ at these walls. In our experiments, we measured $k_{c}\delta \approx O( 10^{-1})$. The contribution to $\bar {\dot {E}}$ from the Stokes layer is a factor $(k_{c} \delta )^{-1}$ larger than that in the bulk whose contribution we neglect. Hence, the end-wall damping rate in a volume $\lambda _c^2 W$ can be evaluated as

(3.3)\begin{equation} \gamma \sim \nu_{m} \frac{(u/ \delta)^2 (W \lambda_{c} \delta)}{(u)^2 (W \lambda_{c}^2)} =\omega \delta /\lambda_{c} \end{equation}

(Milner Reference Milner1991; Christiansen et al. Reference Christiansen, Alstrøm and Levinsen1995), which differs from the bulk damping rate in both limits of weak viscous effects and deep layers (Kumar Reference Kumar1996) and thin layers (Lioubashevski et al. Reference Lioubashevski, Fineberg and Tuckerman1997). Using this expression, we can recast (3.1) as

(3.4)\begin{equation} a_{c} \sim \frac{(\rho_{l}+\rho_{u})}{\Delta \rho} \sqrt{\nu_{m}} \omega^{3/2}. \end{equation}

The frequency dependence in $\omega ^{3/2}$ (solid line) is in close agreement with the experimental data in figure 6(a) which has a power exponent of $1.46 \pm 0.01$ (dashed line). We attribute the slight discrepancy with the experiments to imperfect collapse of the datasets associated with the two lowest values of $\nu _{m}$. The expression (3.4) also indicates dependence of the critical forcing acceleration on the square root of the mean viscosity. We test the theoretical prediction given by (3.4) by plotting it against the experimentally measured values of $a_{c}$ in figure 6(b), using the same symbols for different values of $\nu _{m}$ as in figure 6(a). Note that we are not able to vary the density difference between the fluids significantly. The experimental data collapse satisfactorily on a straight line and a least-squares proportional fit yields a slope of $1.396\pm 0.022$, which depends on the prefactor of the damping rate. This result indicates that the SWs excited near the end walls of our horizontally vibrated container are driven through a Faraday instability dominated by a wall-damping mechanism.