Harmonic and subharmonic waves on the surface of a vibrated liquid drop

Liquid drops and vibrations are ubiquitous in both everyday life and technology, and their combination can often result in fascinating physical phenomena opening up intriguing opportunities for practical applications in biology, medicine, chemistry, and photonics. Here we study, theoretically and experimentally, the response of pancake-shaped liquid drops supported by a solid plate that vertically vibrates at a single, low acoustic range frequency. When the vibration amplitudes are small, the primary response of the drop is harmonic at the frequency of the vibration. However, as the amplitude increases, the half-frequency subharmonic Faraday waves are excited parametrically on the drop surface. We develop a simple hydrodynamic model of a one-dimensional liquid drop to analytically determine the amplitudes of the harmonic and the first superharmonic components of the linear response of the drop. In the nonlinear regime, our numerical analysis reveals an intriguing cascade of instabilities leading to the onset of subharmonic Faraday waves, their modulation instability, and chaotic regimes with broadband power spectra. We show that the nonlinear response is highly sensitive to the ratio of the drop size and Faraday wavelength. The primary bifurcation of the harmonic waves is shown to be dominated by a period-doubling bifurcation, when the drop height is comparable with the width of the viscous boundary layer. Experimental results conducted using low-viscosity ethanol and high-viscocity canola oil drops vibrated at $70\mathrm{Hz}$ are in qualitative agreement with the predictions of our modeling.

Figure 1

Stability diagram for the subharmonic Faraday waves onset in 2-mm-thick ethanol film with $\rho =789\mathrm{kg}/{\mathrm{m}}^3, \sigma =0.022\mathrm{N}/\mathrm{m}, \mu =0.0012\mathrm{Pa}\mathrm{s}$, vibrated at $20\mathrm{Hz}$. The solid line is obtained from the reduced model Eq. (9), and the dashed line corresponds to the exact stability threshold [31]. The arrow indicates the critical wave vector from Eq. (11).

Figure 2

(a) Profile of the drop with $\beta =0.05$ and horizontal size $W\approx 48.5$, vibrated with amplitude $a=0.1$. Domain size is $2L=24\pi$. (b) Temporal evolution of the harmonic waves on the surface of the drop over one period of the vibration $T=\pi$. The arrows indicate the propagating and standing waves. (c) Excess Laplace pressure $p_e=-\mathrm{\Gamma }{\partial }_{xx}h(x,t)+\mathrm{\Gamma }{\pi }^{-1}{\int }_t^{t+\pi }{\partial }_{xx}h(x,t)dt$ for the drop in (a). The dashed line in (c) is obtained by fitting the function $\sim (x-x_0)e^{(x-x_0)/b}$.

Figure 3

Temporal evolution of the deformation of the drop surface, computed from Eq. (17) over one vibration period $T=\pi$ with parameters as in Fig. 2.

Figure 4

Average response amplitude of the ethanol drop from Fig. 2, vibrated at $f=20$ Hz, as a function of $a$. The amplitude $a$ is gradually increased with the step $\mathrm{\Delta }=0.005$. The harmonic waves lose their stability via a subcritical period-doubling bifurcation (pd) at $a=0.23$. The insets labeled as (1), (2), and (3) show the temporal and spatial spectra $S_f$ and $S_k$ in the logarithmic scale for the selected values of $a$ indicated by the arrows.

Figure 5

Average response amplitude of the ethanol drop, vibrated at $f=21\mathrm{Hz}$, as a function of $a$. Continuation by gradually increasing $a$ with step $\mathrm{\Delta }=0.005$. Harmonic waves lose their stability via supercritical period-doubling bifurcation (pd) at $a=0.08$. Modulation instability sets in via a torus bifurcation (tr) at $a=0.15$. Insets labeled (1), (2), and (3) show the temporal and spatial spectra $S_f$ and $S_k$ in the logarithmic scale for selected values of $a$, as indicated by arrows.

Figure 6

Average response amplitude of the ethanol drop, vibrated at $f=28$ Hz, as a function of $a$. Gradually increasing (asterisk) and gradually decreasing (circle) $a$ with step $\mathrm{\Delta }=0.0015$. sn and tr correspond to the saddle-node and torus bifurcations, respectively. The insets labeled (1), (2), (3), and (4) show the temporal and spatial spectra $S_f$ and $S_k$ in the logarithmic scale for the selected values of $a$ indicated by the dashed lines.

Figure 7

Frequency-amplitude phase diagram of the harmonic (h) and subharmonic (sh) responses for a fixed volume ethanol drop. The response is subharmonic in the shaded area. The boundary between the harmonic and subharmonic regions is obtained by gradually increasing the vibration amplitude $a$ by step 0.01 (size of the error bars) when the frequency $f$ is kept fixed.

Figure 8

Increasing the viscosity. Dashed line: Part of the subharmonic branch between the sadle-node (sn) and the torus (tr) bifurcation points for three different values of the viscosity: $\mu , 3\mu$, and $6\mu$, where $\mu$ is the viscosity of the ethanol. The vibration frequency is $f=28$ (Hz) and other parameters as in Fig. 6. The arrows indicate points of transitions from the harmonic (solid line) to subharmonic (dashed line), when $a$ is gradually increased or decreased.

Figure 9

Schematic of the experimental setup. A subwoofer covered by a thin Teflon plate is used as the source of vertical vibration. The sinusoidal vibration signal of frequency $f=70\mathrm{Hz}$ is synthesizes with a digital signal generator and amplified with an audio amplifier. The response of the liquid drop sitting on top of the Teflon plate is measured by using a continuous wave red laser diode and a photodetector. The detected signals are visualized with an oscilloscope and sent to a laptop for postprocessing.

Figure 10

(a) Experimental average response of the ethanol drop subjected to vertical vibration at $70\mathrm{Hz}$ frequency plotted as a function of the vibration amplitude. The inset shows the power spectra obtained by Fourier-transforming the measured time-domain signals. The labels in parenthesis correlate the spectra to the experimental points in the main panel. (b) Experimental average response of the canola oil drop subjected to vertical vibration at $70\mathrm{Hz}$ frequency plotted as a function of the vibration amplitude. Note the difference between the maximum vibration amplitude in (a) and (b). Also note the presence of the modulation sidebands (the spectrum label 7 for the ethanol drop) and their absence in the spectra of the canola oil drop. The shaded regions in the main panels correspond to the regime of chaotic oscillations resulting in strong diffuse scattering leading to a decrease in the optical intensity of the detected signal.

Figure 11

Closeups of the subharmonic peak at $2\pi f=1$ corresponding to $f=35\mathrm{Hz}$ in the experiment. The modulation sidebands can be seen in the spectrum of the ethanol drop (dashed line), but they are absent in the spectrum of the canola oil drop (solid line).