Faraday waves on band pattern under zero gravity conditions

Observations performed with $\mathrm{C}{\mathrm{O}}_2$ near its critical point onboard sounding rockets show that periodical patterns (bands) develop in a two-phase, liquid-vapor system under tangential vibrations in microgravity conditions. Fluids are slightly below their liquid-vapor critical point where liquid and vapor densities are close, surface tension is low, and they exhibit a scaled, universal behavior. With the increase of vibration amplitude, an instability can develop on the band pattern, leading to the appearance of Faraday waves. Theoretical and numerical investigations of the Faraday instability onset and development are carried out taking into account the interaction between the bands. The critical parameters for the onset of instability are determined. The comparisons between theoretical analysis, two-dimensional direct numerical simulation, and original experimental data show good agreement.

Figure 1

Interface between two fluids: (a), (b) in a gravity field, and (c)–(f) in zero gravity conditions; (a), (c), (e) base states, (b), (d), (f) supercritical regimes: (b) frozen waves at $a\omega >{\left(a\omega \right)}_c$, (d) band pattern, (f) Faraday waves on bands at $a>a_c$.

Figure 2

Dependence of the dimensionless vibrational parameter $B={\left(a\omega \right)}^{2}h\left({\rho }_2-{\rho }_1\right)/\phantom{{\left(a\omega \right)}^{2}h\left({\rho }_2-{\rho }_1\right)4\sigma }4\sigma$ on the dimensionless wave number $k$ for $\rho =1.15$ (taken from [5]). Interrupted curve: neutral perturbations with $\lambda =0$ in an inviscid fluid; full curve: perturbations with maximal growth rate in an inviscid fluid.

Figure 3

Experimental arrangement of the $\mathrm{C}{\mathrm{O}}_2$ experiment (schematic). The double arrow indicates the vibration direction. The scale is in centimeters.

Figure 4

(a) $\mathrm{C}{\mathrm{O}}_2$ sample (10.0 mm diameter and 2.189 mm thickness) under $a=0.7$ mm, $f=25.25$ Hz vibrations with direction indicated by a double arrow. The window is enlarged in (b)–(e). The black lines correspond to the interfaces between the liquid and vapor phases, which order as “bands” perpendicular to the vibration direction. (b)–(e) Different times in units of vibration period T (see text) showing subharmonicity by the interface deformation phase as highlighted by the white dotted line). Time origin corresponds to amplitude maximum on the right. Interface deformation occurs on a half period on one side of the band (b), (d), (e) and on the other half period on the other side (c).

Figure 5

Faraday waves for different parameters of vibrations: (a) $f=5$ Hz, $a=2.5$ mm, band pattern period $=0.8$ mm; (b) $f=25$ Hz, $a=1.1$ mm, band pattern period $=0.4$ mm; (c) $f=20$ Hz, $a=1.6$ mm, $\mathrm{band}\mathrm{pattern}\mathrm{period}=0.1\mathrm{mm}$.

Figure 6

Fourier analysis of velocity evolution in the central point of the cavity for frequency $f=5$ Hz and $a=2.5$ mm; A (in arbitrary units) is the amplitude of the Fourier component.

Figure 7

Band patterns for different phases of Faraday waves period for $f=5\mathrm{Hz}$, $a=2.5\mathrm{mm}$, $\mathrm{band}\mathrm{period}=0.8\mathrm{mm}$: (a) $\phi =0$, (b) $\phi =\pi /8$, (c) $\phi =\pi /4$.

Figure 8

Scheme of linear stability analysis of Faraday waves on bands.

Figure 9

The dependences of critical vibration amplitude on the dimensionless layer depth $H$ for (1) $f=10$ Hz, (2) $f=15$ Hz, (3) $f=20$ Hz according to Eq. (36). The parameters correspond to $\mathrm{C}{\mathrm{O}}_2$ experiments, see Table 1. Diamonds correspond to the stability boundary obtained in the numerical simulations for $f=20\mathrm{Hz}$.

Figure 10

Stability map for experimental, analytical, and numerical results. Circles are experimental results (open circles: stable states; filled circles: Faraday instability). The gray line shows numerical data (the linewidth corresponds to the accuracy of the stability boundary determination). The dashed line corresponds to analytical stability estimation given by Eq. (36).