In 1831, Michael Faraday discovered that a liquid submitted to a vertical oscillation can become unstable and exhibit standing surface waves.1 This instability occurs above a vibration amplitude threshold that increases with the liquid viscosity. Using this property, we triggered the Faraday waves in drops of a fluid of low viscosity floating on a more viscous bath (Figure 1(a)). When this experiment is performed with many pairs of immiscible fluids, only two archetypes of behavior are observed.2,3

FIG. 1.

(a) A liquid drop is deposited on a viscous bath and the system is forced to oscillate vertically. Lateral (b) and top (c) views of the elongated drop in the first archetype of behavior. ρ = 785 kg/m3, f = 65 Hz, A ≈ 0.5 mm, σ = 20.3 mN/m, R ≈ 1 cm. Source: APS-DFD http://dx.doi.org/10.1103/APS.DFD.2014.GFM.V0038.

FIG. 1.

(a) A liquid drop is deposited on a viscous bath and the system is forced to oscillate vertically. Lateral (b) and top (c) views of the elongated drop in the first archetype of behavior. ρ = 785 kg/m3, f = 65 Hz, A ≈ 0.5 mm, σ = 20.3 mN/m, R ≈ 1 cm. Source: APS-DFD http://dx.doi.org/10.1103/APS.DFD.2014.GFM.V0038.

Close modal

In the first archetype,4 the drop reaches a stable elongated shape as a result of the mutual adaptation between the wave pattern and the drop boundaries. In this elongated regime, we observe standing waves with wave-vector parallel to the long axis (Figures 1(b) and 1(c)). The resulting stable shape is explained by the balance between the radiation pressure and the Laplace pressure,

$ρ ( ω A ) 2 ≈ σ / R ,$
(1)

where ρ is the drop density, ω = 2πf is the wave frequency, A is the wave amplitude, σ is the drop surface tension, and R is the horizontal radius of the drop at rest.

In the second archetype,5 because of the much lower drop surface tension, the wave radiation pressure exceeds the capillary response of the drop border

$ρ ( ω A ) 2 ≫ σ / R$
(2)

and the drop elongates rapidly to a worm-like shape that is unstable (Figure 2(a)). At high forcing amplitudes, fingers emerge in all directions from the initially circular drop (Figure 2(b)). After a few minutes, the fingers detach and move spontaneously on the surface, which becomes covered with fragments that keep colliding, merging, and splitting. Stable shapes of small fragments usually observed are the elongated one (Figure 2(c)), the self-propagating “croissant” and the ring, here showed just before a collision (Figure 2(d)).

FIG. 2.

Worm-like shape of a drop elongated by Faraday waves in the second archetype of behavior (a). “Fingering” at high forcing (b) that generates fragments having different shapes ((c) and (d)). ρ = 789 kg/m3, f = 50 Hz, A ≈ 0.5 mm, σ = 0.7 mN/m, R ≈ 1 cm. Source: APS-DFD http://dx.doi.org/10.1103/APS.DFD.2014.GFM.V0038.

FIG. 2.

Worm-like shape of a drop elongated by Faraday waves in the second archetype of behavior (a). “Fingering” at high forcing (b) that generates fragments having different shapes ((c) and (d)). ρ = 789 kg/m3, f = 50 Hz, A ≈ 0.5 mm, σ = 0.7 mN/m, R ≈ 1 cm. Source: APS-DFD http://dx.doi.org/10.1103/APS.DFD.2014.GFM.V0038.

Close modal

In this work, we have shown how a hydrodynamic instability forming in a domain with deformable boundaries can lead to a phenomenon of self-adaptation with also the possibility of self-propagation. More details can be found in Refs. 2–5. Related phenomena have been studied more recently involving in particular the elongation of a sessile drop on a solid surface6 and self-propagating “swimming” droplets7 with Faraday waves.

We thank M. Receveur and L. Rea for technical assistance and Universita Italo-Francese (UIF) for its support. ANR 06-BLAN- 0297-03 and ANR FREEFLOW also supported this work. M.B.A. acknowledges the support of Institut Universitaire de France.

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