Rayleigh-Taylor instability of a thin liquid film

Master’s semester project of: Bertil Trottet

Supervisor: Nicolas Kofman

How to prevent dripping when painting a ceiling ? The mechanism of drop formation is here the Rayleigh-Taylor instability of a dense liquid overhanging a lighter one. Several elements have been shown to influence this instability, some of which we revisit or explore in this experimental project.

The set-up is composed of a PMMA plate mounted on a support that allows to turn the surface upside-down. The first step is to prepare a fluid layer on the plate by pouring a given volume V in one point so that it spreads over time. We have used mainly silicon oil as a working fluid because it spreads very well due to its low surface tension. After some time, the plate is reversed and the instability grows. Figure 1 presents typical pictures obtained at different stages of the evolution. The first modulations appear at the boundaries and then evolve into a nice hexagonal pattern of drops. It is not perfectly symmetric due to the perturbations generated by reversing the plate. Let us describe now some variation around this simple experiment.

Effect of viscosity and film thickness : The growth rate of the instability can be computed analytically : it is inversely proportionnal to viscosity and proportionnal to film thickness cubed. We have observed indeed that the instability is damped when the fluid is more viscous or thinner. Moreover, there is no dripping on site whenever the film thickness is smaller than roughly 0.3 mm. Below this critical value, there is not enough volume per wavelength for the drops to detach. However, dripping still occurs in our experiments because of coalescence events : indeed, the drops all move in the same direction due to a small tilt angle.

Effect of surface tension and wettability : Surface tension influences the instability growth rate but also the most amplified wavelength which is proportionnal to the capillary length. With glycerol instead of silicon oil, we have found that the distance between droplets is indeed higher because of higher surface tension. In addition, the long-time evolution in the dripping regime reveals another interesting observation : with silicon oil, there always remains a very thin film between the drops whereas the film completely dewetts when using glycerol solution.

Effect of the boundary conditions : As the first stages of the instability are driven by the boundary conditions, we wanted to test also different plate shapes. Technically, these shapes have been manufactured by cutting groove into the PMMA plates. Figure 2 confirms that the shape of the boundaries imposes the obsverved drop pattern.

Effect of evaporation : Eventually, we have investigated the effect of evaporation which is crucial for painters ! We have used for that purpose different varnish samples supplied by the paint company Jallut. These varnishes are composed of a solvant which evaporates and a resin more viscous and which solidifies rapidly when put in contact with the oxygen in the air. This effect stabilizes the instability as the liquid viscosity increases and the layer thickness decreases during drying. We have observed indeed that, for a given volume, the system can be stable or unstable depending on the waiting time before the plate is turned upside-down. Moreover, the experiments have revealed some exotic phenomena (Figure 3) due to the triple-line motion (cracks, dendrites) or the paint viscoelastic behaviour (stalactite).

Figure 1 : Experimental pictures showing the pattern development (V = 2 ml, silicon oil 500 cSt)

Figure 2 : Influence of different boundary conditions

(a)                                               (b)                                                   (c)

Figure 3 : Some exotic phenomena arising during the drop evaporation : cracks after
drying (a), dendrites oscillating in time (b), stalactite during the dripping process (c)