A drop does not fall in a straight line: a rationale for the width of stalagmites.

Drops loaded in calcium ions detach from stalactites and impact the underlying stalagmites, thereby allowing these latter to grow through calcite precipitation. Nevertheless, little is known about the influence of the drop free fall and splash dynamics on stalagmite shape and width. Through high-speed imaging of impacting drops on stalagmites from several caves, we observed that the impact point position of the drops is scattered, sometimes over several centimetres. We show that this dispersal has no external cause and must, therefore, be self-induced. Using a Langevin-like equation, we then propose a prediction of the impact point dispersal as a function of the falling height travelled by the drops. We finally show that measured stalagmite widths are correlated to the dispersal in the impact point position of the drop.

Fluid fragmentation shapes rain-induced foliar disease transmission.

Plant diseases represent a growing threat to the global food supply. The factors contributing to pathogen transmission from plant to plant remain poorly understood. Statistical correlations between rainfalls and plant disease outbreaks were reported; however, the detailed mechanisms linking the two were relegated to a black box. In this combined experimental and theoretical study, we focus on the impact dynamics of raindrops on infected leaves, one drop at a time. We find that the deposition range of most of the pathogen-bearing droplets is constrained by a hydrodynamical condition and we quantify the effect of leaf size and compliance on such constraint. Moreover, we identify and characterize two dominant fluid fragmentation scenarios as responsible for the dispersal of most pathogen-bearing droplets emitted from infected leaves: (i) the crescent-moon ejection is driven by the direct interaction between the impacting raindrop and the contaminated sessile drop and (ii) the inertial detachment is driven by the motion imparted to the leaf by the raindrop, leading to catapult-like droplet ejections. We find that at first, decreasing leaf size or increasing compliance reduces the range of pathogen-bearing droplets and the subsequent epidemic onset efficiency. However, this conclusion only applies for the crescent moon ejection. Above a certain compliance threshold a more effective mechanism of contaminated fluid ejection, the inertial detachment, emerges. This compliance threshold is determined by the ratio between the leaf velocity and the characteristic velocity of fluid fragmentation. The inertial detachment mechanism enhances the range of deposition of the larger contaminated droplets and suggests a change in epidemic onset pattern and a more efficient potential of infection of neighbouring plants. Dimensionless parameters and scaling laws are provided to rationalize our observations. Our results link for the first time the mechanical properties of foliage with the onset dynamics of foliar epidemics through the lens of fluid fragmentation. We discuss how the reported findings can inform the design of mitigation strategies acting at the early stage of a foliar disease outbreak.

From a bouncing compound drop to a double emulsion.

We show that a double emulsion (oil in water in oil) can be created starting from a compound droplet (surfactant solution in oil). The compound drop bounces on a vertically vibrated liquid surface. When the amplitude of the vibration exceeds a threshold value, the oil layer penetrates the water content and leaves a tiny oil droplet within. As this phenomenon occurs at each vigorous impact, the compound drop progressively transforms into a double emulsion. The emulsification threshold, which is observed to depend on the forcing frequency but not on the drop size, is rationalized by investigating the impact of compound drops onto a static liquid surface. The droplet creation occurs when the kinetic energy released at impact is larger than the energy required to deform the compound drop, namely when the Weber number is higher than a given threshold value.

Droplets sliding on fibres.

We present the results of a combined experimental and theoretical investigation of oil droplets sliding on fibres. First, both the axisymmetric shape and the motion of a droplet on a vertical fibre are described. The motion is shown to result from a balance between the droplet weight and the viscous stresses. On a long-term range, the droplet loses some mass through coating the fibre, which decreases its velocity. In a second time, we rationalize the behaviour of a droplet that encounters a junction between vertical and horizontal fibres. Depending on its size, the droplet may cross the junction or remain blocked. The transition is well described by an ordinary differential equation equivalent to a damped harmonic oscillator truncated to the neighbourhood of the horizontal fibre. This simple system is the basic element for more complex fiber networks that would be useful in microfluidic applications involving droplets.

Completely inelastic ball.

This Rapid Communication presents an analytical study of the bouncing of a completely inelastic ball on a vertically vibrated plate. The interplay of saddle-node and period-doubling bifurcations leads to an intricate structure of the bifurcation diagram with uncommon properties, such as an infinity of bifurcation cascades in a finite range of the control parameter Gamma. A pseudochaotic behavior, consisting in arbitrarily long and complex periodic sequences, is observed through this generic system.

Chaotic bouncing of a droplet on a soap film.

We examine the complex dynamics arising when a water droplet bounces on a horizontal soap film suspended on a vertically oscillating circular frame. A variety of simple and complex periodic bouncing states are observed, in addition to multiperiodicity and period-doubling transitions to chaos. The system is simply and accurately modeled by a single ordinary differential equation, the numerical solution of which captures all the essential features of the observed behavior. Iterative maps and bifurcation diagrams indicate that the system exhibits all the features of a classic low-dimensional chaotic oscillator.

Dynamics of a bouncing droplet onto a vertically vibrated interface.

Low viscosity (<100 cSt) silicon oil droplets are placed on a high viscosity (1000 cSt) oil bath that vibrates vertically. The viscosity difference ensures that the droplet is more deformed than the bath interface. Droplets bounce periodically on the bath when the acceleration of its sinusoidal motion is larger than a threshold value. The threshold is minimum for a particular frequency of excitation: droplet and bath motions are in resonance. The bouncing droplet has been modeled by considering the deformation of the droplet and the lubrication force exerted by the air layer between the droplet and the bath. Threshold values are predicted and found to be in good agreement with our measurements.

Controlling the partial coalescence of a droplet on a vertically vibrated bath.

A method is proposed to stop the cascade of partial coalescences of a droplet laid on a liquid bath. The strategy consists of vibrating the bath in the vertical direction in order to keep small droplets bouncing. Since large droplets are not able to bounce, they partially coalesce until they reach a critical size. The system behaves as a low pass filter: droplets smaller than the critical size are selected. This size has been investigated as a function of the acceleration and the frequency of the bath vibration. Results suggest that the limit size for bouncing is related to the first mode of the droplet deformation.

Critical parameters for the partial coalescence of a droplet.

The partial coalescence of a droplet onto a planar liquid-liquid interface is investigated experimentally by tuning the viscosities of both liquids. The problem mainly depends on four dimensionless parameters: The Bond number (gravity vs surface tension), the Ohnesorge numbers (viscosity in both fluids vs surface tension), and the density relative difference. The ratio between the daughter droplet size and the mother droplet size is investigated as a function of these dimensionless numbers. Global quantities such as the available surface energy of the droplet have been measured during the coalescence. The capillary waves propagation and damping are studied in detail. The relation between these waves and the partial coalescence is discussed. Additional viscous mechanisms are proposed in order to explain the asymmetric role played by both viscosities.