# Inclined to splash: triggering and inhibiting a splash with tangential velocity

James C Bird, Scott S H Tsai, and Howard A Stone

New Journal of Physics 11 (2009) 063017

wiki entry by Emily Russell, Fall 2010

The article can be found here

The phenomenon of a splashing droplet is a common everyday one, and yet the rapid and complex dynamics of a splash are still not understood in full detail. This paper forays beyond the scope of previous models to consider the even-more-realistic case of a droplet splashing on a surface with a relative tangential velocity. Experimental observations yield pretty pictures showing asymmetric splashing, in which one side of the droplet spreads on the surface, while the other side splashes. A fairly simple model of the dynamics of the lamella, the film that spreads out from the droplet as it impacts and which sometimes forms into a splash, provides a convincing scaling law for the splash threshhold.

## Salient Features of the Model

As in previous models of droplets splashing upon normal incidence, the lamella begins to spread simply because the fluid of the bottom of the drop must be displaced as the drop impacts a solid surface. The speed at which the lamella expands is arrived at by simple geometric arguments on the rate at which the fluid is displaced.

Whether the lamella continues to spread, or splashes, depends on the balance of energy between the kinetic energy of the lamella, and the surface energy of the new fluid-vapor interface. The surface energy will tend to prevent deformations of the lamella, inhibiting splashing; in order for a splash to occur, the kinetic energy must be much greater than this surface energy. In the case of a droplet with a tangential velocity relative to the surface, the relevant kinetic energy is modified.

The final piece of the puzzle is to ignore the short-time behavior (where, for example, the velocity of the lamella diverges under this simple model), justified by the argument that so long as the contact radius of the drop is expanding faster than the lamella, it does not make sense to talk about an independent lamella which can undergo splashing. Hence, the model comes into play only after some critical time. It is at this critical time that the spreading or splashing behavior of the lamella is determined, depending on the balance of kinetic and surface energy.

These relatively straightforward arguments lead to a definite prediction as to the scaling behavior of the splash threshhold, the line on the phase diagram separating splashing behavior from spreading behavior. Previous models of normal (perpendicular) splashing predicted that splashing would occur for $We Re^{1/2} > K$, with K a constant depending on the surface and atmosphere, and the Weber number $We = \frac{\rho V^2 R}{\gamma}$ measuring the balance between inertial and surface forces, while the Reynolds number $Re = \frac{V R}{\nu}$ measuring the balance between inertial and viscous forces. The model developed in this paper reduces to this previous prediction for normal impact, but derives the more general threshhold $We Re^{1/2} (1 - k \frac{V_t}{V_n} Re^{-1/2})^2 > K$, with k a constant of order 1, and $V_t$ and $V_n$ the tangential and normal velocities, respectively.

## Results and Discussion

Of course, a mere prediction based on a model is little use if it is not compared to experimental data. After dropping many drops of different sizes from different heights and onto surfaces both tilted and moving, the authors present a phase diagram which shows that when the experimental parameters are appropriately non-dimensionalized, the data collapse into regions separated by a splash threshhold which compares very well to their prediction. The prediction also nicely separates the regions of total splashing, total spreading, and asymmetric splashing.

The paper does not go into inordinate mathematical detail, but presents geometrical and energy arguments to develop a relatively simple yet effective model for the complex phenomenon of splashing, which combines inertial, viscous, and surface effects to describe a phenomenon based on instabilities in thin films of liquid, that is, the lamella. The figures and diagrams are illuminating and convincing, and the subject matter is interesting both to the fluid mechanic or soft matter physicist, and to many a layman who has been splashed while running through falling rain.