# Precursors to droplet splashing on a solid surface

Original entry by Joerg Fritz, AP225 Fall 2009

## Source

Shreyas Mandre, Madhav Mani, and Michael P. Brenner: Physical Review Letters, 2009, 102, pp 134502-1 to 134502-4.

## Summary

Fig.1 A corona splash.

Under the right conditions a drop impacting on a solid surface creates a splash, a process everyone in a city as rainy as Boston is very familiar with. The origin of this splash is traditionally thought to be a thin sheet of fluid that is ejected parallel to the surface near the point of impact. Motivated by recent experiments that show a dependence of the splashing behavior on the ambient pressure this paper explores an alternative explanation. In this picture the drop does not contact the surface at all, but instead compresses the air below it into a thin layer without making contact. During this process regions with a very high curvature at the water-air interface are created. These are the origin of capillary waves, which could ultimately provide a different way to produce the splash.

## The classical picture

If we associate the splash with the ejection of a thin sheet of fluid very shorty after impact, we first have to explain where this sheet comes from. The classic explanation of this effect assumes that the drop essentially does not deform before impact. If $t=0$ is the time of contact between the drop and the surface then the drop falling with V would have penetrated the surface to a depth $a=V t$ at time t. Thus the area of contact between drop and water has a radius of r determined by the geometry

$\left (R-V t \right )^2 + r^2 = R^2$

where R is the radius of the drop. For small times this means

$r \approx \sqrt{2 R V t}$

and thus the speed of the rim of the wetted area scales like

$dr/dt \approx \sqrt{R V / (2 t)}$

which at first will be faster than the speed of sound in the medium, thus water molecules at the edge have initially no information that the drop has made contact. Due to the $1/t$ scaling, the rim velocity will eventually fall below the speed of sound. This would lead to a very sudden increase in pressure at the rim, which we can imagine to be the source of the ejection for the observed sheet.

This framework of thought has several problems

1. it predicts that the sheet of liquid would be ejected parallel to the surface, whereas the experimentally observed sheets are launched at an angle
2. if the sheet initially starts at the surface there is not only no mechanism that could explain a detachment from it, the sheet would also be slowed down by friction with the surface comparatively quickly
3. there is no reason why the ambient pressure should have any influence on the formation of the sheet, again in contrast to experimental observations

These shortcoming of the classical "theory" are the motivating backgroud for the idea presented in the paper.

## An alternative idea

Fig.2 The height and pressure profile of the gas layer for early times. Notice the formation of a dimple and the development of the two minima/maxima in the height/pressure.
Fig.3 The full height of the gas layer and a magnified look at the region of the height minimum for later times. Notice the emission of capillary waves.

Recent experiments have shown that when the ambient pressure is reduced below a certain threshold, splashing can be completely suppressed. This threshold pressure scales with the molecular weight of the gas in which the drop is falling, strongly suggesting that the compressibility of the gas is responsible for this surprising behavior.

One possible explanation for this is the hypothesis that the liquid sheet originates from the interaction of the liquid with the compressible gas before impact. To study this idea the authors propose a simple, 2-dimensional model. It couples the flow in an incompressible, inviscid liquid cylinder to a compressible lubrication gas layer underneath. This problem is essentially governed by the interplay of four key players: the inertia and surface tension of the drop and the pressure and viscosity of the gas layer.

A numerical simulation of the model described above shows the following main results:

• while the drop is falling the pressure in the gas layer below is rising and eventually the drop shape develops a dimple, at a critical distance from the solid surface (see figure 1)
• subsequently the pressure develops two maxima where the drop shape deforms strongly with a rapidly increasing curvature (see figure 1)
• the curvature at these points diverges faster than the pressure, which avoids contact between the drop and the surface and creates a kink in the profile of the drop which moves at constant velocity (see figure 2)
• the pressure saturates at the point of minimal gas layer height and these points become the source of capillary waves (see figure 2)

## Interpreting the results

These simulation results indicate that the nature of splashing problem might be very different from what has been assumed up to this point. It appears probable that splashing actually results from the rich interaction between the drop and the gas layer for very small separations of the drop from the solid surface.

The most surprising of these elements is probably that the fluid does not initially contact the solid at all. The calculation neglected intramolecular attractions between the fluid and the solid which will eventually lead to contact for the gap thicknesses, but over time scales much longer than those the paper is concerned with.

The technical difficulties in continuing the calculation in the framework presented here for larger times appears to be very high. So further predictions that connect the emission of the capillary waves more directly with the ejection of a thin film might be problematic. But the model has all the right ingredients that the traditional theory is lacking. It very plausibly explains the ambient pressure dependence of splashing. And if capillary waves lead to the ejection of a sheet it will not be parallel to the surface, which is consistent with observations.

Finally, this paper makes precise predictions about the film thickness of the gas layer as a function of the governing parameters. The highest values of these are just on the borderline of what could be measured with equipment currently available. So an experimental investigation into the phenomenon described here will only be a matter of time.