Fundamental Phenomena and Principles 229

3.2.5Interaction, Spreading and Splashing of Multiple Droplets on a Surface

Interactions of multiple droplets on a surface during impact are of practical interest in many applications. Multiple droplets colliding with a non-flat surface (Fig. 3.29) may cause breakup and bounce of liquid, and formation of voids within liquid due to the entrainment of air/gas.[389] The ejection and bounce of liquid, as well as the void formation may be largely depressed by rapid solidification in surface impact of melt droplets. Experimental, theoretical, and analytical studies by Yarin and Weiss[357] showed that at low impact velocities, a fully liquid droplet in a droplet train striking onto a preceding, flattening droplet on a substrate surface results in spreading in the form of a thin liquid film and propagation of capillary waves (Fig. 3.30), with the impact energy totally absorbed by the waves; At high impact velocities, a cloud of small secondary droplets emerge and splashing of liquid sets in. The pattern of the capillary waves is predicted to be self-similar. The splashing threshold corresponds to the onset of a velocity discontinuity propagating over the liquid layer on the substrate surface. This discontinuity shows several aspects of a shock. In an incompressible liquid, such a discontinuity can only occur in the presence of a sink at its front. The sink results in a circular crown-like sheet virtually normal to the substrate surface, and propagating with the discontinuity radially from the center of impact. The crown is unstable owing to the formation of cusps at the free rims at its top edge under the action of surface tension while the free rims propagate with a velocity independent of their local curvature. At the cusp sites, thin liquid jets are formed, which are unstable, again owing to the action of surface tension, i.e., capillary instability of the Rayleigh type, leading to splashing. The thin liquid jets break up eventually and form very small secondary droplets. In the splashing phenomenon, the role of gravity is negligible owing to the small scale involved, and only inertia and surface tension are of importance.

Fundamental Phenomena and Principles 231

Yarin and Weiss[357] discussed the underlying mechanism governing the spreading and splashing during impingement of a droplet train on a substrate surface, and indicated that the phenomena may be represented by a relatively simple quasi one-dimensional (1-D) model. They derived the governing equations for the surface tension dominated flow in the thin liquid film on the substrate surface, discussed the propagation of capillary waves over such a film, and found the self-similar regimes. They also derived the splashing conditions, explained the physical nature of splashing in terms of a kinematic discontinuity formation in an incompressible liquid, and addressed the structure of this discontinuity. The splashing threshold was described by a correlation between the Capillary number, Ca, and a non-dimensional viscous length λ v such that:

Eq. (48)			Ca = cλ v−3/ 4
where:	Ca =			We/Re = u0 µL /σ
	λ	v	=	(v /f )1/2σ /(ρ v 2)
				L		L L
	c =			u	(ρ /σ)1/4	v -1/8 f -3/8	≈17–18
				0	L	L
		f =		frequency of droplet train

The coefficient c, termed dimensionless impact velocity, was found to depend only slightly on the surface roughness as long as the roughness is much smaller than droplet diameter. Plotting Ca versus λ v in logarithmic scales generates a straight line (Fig. 3.31). In the regime above the line, splashing occurs, whereas in the regime below the line, only spreading takes place during surface impact of a droplet train. The splashing threshold at c ≈ 17–18 corresponds to a developed crown instability, strong enough to produce a cloud of secondary droplets, while crown formation begins slightly below the splashing threshold. Compressibility effect is of importance at the very early stage of splashing, leading to droplet jetting over the substrate surface. However, it is impossible to correlate the onset of splashing with the Mach number of the liquid based on its sonic speed, showing that the early events related to compressibility occur almost instantaneously in the time scale of the whole splashing process.