Drop Fragmentation at Impact onto a Bath of an Immiscible Liquid

The impact of a drop onto a deep bath of an immiscible liquid is studied with emphasis on the drop fragmentation into a collection of noncoalescing daughter drops. At impact the drop flattens and spreads at the surface of the crater it transiently opens in the bath and reaches a maximum deformation, which gets larger with increasing impact velocity, before surface tension drives its recession. This recession can promote the fragmentation by two different mechanisms: At moderate impact velocity, the drop recession converges to the axis of symmetry to form a jet which then fragments by a Plateau-Rayleigh mechanism. At higher velocity the edge of the receding drop destabilizes and shapes into radial ligaments which subsequently fragment. For this latter mechanism the number $N\propto {\mathrm{We}}^3$ and the size distribution of the daughter drops $p(d)\propto d^{-4}$ as a function of the impact Weber number We are explained on the basis of the observed spreading of the drop. The universality of this model for the fragmentation of receding liquid sheets might be relevant for other configurations.

Figure 1

Subsurface view of a water drop ($d_0=2.92\mathrm{mm}$) impact onto a deep silicone oil bath (${\nu }_B=10{\mathrm{mm}}^2{\mathrm{s}}^{-1}$) showing the drop recession after spreading, for increasing impact velocities: (a) $V=0.73{\mathrm{m}\mathrm{s}}^{-1}$, (b) $V=2.26{\mathrm{m}\mathrm{s}}^{-1}$, (c) $V=2.96{\mathrm{m}\mathrm{s}}^{-1}$. For each sequence, the images are taken at the same time $t$ after the first contact.

Figure 2

Number of drops $N$ produced per impact versus Weber number We, for ${\nu }_B=10{\mathrm{mm}}^2{\mathrm{s}}^{-1}$. The black vertical lines separate the no-fragmentation regime (1), the axial (2), and azimuthal (3) fragmentation regimes. The red line is $N\sim {\mathrm{We}}^3$ according to equation (2). Inset: Diameter $d$ of the largest (filled circle) and second largest daughter drops (hollow circle).

Figure 3

Bottom view of the flattened drop recession after impact emphasizing the azimuthal destabilization of the receding drop edge. The white line follows the drop contour.

Figure 4

Maximal area ${\Sigma }_m$ covered by the flattened drop versus We for increasing bath viscosities ${\nu }_B$. The black line has slope one. Inset: Threshold Weber number for axial fragmentation (hollow symbols) and for azimuthal fragmentation (filled symbols) regimes. ${\mathrm{We}}_c>1250$ for ${\nu }_B=200{\mathrm{mm}}^2{\mathrm{s}}^{-1}$.

Figure 5

Mechanism explaining the broad distribution of daughter drop sizes. Time increases from top to bottom.

Figure 6

Normalized distribution functions of the daughter drop diameters $d$ for $\mathrm{We}=550$ (blue circles), 800 (green triangles), 970 (orange squares), and 1170 (red inverted triangles) with ${\nu }_B=10{\mathrm{mm}}^2{\mathrm{s}}^{-1}$. The black line is $p(x)=(8/9)x^{-4}$ from equation (5). Inset: Mean drop diameter $⟨d⟩$. The black line is $⟨d⟩\propto {\mathrm{We}}^{-1}$ according to (2).