Universal Rim Thickness in Unsteady Sheet Fragmentation

Unsteady fragmentation of a fluid bulk into droplets is important for epidemiology as it governs the transport of pathogens from sneezes and coughs, or from contaminated crops in agriculture. It is also ubiquitous in industrial processes such as paint, coating, and combustion. Unsteady fragmentation is distinct from steady fragmentation on which most theoretical efforts have been focused thus far. We address this gap by studying a canonical unsteady fragmentation process: the breakup from a drop impact on a finite surface where the drop fluid is transferred to a free expanding sheet of time-varying properties and bounded by a rim of time-varying thickness. The continuous rim destabilization selects the final spray droplets, yet this process remains poorly understood. We combine theory with advanced image analysis to study the unsteady rim destabilization. We show that, at all times, the rim thickness is governed by a local instantaneous Bond number equal to unity, defined with the instantaneous, local, unsteady rim acceleration. This criterion is found to be robust and universal for a family of unsteady inviscid fluid sheet fragmentation phenomena, from impacts of drops on various surface geometries to impacts on films. We discuss under which viscous and viscoelastic conditions the criterion continues to govern the unsteady rim thickness.

Figure 1

Dimensionless dispersion relation of the coupled Rayleigh-Plateau (RP) and Rayleigh-Taylor (RT) instabilities for different Bond number Bo compared to that of the Rayleigh-Plateau instability. Inset: fluid cylinder of diameter $b$ subject to an acceleration $-\ddot{R}$.

Figure 2

(a) Unsteady sheet fragmentation upon drop impact on a target of comparable size which ensures a horizontal expanding sheet [22]. Scale bar is 6 mm. Upper inset: rim-ligament separation by our image processing algorithm. Lower inset: schematic diagram of a local corrugation. (b) Time evolution of the sheet rim thickness compared to the local instantaneous capillary length $l_c$ based on the rim acceleration $-\ddot{R}$ for $\mathrm{We}=693$. Time is nondimensionalized by the global capillary time ${\tau }_{\mathrm{cap}}=\sqrt{\rho d_0^3/8\sigma }$ of sheet expansion, where $d_0$ is the diameter of the impacting drop. Inset: local, instantaneous Bond number of the rim $\mathrm{Bo}=1$, robust for three different Weber numbers. (c) Sheet fragmentation from (i) impact near a surface edge, causing the sheet’s left part to expand in the air, and (ii) drop impact on a thin liquid film with a crown rising upward. (iii) Criterion of $\mathrm{Bo}=1$ holds for the rim in the air in case (i) and for the crown rim in case (ii).

Figure 3

(a) Sheet expansion upon drop impact on pole for fluids of different viscosity and elasticity. Scale bars are 6 mm. (b) Time evolution of the local instantaneous rim Bond number for the fluids shown in (a). (c) Regime diagram in terms of the local Reynolds number $\widetilde{\mathrm{Re}}$ and Deborah number $\widetilde{\mathrm{De}}$ defined in (5). Circles represent the experimental conditions for which $\mathrm{Bo}=1$ holds, and squares the conditions for which it fails, for the rim of unsteady expanding sheet upon drop impact on a small surface. The criterion to determine if $\mathrm{Bo}=1$ holds is that $0.8<⟨\mathrm{Bo}⟩<1.2$.

Figure 4

(a) Time evolution of the local Bond number of the rim upon drop impact on a small smooth surface, compared with that upon impact on a rough surface for $\mathrm{We}=693$. The insets show the sheet at time $t=0.06{\tau }_{\mathrm{cap}}$, when the fragmentation is occurring on the rough surface while it has not yet started on the smooth one. (b) Time evolution of the local Bond number of the rim from drop impact on a deep pool for $\mathrm{We}=824$ (circles) and from impact on a solid surface for $\mathrm{We}=543$ (squares), showing that $\mathrm{Bo}=1$ fails in these two cases. (c) Time evolution of the local Bond number of the rim from bubble bursting. The capillary time in this case is ${\tau }_{\mathrm{cap}}^{\prime }=\sqrt{\rho (rh{)}^{3/2}/\sigma }$ with $r=5.6\mathrm{mm}$ and $h=7.4\mu \mathrm{m}$ the bubble cap radius and thickness, respectively. The Bond number is computed based on the centripetal acceleration measured at the early stage of the hole retraction on the bubble cap.