Threshold for discretely self-similar satellite drop formation from a retracting liquid cone

Predicting the size of droplets that pinch off from a liquid jet is important to applications ranging from bubble-initiated atmospheric aerosols to inkjet printing. These predictions are complicated by smaller satellite drops that form when a thin liquid thread develops and breaks up faster than its ends fully retract. Typically this process is modeled by perturbing a cylindrical liquid thread with an amplitude that is small relative to the cylinder diameter. Yet early on in the pinch-off process, the ends of the liquid thread are conical and lack a characteristic length scale from which to normalize a finite perturbation. Here we numerically simulate the retraction of nearly inviscid conical filaments and introduce self-similar perturbations to drive the system into a discretely self-similar retraction that can enable breakup without biasing a particular length scale. We find that for most cone angles, the perturbation amplitude must exceed a threshold for satellite drops to form. We show that this critical perturbation amplitude depends on the cone angle and can be accurately predicted by an argument based on the static stability of the initial perturbed cone.

Figure 1

Examples and schematics of satellite and subsatellite droplet formation at various times $t$. (a) Two snapshots of the jet and drops ejected by a bubble bursting at an air-water interface. After the jet pinches off at the base, the remaining filament pinches off into smaller satellite drops, highlighted by asterisks (*). (b) Sequence of images of a stream of water released from a syringe, with a filament breaking up into satellite and subsatellite drops after pinching off from the main drop. (c) Schematic of a cylindrical ligament of liquid, before and after breaking up into droplets. (d) Schematic of pinch-off process of a ligament leading to a sequence of satellite drops. The first pinch-off event (top right) is stable and does not create satellite drops. However, it initiates capillary waves, which perturb the remaining conical filament as it pinches off (top left). For a significant enough perturbation, the remaining filament could break up into satellite and subsatellite drops.

Figure 2

Schematic of the initial geometry for cone angle $\theta =0.15$ rad ($8.{59}^{\circ }$), perturbation amplitude $\epsilon =0.5$, and relative wavelength ${\lambda }^{*}=2\pi$. (a) To numerically simulate the cone evolution, a maximum radius $R$ and half-length $Z$ are defined. Zooming in along successively smaller regions [(b)–(d)] demonstrates the cyclic self-similarity of the profile. The self-similarity is truncated after the seventh wave by a spherical cap.

Figure 3

Profile snapshots for $\theta =0.15$ with no perturbation ($\epsilon =0$, top) and with $\epsilon =0.5$ (bottom). Successive times are plotted both in the original coordinate system (left) and with the self-similar scaling (right). (a) With no perturbation, the tips grow as they retract. (b) The profiles collapse when scaled in the self-similar form $\overline{r}\equiv r/{(t^2\gamma /\rho )}^{1/3}$ and $\overline{z}\equiv z/{(t^2\gamma /\rho )}^{1/3}$, and they compare well with the inviscid self-similar solution of Ref. [22]. (c) For $\epsilon =0.5$, profiles are offset from each other based on time, with $t$ increasing from top to bottom. The times plotted are $t_0/r_t^i$ for $i=0–4$ for the blue curves and $t_0/r_t^{i+1/3}$ and $t_0/r_t^{i+2/3}$ for the red and green curves, respectively. Dotted boxes highlight the red profiles with drops about to pinch off taking the same shape. (d) When plotted in self-similar form, the blue, red, and green curves all collapse, illustrating the discrete self-similarity of the retracting cone. The black arrows emphasize the repeated thinning and breakup of the neck near $\overline{z}=5$.

Figure 4

(a) Phase diagram showing the critical perturbation amplitude for a liquid cone to break up into drops as a function of cone angle. Red crosses and cyan circles respectively correspond to simulations with and without breakup. The radius of the circle corresponds to the equivalent radius of the largest drop from the simulation in scaled coordinates $\overline{r}$ and $\overline{z}$. The black curve is the geometric stability threshold ${\epsilon }_g\left(\theta \right)$, above which the initial geometry of the perturbed cone exhibits a local maximum in pressure, as illustrated schematically. (b) Pressure and interface profiles for a single wave from the initial geometry of the perturbed cone described by Eq. (1) with $\theta ={10}^{\circ }$ and $\epsilon$ between 0 and 0.5, with $\mathrm{\Delta }\epsilon =0.05$ between curves. Pressure profiles are normalized by $p_{n+1}\equiv \gamma cos\theta /(Z_{n+1}tan\theta )$, the pressure at $Z_{n+1}$ for an unperturbed cone. Black crosses denote local maxima in pressure and minima in radius, highlighting the transition from a stable monotonically decreasing pressure profile for $\epsilon \le 0.35$ to an unstable profile with a local maximum for $\epsilon =0.4$, 0.45, and 0.5.

Figure 5

Scaled profiles of unperturbed cones ($\epsilon =0$) with angles (a) $\theta =0^{\circ }$ and (b) $1^{\circ }$. (a) Ten profiles are plotted, corresponding to times logarithmically spaced between $0.195\tau$ and $0.962\tau$, with earlier times corresponding to lighter colors. The self-similar solution for an inviscid cone, calculated using the one-dimensional model from Ref. [23], is shown for comparison. Our simulation is clearly not self-similar for $\theta =1^{\circ }$, instead breaking up into drops. (b) For $\theta =2^{\circ }$, the ten plotted times are log-spaced between $0.0498\tau$ and $0.962\tau$. The simulation appears to be converging to a self-similar solution, though it differs notably from the solution from Ref. [23], with a stretched first lobe at the tip.

Figure 6

Comparison between the inviscid self-similar solution for a retracting cone [22] and scaled profile snapshots from simulations with the same angle ($\theta =0.15$), $\epsilon =0$, and various Ohnesorge numbers. Eight times log spaced between $0.0179\tau$ and $0.497\tau$ are plotted from each simulation with earlier times corresponding to lighter colors, as in Fig. 3. [Note that Figs. 3 and 3 used the same eight times along with six earlier ones, with the earliest being $0.00104\tau$.]

Figure 7

Comparison of profile snapshots for $\theta =0.15$, $\epsilon =0.5$ between simulations with the standard mesh refinement and with a finer mesh refinement, plotted with the self-similar scaling. The times plotted correspond to those from Fig. 3 along with two earlier times for each frame, making the earliest time in the top frame $t_0{r}_t^2=8.57\times {10}^{-5}\tau$ or $\overline{t}\equiv logt/t_0=-2T$ (note that $r_t=0.241$ for $\theta =0.15$). A vertical offset proportional to the logarithmic time $\overline{t}$ has been applied such that the earliest times are highest in order to allow direct comparison between refinement levels. The constants used to specify the offsets are $\alpha =0.3/T$, ${\overline{t}}_a=4T$, ${\overline{t}}_b={\overline{t}}_a+T/3$, and ${\overline{t}}_c={\overline{t}}_a+2T/3$.