# Universality of satellites in the breakup of a stretched fluid bridge

Anna Frishman1,2 Daniel Lecoanet3,4,2 1Technion Israel Institute of
Technology, 32000 Haifa, Israel  2Princeton Center for Theoretical Science,
Princeton University, Princeton, New Jersey 08544, USA  3Department of
Engineering Sciences and Applied Mathematics, Northwestern University,
Evanston 60208, IL, USA  4CIERA, Northwestern University, Evanston 60201, IL,
USA

###### Abstract

As a fluid object breaks, it often leaves behind satellite fragments. Here we
show that satellite formation can follow universal dynamics, leading to robust
satellite sizes. Specifically, we consider the breakup of a slowly stretched
fluid bridge, which we realize experimentally using a soap-film bubble
suspended between two plates. Combining experiments and one-dimensional
simulations, we show that a main satellite bubble always forms as the bridge
breaks. We discover that the size of the bubble is highly reproducible and can
be dramatically increased by stretching the bridge faster or increasing its
volume. The satellite size is a simple function of two non-dimensional
parameters: the normalized volume of the bridge and the Weber number,
measuring inertia due to stretching as compared to surface tension. These
observations can be explained by tracing the bridge evolution over a series of
dynamical stages in which the bridge: (i) closely follows a sequence of
equilibrium bridge configurations; (ii) stretches as it begins to breakup
after reaching an unstable equilibrium; and (iii) follows a universal breakup
solution. The last stage takes place over a finite region, the corresponding
length scale determined by stretching during the previous stage. This length
scale controls the satellite size, and the universality of the dynamics makes
the system highly reproducible. This work suggests universal satellite
formation dynamics may provide a route for understanding satellite bubble
sizes in turbulent flows.

##  I Introduction

The fragmentation of fluid objects is central to many industrial and natural
processes, from ocean-atmosphere exchange in breaking waves [1], to the
preparation of emulsions [2]. Small satellite drops are often generated in the
process, either in individual breakup events or through their sequence.
Predicting the size and number distribution of satellites in fragmentation is
thus a fundamental problem of practical importance. For breakup where viscous
effects dominate over inertia, the sizes of satellites based on individual
breakup events have been studied in detail [3, 4]. Their formation can be
controlled to a high degree by controlling the external flow [5]. When viscous
forces are comparable to inertia, it was recently discovered that satellite
sizes can be universal [6]. However, when viscous forces are negligible
compared to inertia (high Reynolds number flow) [7, 8, 9], initial conditions
and external perturbations seem to be important. A well known example is the
Rayleigh-Plateau instability, responsible for the breakup of a jet into drops.
The size and number of fragments then depends on the initial perturbation,
which determines the relevant unstable mode [7, 10]. For high Reynolds number
flows, especially for breakup driven by turbulence [11, 12, 13, 14], such
initial conditions are often unknown and control of the external flow can be
difficult.

While the size of satellites can be sensitive to the precise flow conditions,
the dynamics around pinch-off points are often universal due to their singular
nature [15, 16, 17]. Namely, the dynamics become independent of the initial
conditions and spatial details away from the pinch-off point.111Exceptions to
universal behaviour have also been observed [18, 19], e.g. due to asymmetries
[20]. The universal pinch-off behavior occurs on small lengthscales, in many
cases, much smaller than the size of satellites. This suggests satellite sizes
may be determined by processes unrelated to pinch-off.

A prototypical system that exhibits fragmentation is the fluid bridge, which
has been studied since at least Plateau [21]. Fluid bridges remain a subject
of continued interest due to their ubiquity and practical and fundamental
importance, see [22] and references therein. Refs. [23, 24, 25, 26] studied
fragmentation of soap film bridges between two open rings. The bridge
fragments, forming one main satellite bubble, when the separation between the
rings is larger than a critical value. While experiments and simulations show
the size of the satellite bubble is reproducible, these works do not determine
what sets this size. For inertia-driven breakup, if the system is reflection-
symmetric around the initial necking point, the breakup generically produces a
satellite bubble instead of pinching off at a single point [27, 24, 28, 29].
The satellite forms because the inertia of the fluid builds up near the
necking point, flattening the surface until it overturns, and sealing a
satellite bubble upon breakup.

We extend this work to consider a fluid bridge between two solid plates, such
that the volume of the bridge is conserved. It is then driven to breakup as it
is slowly stretched by the movement of its end plates. This setup includes two
new dimensionless control parameters—the non-dimensionalized volume and
speed—and we determine the satellite size as a function of these parameters.
We realize this set-up using a soap bubble placed between two solid plates,
forming a soap-film bridge. As the evolution is controlled by the dynamics of
the air inside the bridge [25], we accompany the experiments with simulations
of a general fluid bridge, where only the air dynamics are modeled. These
minimal simulations reproduce the main findings of our experiments,
demonstrating the relevance to e.g. a liquid bridge. While previous works have
studied fragmentation of liquid bridges and fixed-volume fluid bridges
experimentally, e.g., [30, 27, 31], and numerically, e.g., [32, 33], none
addressed satellite formation.

Each of our experiments is initiated with a cylindrical soap-film bridge, as
shown in Fig. 1$(a)$. The initial aspect ratio is denoted by $L_{0}$. The
plates are pulled apart as described in Appendix A. At the plates, the bridge
remains practically pinned to its initial height, denoted by $h_{0}$. We non-
dimensionalize all length-scales using $h_{0}$. The initial volume of air
inside the bridge, $V$, is given by $V/(h_{0}^{3})=\pi L_{0}$ and it remains
fixed throughout the evolution (Fig. 1$(b)$). Thus, $L_{0}$ is a non-
dimensional control parameter for our experiments.

The breakup of the bridge is driven by surface tension. To quantify the
relative importance of inertia and surface tension we introduce the Weber
number $\text{We}=\rho v_{p}^{2}h_{0}/\sigma$, where $2v_{p}$ is a
representative speed with which the plates separate, $\rho$ is the density of
air inside the bridge and $\sigma$ is the surface tension. The We number is
our second non-dimensional control parameter. Our experiments are performed in
the regime of low Weber number, high Reynolds number,222Viscosity becomes
important only at late stages of the breakup and around the pinch-off points,
a known effect, e.g. [34, 16, 35] low Ohnesorge number (small viscosity
compared to surface tension effects) and low Bond number333The combination of
low Weber number, high Reynolds number, and low Bond number is easily achieved
for a soap film bridge. This regime was also achieved for liquid bridges in
Ref. [30], which also formed satellite drops. (see Appendix E).

As a fixed-volume bridge is stretched, it eventually breaks, always leaving
behind two spherical caps at the plates and one central satellite
bubble,444Upon breakup the central bubble is attached to the two caps via
fluid strings. Those strings then break into a multitude of very small
satellite bubbles, which we do not consider here. as in Fig. 1$(a)$ and Fig.
1$(b)$. Here, varying We and the non-dimesionalized volume, we reveal that the
formation of the satellite follows a universal route, resulting in robust
satellite sizes, see Fig. 1. The corresponding universal dynamics are novel,
occurring on a length scale which remains finite in time, see the third panel
in Fig. 1$(b)$, and are unrelated to the dynamics at the pinch-off points.
Moreover, we show that the volume of the resulting satellites has a simple
functional dependence on the two control parameters and is an increasing
function of both. This provides the first example in the high Reynolds number
regime where satellite sizes can be characterized without knowledge of the
exact flow conditions and their control.

Figure 1: $(a)$ Soap-film bridge experiments. Left panels show initial
conditions with different $L_{0}$, the ratio of the initial length to the
radius $h_{0}$ of the cylindrical bridge. Right panels show the final state
with a central satellite bubble for experiments with different $L_{0}$ and
Weber number ${\rm We}$. We study bridges with typical radii $\sim 1\,{\rm
cm}$. $(b)$ Snapshots of a bridge with $L_{0}=2.4$ pulled apart with ${\rm
We}=0.02$ to form a central satellite bubble. The snapshots are evenly spaced
in time. $(c)$ The volume of the satellite bubble normalized to the initial
bubble volume as a function of ${\rm We}$; different colors show different
$L_{0}$. Each dot represents a single experiment. $(d)$ Volume of the
satellite bubble normalized by $h_{c}^{3}$, where $h_{c}$ is the radius of the
center of the bridge when it is flat (see panel $(b)$). We find that
$V_{b}\approx 13h_{c}^{3}$ in both experiments and simulations. $(e)$ $h_{c}$
as a function of $\sqrt{{\rm We}}\,L_{0}$, in experiments and simulations. In
all cases, we find $h_{c}$ is linearly proportional to $\sqrt{{\rm
We}}\,L_{0}$ plus a constant term. In both plots, dots show the ensemble
average of experimental results with error bars representing one standard
deviation; lines show results from simulations.

##  II Numerical Modelling

We aim to capture the phenomenology of the breakup dynamics within the
simplest possible model, focusing on satellite formation. This will allow us
to probe the generality and robustness of the obtained results, and reveal
their physical origin. In particular, we focus on the air dynamics within the
bridge, and do not model the soap film itself. The bridge in our experiments
remains axisymmetric for the bulk of the dynamics, as they are performed in
the low Bond number regime. Its shape is thus captured by the radius along the
shape, equivalent to the height $h(z)$ as measured from the axis of symmetry
in the images, see Fig. 1$(b)$. To further simplify the model, we use the
slender jet approximation [36], which here can be justified both by the
axisymmetry of the bridge, and by the fact that it becomes increasingly
slender as the dynamics proceeds (Appendix E). In non-dimensional form,
normalizing time by the surface-tension time-scale $t_{\sigma}=\sqrt{\rho
h_{0}^{3}/\sigma}$, the inviscid equations read

$\displaystyle\partial_{t}v+vv_{z}=-p_{z}\qquad$ $\displaystyle
p=\frac{1}{h\sqrt{1+h_{z}^{2}}}-\frac{h_{zz}}{(1+h_{z}^{2})^{3/2}}$ (1)
$\displaystyle\partial_{t}h+vh_{z}=-v_{z}\frac{h}{2}$ (2) $\displaystyle
h\left(t,\frac{\pm L(t)}{2}\right)=1\qquad$ $\displaystyle v\left(t,\frac{\pm
L(t)}{2}\right)=\pm\sqrt{\text{We}}$ (3)

where $v(z)$ is the axial velocity of the air inside the bridge, the non-
dimensional bridge length is $L(t)=L_{0}+2t\sqrt{\text{We}}$, and the non-
dimensional speed of the boundary is $v_{p}=\sqrt{\text{We}}$. Partial
derivatives with respect to $z$ are denoted by a subscript. The momentum
balance appears in equation (1), with inertia on the one hand and the pressure
difference due to surface tension, proportional to the mean curvature, on the
other. Viscous effects are neglected (Appendix E). Equation (2) represents
conservation of mass $\propto h^{2}$ inside the bridge. We use pinned boundary
conditions for the height in our simulations, Eq. 3, since we do not observe
significant slippage in experiments (Appendix F.1). We solve Eqs. 1-3
numerically (Appendix B).

##  III Main results: satellite sizes

Our first main result is that the volume of the satellite bubble, formed as a
by-product of the breakup, increases with We and $L_{0}$, as can be seen in
Fig. 1$(a)$ from the experimental data (the same trends are found in
simulations). We quantify the increase in size of the satellite in Fig. 1$(c)$
where the satellite volume normalized by the initial volume, $V_{b}/V$, is
plotted as a function of We for three values of $L_{0}$. For each $L_{0}$ and
We number we performed 10 experiments, the results of which fall on top of
each other as seen in Fig. 1$(c)$. Thus, the volume of the final bubble is
extremely reproducible and robust. In particular, special care to control the
airflow around the bridge or have the same precise velocity profile for the
plates is not required to reproduce the same values.

Figure 2: The fractional bubble volume, $V_{b}/V$ as a function of ${\rm
We}\,L_{0}^{4/3}$ in $(a)$ experiments, and $(b)$ simulations. The colors
denote different initial aspect ratios $L_{0}$, and the dashed lines represent
our theoretical formula, Eq. 5. The inset shows the general agreement between
experiments (symbols) and simulations (lines).

Denoting by $h_{c}$ the radius at the mid point of the bridge when it is flat,
as shown in Fig. 1$(b)$, we find that $V_{b}\approx 13h_{c}^{3}$ in
simulations and experiments, demonstrated in Fig. 1$(d)$ where lines show
results from simulations and dots show an ensemble average over experimental
results. Thus, $h_{c}$ is a single characteristic length scale determining the
final satellite volume. Furthermore, for $h_{c}$ we find the simple linear
dependence:

$h_{c}=h_{c}^{0}+a\sqrt{\text{We}}\,L_{0}$ (4)

as shown in Fig. 1$(e)$. The values of $h_{c}^{0}$ and $a$ can be obtained
from a linear fit of $h_{c}$ as a function of $\sqrt{\text{We}}L_{0}$. We find
that $a\approx 0.2$ ($a\approx 0.15$ in simulations), while
$h_{c}^{0}=AL_{u}(L_{0})$ with $A\approx 0.014$ in simulations and
experiments. As described below, $L_{u}(L_{0})$ is the critical bridge length
for a bridge of volume $L_{0}$ beyond which there is no equilibrium bridge
shape [37].

Therefore, $V_{b}/V$ is captured by the detailed functional form:

$\begin{split}\frac{V_{b}}{V}&=\frac{13\left(AL_{u}+a\sqrt{\text{We}}\,L_{0}\right)^{3}}{\pi
L_{0}}\\\
&=\frac{13}{\pi}\left(AL_{u}L_{0}^{-1/3}+a\sqrt{\text{We}\,L_{0}^{4/3}}\right)^{3}\end{split}$
(5)

The factor $L_{u}(L_{0})L_{0}^{-1/3}$ can be directly computed from the known
equilibrium shapes. While it generally depends on $L_{0}$, for the range
considered here the variation is not large, and, since $A/a$ is small, will be
apparent only for $\text{We}\,L_{0}^{4/3}<10^{-2}$. Indeed, the fractional
final volume $V_{b}/V$ roughly collapses as a function of the combined
parameter $\text{We}\,L_{0}^{4/3}$, as shown in Fig. 2$(a)$ for the
experiments and in Fig. 2$(b)$ for the simulations. A slight spread with
$L_{0}$ is observable for simulations with $\text{We}\,L_{0}^{4/3}<10^{-2}$.
In the same figures, formula (5) is shown as a dashed line for $L_{0}=1.8$,
using the corresponding inferred constants $a$ and $A$. It captures the
satellite volume across roughly two orders of magnitude extremely well, both
in experiments and simulations.

Below we will offer an explanation for these findings, suggesting that formula
(5) results from three dynamical stages. Initially, the bridge evolves quasi-
statically: its shapes closely matching a sequence of static equilibrium
shapes (Sec. IV.1). When the aspect ratio reaches $L_{u}$, an instability at
the center of the bridge develops (Sec. IV.2). We will show that the duration
of this stage is independent of $\rm We$ and is proportional to $L_{0}$. The
length of the central region is thus increased by a factor proportional to
$\sqrt{\text{We}}\,L_{0}$ during this stage, due to the bridges stretching. In
the third stage, the dynamics follow a universal evolution controlled by a
single length scale, proportional to $h_{c}$ (Sec. IV.3). These two
observations combine to give Eq. 5 for the final satellite volume, as
described in Section V. We summarize and discuss possible implications of our
findings in Sec. VI.

Figure 3: Sketch of stages of the dynamics. Data is taken from the evolution
of a simulated liquid bridge with ${\rm We}=0.07$ and $L_{0}=1.8$; all
variables are non-dimensionalized with $h_{0}$ and the surface-tension time-
scale. The line shows $h_{zz}$ at the center of the bridge; at representative
points denoted with dots, we plot the full shape of the bridge. In the first
stage of evolution (blue), the bridge follows a sequence of equilibrium
shapes. The unstable equilibrium is reached once $L(t)=L_{u}$. This begins the
second stage of evolution (yellow) which lasts for $\Delta
t_{b}=\mathcal{O}(1)$. In the second stage, the central region of the bridge
stretches as the breakup process begins. Finally, the central region
approaches a universal breakup trajectory, which is the third stage of
evolution (red). The universal trajectory is plotted as a dashed line.

##  IV Stages of the breakup dynamics

As the bridge stretches, it undergoes three distinct phases prior to
fragmentation: quasi-static dynamics, stretching and instability onset, and
universal breakup dynamics, as summarized in Fig. 3.

###  IV.1 Quasi-static dynamics

Our experiments and simulations are in the regime $\text{We}\ll 1$, where the
plate separation is slow compared with the surface-tension time scale. Thus,
as the length of the bridge $L(t)$ slowly increases with time, the bridge
takes a surface-area minimizing shape. Such constant mean curvature shapes
have been extensively studied in the past [37, 38]. In particular, for pinned
boundaries, zero Bond number and a conserved volume $V$, there is a unique
stable, connected equilibrium shape for each pair $V,L$ up to a critical
aspect ratio $L_{u}(V)$ (Appendix D). We show the close agreement between the
shapes in simulations and the sequence of equilibrium bridge shapes when
$L\lesssim L_{u}$ in Appendix C. At $L(t)=L_{u}$ the equilibrium solution
becomes unstable, while for $L>L_{u}$ a constant mean curvature solution with
the given volume no longer exists [37]. Quasi-static dynmaics ends once the
bridge reaches this critical length $L_{u}(L_{0})$. This is typically the
longest dynamical phase.

As breakup starts at $L(t)=L_{u}$, the length of the bridge at pinch-off,
$L_{f}$, approaches the critical value $L_{f}\to L_{u}(L_{0})$ in the limit
${\rm We}\to 0$. In Fig. 4$(a)$ we plot $L_{f}$ as a function of
$\sqrt{\text{We}}$ for the experimental data (the same dependence is observed
in simulations), where dots represent an ensemble average over experiments and
error bars show one standard deviation. A linear fit is overlaid, and
experiments with different $L_{0}$ cross the y-axis at different points,
corresponding to $L_{f}^{0}\equiv L_{f}(\text{We}=0)$. As expected, we find
$L_{f}^{0}\approx L_{u}$ (see Appendix C).

Figure 4: $(a)$ The final bridge length $L_{f}$ as a function of $\sqrt{{\rm
We}}$. We find $L_{f}$ is linearly proportional to $\sqrt{{\rm We}}$, but with
a slope and constant offset $L^{0}_{f}$, extracted through a linear fit, both
of which depend on $L_{0}$. Dots show the ensemble average of experimental
results with error bars representing one standard deviation. $(b)$
$L_{f}-L_{f}^{0}$ as a function of $\sqrt{\rm We}\,L_{0}$ for experiments with
different initial aspect ratios $L_{0}$. The inset includes simulation data,
depicted with lines for different $L_{0}$. While $L_{f}-L_{f}^{0}$ is linearly
proportional to $\sqrt{\rm We}\,L_{0}$ in both experiments and simulations,
the constant of proportionality is larger in experiments than in simulations.

###  IV.2 Stretching and instability onset

Once $L(t)=L_{u},$ the bridge becomes unstable and starts to breakup. In this
stage, the bridge stretches before it is attracted to the universal breakup
solution (described in the next section). This stretching phase starts when
$L(t)=L_{u}$, and ends when the central region of the bridge becomes flat
(e.g., Fig. 1$(b)$ and Fig. 3).

We find the duration of this stretching phase, $\Delta t_{b}$, is independent
of We, and increases with $L_{0}$. As the bridge is stretched at a constant
velocity $2\sqrt{\text{We}}$, and the duration of the final breakup phase is
very short, we expect that $L_{f}\approx L_{f}^{0}+2\sqrt{\text{We}}\Delta
t_{b}$. We find that $L_{f}-L_{f}^{0}$ increases linearly with $\sqrt{\rm
We}$, as shown in Fig. 4$(a)$, implying the time it takes for the bridge to
break $\approx\Delta t_{b}$, is independent of ${\rm We}$. We also notice that
experiments with different $L_{0}$ have a different slope, i.e. different
$\Delta t_{b}$. In particular, Fig. 4$(b)$ shows that $\Delta t_{b}\propto
L_{0}$, giving the functional dependence
$L_{f}=L_{f}^{0}+b\sqrt{\text{We}}\,L_{0}$.

The same features are observed in simulations, presented alongside experiments
in the inset of Fig. 4$(b)$. Note that the slope $b$ differs between
simulations and experiments (Fig. 4$(b)$). We obtain $b=2.4$ in experiments
and $b=1.3$ in simulations, implying this stage is twice as long in the
experiments than in the simulations. This may be related to differences in the
bridge dynamics near the boundaries. However, these details do not alter the
scaling of the satellite volume with $\sqrt{\text{We}}\,L_{0}$, which is the
robust feature of interest here.

We have demonstrated that the bridges breakup time is unaffected by the
stretching of the bridge. More generally, the dynamics are similar to that of
a soap film bridge between two open rings held at its critical length [24].
Initially, the fluid accelerates away from the necking point due to the
pressure imbalance, and both the advection term and the inertia induced by the
stretching boundaries are negligible. As in Ref. [24], this process of fluid
acceleration continues until advection from the central region becomes as
important as pressure gradients. The inertia induced by stretching, which is a
small effect when $\text{We}\ll 1$, is then subdominant compared to the
inertia that has developed due to the instability, as we demonstrate in
Appendix D. As advection becomes important, fluid starts to escape the entire
central region, causing the height to flatten, corresponding to the start of
the universal breakup stage. In this next stage the surface will overturn,
sealing a satellite.

While the onset of the instability appears to be insensitive to the stretching
of the bridge, stretching does enter in two important ways. First, close to
the boundaries of the bridge the induced inertia cannot be neglected. The
induced We-dependent flow is thus negligible only in a finite region around
the bridge center, corresponding to the breakup region, whose extent we denote
by $z_{b}$. Second, the stretching of the bridge changes the bridges length,
thus changing the local curvature, and so the pressure, everywhere—including
in the central region, see Appendix D. Our finding that the duration of the
breakup is roughly independent of We is therefore surprising, and is a feature
that remains to be explained.

Figure 5: $(a)$ Bridge shape at the pinch-off time in experiments (dashed
lines), and simulations (solid lines), for $L_{0}\approx 1.1$ and various
${\rm We}$. When normalized by $h_{b}$, the bridge shapes of the experiments
and simulations at different ${\rm We}$ collapse near the central region.
$(b)$ Bridge shapes in simulations at the pinch-off time, with different
$L_{0}$ (colors) and ${\rm We}$ (high curves correspond to lower ${\rm We}$).
The central region of the bridges collapses to a universal shape independent
of $L_{0}$ and ${\rm We}$. $(c)$ Bridge shapes in simulations at $t_{c}$, the
time at which $h_{zz}(z=0)=0$, with different $L_{0}$ and ${\rm We}$. $(d)$
$h_{zz}(z=0)$ as a function of time, both normalized using $h_{c}$. Dots
represent the trajectories of different simulations with $L_{0}\approx 1.1$,
colored by their ${\rm We}$. For the highest ${\rm We}$, we plot a black cross
at the time of the last stable equilibrium when $L(t)=L_{u}$. While
simulations with different ${\rm We}$ initially have different curvatures,
they are attracted to a universal breakup solution prior to $t_{c}$ where
$h_{zz}=0$.

###  IV.3 Universal breakup dynamics

The dynamics in the central region of the bridge (away from the plates) become
universal towards the end of the breakup. We first demonstrate the
universality of the shape of the bridge in the central region at the pinch-off
time $t_{f}$. In Fig. 5$(a,b)$, we plot the height $h(z)$ at $t_{f}$, with
both lengths rescaled by $h_{b}$, the height at the center of the bridge at
this time. We see a very good collapse of the shape in the central region,
independent of both $L_{0}$ and We. The collapse is only seen in the central
region, not near the boundaries. While there is good agreement between
experiments and simulations in the central region in Fig. 5$(a)$, the bridges
take quite different shapes outside this region.

To show that the collapse is not simply dictated by properties of the shape at
the final time, in Fig. 5$(c)$ we present the shape of the bridge at the time
$t_{c}$ when the surface flattens ($h_{zz}(t_{c},0)=0$, see Fig. 3). This is
the beginning of this universal phase. We again rescale both axes by $h_{b}$;
the collapse of the central region is evident. In particular, as expected
$h_{b}\propto h_{c}=h(t_{c},z=0)$.

Using results from numerical simulations, we now wish to demonstrate that the
breakup dynamics follow a universal time-dependent trajectory. That is, a
finite region forms near the center of the bridge where, if both time and
length-scales are properly rescaled, the height and velocity dynamics follow a
universal trajectory independent of We and $L_{0}$. We have already
demonstrated that the central region of the bridge takes a universal form when
rescaled by the length-scale $h_{c}$, at times $t_{c}$ and $t_{f}$. Taking
$h_{c}$ as the relevant length scale and assuming that breakup is driven by
surface tension, from dimensional analysis we expect the temporal scaling to
be $(t_{f}-t)\propto h_{c}^{3/2}$. This is the well known surface tension-
inertia scaling, as in Ref. [16], based on the length scale $h_{c}$. In Fig.
5$(d)$ we plot the curvature at the center of the bridge in such rescaled
coordinates. Indeed, under this rescaling, trajectories with different We all
asymptotically converge to the same universal curve. We observe that
simulations with higher We approach this curve later in (rescaled) time, but
that by the time $t_{c}$ ($h_{zz}=0$), all our simulations follow the
universal dynamics. While the scaling $t-t_{f}\propto h_{c}^{3/2}$ is not new,
to the best of our knowledge this is the first time it is observed for a
universal solution occupying a finite, constant in time, spatial region.
Previous studies of inertia-surface tension driven breakup focused on the
pinch-off points and the similarity dynamics there [34, 24, 39, 16, 35].

We now discuss why universal dynamics emerge in this system. The universal
breakup phase occurs quickly, i.e., much faster than the surface tension time
$t_{\sigma}$ (see Fig. 3). Thus, the length of the bridge is roughly constant
during this phase (equivalent to ${\rm We}=0$). Next, we estimate the typical
height of the central breakup region to be $h_{c}$, which is much smaller than
the height of the bridge at the boundaries $h_{0}=1$.

We consider the ansatz
$h=h_{c}H\left[\frac{t_{f}-t}{t_{c}},\frac{z}{z_{c}}\right]$ and
$v=v_{c}V\left[\frac{t_{f}-t}{t_{c}},\frac{z}{z_{c}}\right]$ in equations (1)
& (2), and require that all terms are of the same order in $h_{c}\ll 1$. We
expect that in the breakup region the inertia and surface tension terms are
comparable, although We based on the plate velocity is assumed to be small.
That is because the pressure imbalance, due to the instability, drives a fast
flow with a local ${\rm We}\sim\mathcal{O}(1)$. Using the ansatz that all
terms in (1) & (2) are of the same order, we find that the horizontal scale
$z_{c}\propto h_{c}$. In addition, $v_{c}\propto z_{c}/t_{c}$ and, as expected
from an inertia-surface tension balance, $t_{c}\propto h_{c}^{3/2}$, in
agreement with our findings in Fig. 5 (a more careful analysis, showing that
the bridges length $L\approx L(t_{f})$ does not enter, is presented in
Appendix D). Note that this temporal scaling is consistent with our assumption
that the bridge is hardly stretched during this stage: since $(t_{f}-t)\propto
h_{c}^{3/2}\ll\Delta t_{b}=O(1)$, the change in $L$ is negligible.

We thus obtain the following universal form

$\begin{split}&h(t,z)=h_{c}H\left[\frac{t_{f}-t}{h_{c}^{3/2}},\frac{z}{h_{c}}\right]\equiv
h_{c}H[T,Z]\\\
&v(t,z)=h_{c}^{-1/2}V\left[\frac{t_{f}-t}{h_{c}^{3/2}},\frac{z}{h_{c}}\right]\equiv
h_{c}^{-1/2}V[T,Z]\end{split}$ (6)

with the functions $H[T,Z]$ and $V[T,Z]$ satisfying equations (1)-(2), which
do not depend on any external parameters. External parameters can enter
through the boundary conditions, but the outlined considerations apply only to
the breakup region, and so the relevant boundary conditions are to be taken at
$z=z_{b}$. To determine those we assume that $z_{b}/h_{c}\gg 1$ so that in
terms of the variable $Z$ the boundary is at infinity. We require that the
solution in the breakup region match the external solution, where we assume
that the velocity is finite, i.e. that $v(t,z_{b})$ does not scale with
$h_{c}$. We thus get the boundary condition
$V\left[T,Z\right]=v(t,z_{b})/v_{c}=h_{c}^{1/2}v(t,z)\to 0$ as $Z\to\infty$.
Next, assuming that the height of the external solution has a finite
curvature, i.e. that $\partial_{zz}h(t,z_{b})$ is independent of $h_{c}$, we
get the condition $\partial_{ZZ}H[T,Z]\to h_{c}\partial_{zz}h(t,z_{b})=0$ as
$Z\to\infty$. The matching is thus to be done at the inflection point of the
external solution. Equivalently, this implies the boundary condition
$H[T,Z]/Z\to O(1)$ as $Z\to\infty$. To summarize, under these assumptions, the
boundary conditions for $H,V$ which replace (3) are

$\displaystyle H[T,Z\to\infty]\propto Z$ $\displaystyle
V\left[T,Z\to\infty\right]=0$ (7)

Equations (1)-(2) which $H,Z$ satisfy together with the boundary conditions
(7) are seen to be universal, independent of external parameters. The breakup
of the bridge should be captured by the attracting dynamical solution of this
system, explaining our findings in Fig. 5.

##  V The satellite bubble volume

Here we combine our insights from the previous section to explain the
dependence of the volume of the satellite bubble on We and $L_{0}$ as shown in
Fig. 2 and quantified in Eq. (5). We expect for the volume of the satellite
that $V_{b}\propto h_{b}^{2}l_{b}$, where $h_{b}$ is the height at the center
of the bridge, and $l_{b}$ the lateral distance between the pinch-off points
at the final time (see Fig. 3). In Figs. 5$(a)-(c)$ we have shown that
$l_{b}\propto h_{b}\propto h_{c}$, explained by the universal breakup solution
(Eq. 6). Thus, in agreement with our observations in Fig. 1$(d)$,
$V_{b}\propto h_{c}^{3}$, with a universal constant of proportionality.

We next explain the linear dependence of $h_{c}$ on $\sqrt{\rm We}$ and
$L_{0}$ (Eq. (4)). As $l_{b}$, the length of the final bubble is proportional
to $h_{c}$, we will equivalently argue why $l_{b}$ has a linear dependence on
$\sqrt{\rm We}$ and $L_{0}$. Without stretching, starting with a bridge of
critical length $L_{u}$, we expect for the satellite length to be proportional
to the total bridge length $l_{b}^{0}\propto L_{u}$. This motivates the
relation

$h_{c}^{0}=A\,L_{u}$ (8)

with $A$ independent of initial conditions and $L_{0}$.

For finite We, the breakup region extends during the stretching phase
(dynamical stage II), so we have $l_{b}=l_{b}^{0}+2v_{b}\Delta t_{b}$, where
$v_{b}\approx v_{p}=\sqrt{\rm We}$ is the stretching velocity of the breakup
region (Appendix C). Since $h_{c}\propto l_{b}$, and $\Delta t_{b}\propto
L_{0}$, we recover the linear dependence of $h_{c}$ on $\sqrt{\rm We}$ and
$L_{0}$ in Eq. 4. Our experiments and simulations have slightly different
values of $a$, the proportionality constant between $h_{c}$ and $\sqrt{\rm
We}\,L_{0}$, which is due to differences in the breakup time and the
stretching velocities of the breakup region. Put together, these correspond to
values of $a$ which are 25% smaller in the simulations than the experiments.
Combining equations (8) and (4) together with the finding $V_{b}=13h_{c}^{3}$
we recover Eq. (5). The differences in $a$ between experiments and simulations
lead to satellite volumes which are smaller by $\sim 50\%$ in the simulations
(inset of Fig 2(b)). Crucially, these differences do not alter the properties
of the universal dynamics or the scaling of the satellites volume.

##  VI Summary and discussion

A fluid bridge that is held between two plates and slowly stretched will
eventually break, undergoing a dramatic topological change. We find that, when
the bridge is stretched at constant speed, the outcome of the breakup is
highly predictable. A small satellite bubble always forms, its volume uniquely
determined by two non-dimensional parameters: the (normalized) volume $L_{0}$
of the bridge and the normalized stretching speed $\sqrt{\text{We}}$. By
varying these two parameters, the volume of the satellite bubble can be made
to vary over two orders of magnitude, reaching up to $10\%$ of the total
bridge volume.

The high reproducibility of the bubble size can be explained by the
universality of the breakup dynamics, revealed by our simulations and
experiments. In the slow stretching regime $\text{We}\ll 1$ considered here,
the stretching has a negligible effect on the breakup dynamics itself, which
are spatially localized to the central part of the bridge. Instead, the
dramatic dependence of the satellite bubble size on We is a consequence of the
stretching of the breakup region, occurring prior to the universal breakup
solution. Correspondingly, we find a simple dependence of the bubble size on
the combination $\sqrt{\text{We}}\,L_{0}$ which arises due to two robust
features. First, the universal breakup solution depends on a single length
scale determining the bubble size. Second, it takes a fixed time, independent
of the stretching and proportional to $L_{0}$, to approach the universal
solution (starting from a bridge with the unstable equilibrium length
$L_{u}(L_{0})$). It is mainly during this time interval that the breakup
region is stretched. The bubble size can thus be increased either by
increasing $L_{0}$, which prolongs the time over which the breakup region is
stretched prior to the convergence to the universal solution, or by increasing
We, increasing the stretching rate of that region during the approach.

Our results also shed light on the universal features of static breakup, i.e.,
the breakup dynamics of a fluid bridge of a fixed critical length
$L_{u}(L_{0})$. In particular, while the universality of pinch-off dynamics is
a well established phenomena, e.g. [16], the novelty here is that the
attracting universal solution spans a finite region in space, remaining finite
even as pinch-off is approached. Thus, it is not a similarity solution.
Instead, we find that the extent of the breakup region is proportional to
$L_{u}(L_{0})$, with a universal proportionality constant. What selects this
constant, determining the disconnected solution that the universal dynamics
converges to (and thus the satellite size), remains to be understood. The
dependence of the duration of the breakup process on the liquid bridge volume
$L_{0}$, is another robust aspect of the breakup process which merits future
investigation.

Our work demonstrates that the singular nature of breakup can lead to
universality of satellite formation, going beyond the universality of pinch-
off dynamics. This opens the possibility of universality of satellite
formation driven by other force balances—as is the case for pinch-off—an
exciting prospect for understanding satellite formation and drop-size
distributions in turbulent flows [13].

## Acknowledgements

The authors thank Albane Smilga for inspiring this project, and Bruno Le Floch
for the introduction. We thank Howard Stone for generously hosting us in his
lab, and a critical reading of this manuscript. We also thank numerous members
of the Stone lab for their generous help: Janine Nunes, Ching-Yao Lai, Suin
Shim and Maksim Mezhericher. We gratefully acknowledge Jens Eggers for useful
conversations and Kinneret Keren for helpful comments on this manuscript. A.F.
would like to thank Yuval Elhanati and Eldad Afik for advice on the image
analysis. A.F. and D.L. are supported by PCTS fellowships, and BSF grant No.
2022107. A.F. is supported by ISF grant No. 486/23. D.L. is supported in part
by NASA HTMS grant 80NSSC20K1280 and NASA OSTFL grant 80NSSC22K1738.
Computations were conducted with support by the NASA High End Computing
Program through the NASA Advanced Supercomputing (NAS) Division at Ames
Research Center on Pleiades with allocation GID s2276.

##  Appendix A Experiments

###  A.1 Experimental setup

For our experiments, we form bubbles using a soap solution, using a 10:1
mixture of distilled water and blue Dawn dish soap. The thickness of the soap
film in our experiment was smaller than $500\,{\rm nm}$ (blue colors could be
observed in the interference patterns on our soap films). Using a syringe we
form a bubble of a fixed volume and place it between two wetted acrylic
plates. To achieve the desired initial aspect ratio, we slowly pull the plates
apart to form a cylindrical soap bubble. During this process, the contact
points slip, corresponding to a fixed contact angle with the plates of
$\pi/2$. For these Neumann boundary conditions, the maximum initial aspect
ratio is $L_{0}=\pi$.

To drive the bubble to breakup, we pull apart the acrylic plates using a
motor. The plates are glued onto two pillars which are mounted onto a low
friction track. The pillars are pulled by strings which are connected to an
electric motor through a pulley, converting the rotation of the motor to
linear motion of the pillars and plates. The speed of the plates is controlled
by changing the input voltage to the motor. We use a high speed camera to
capture the breakup process, using a frame-rate in the range of $4000-6500$
frames per second, depending on We. The breakup process takes $\sim
30-300\,{\rm ms}$.

###  A.2 Data analysis

We analyze the images with the scikit-image Python package [40], using the
canny edge detector to extract the soap-film height and the final bubble
volume. The height is computed as the half-distance between the upper and
lower free surface of the bridge. The final bubble volume is computed after
the bridge breaks, once the bubble relaxes to a spherical form. Then, for a
series of images, we compute the bubble area and extract its radius. The
volume is then computed based on the average (over the series of frames)
radius. We compute the velocity $v_{p}(t)$ from the soap-film bridge length
$\frac{L(t)}{2}$. We fit $\frac{L(t)}{2}$ using a cubic spline (through the
UnivariateSpline scipy package [41]) and then take its derivative.

We estimate $t_{c}$, the time at which the curvature at the center of the
bubble is zero, by finding the first time the curvature in the central region
is negative. We define the central region using a We-dependent height
threshold. There are sometimes oscillations of the location of the minimum
before the curve overturns at the center, which typically leads to an
overestimation of $h_{c}$. To mitigate the effects of these oscillations on
our analysis, we also require that in the next image the curvature in the same
region remains negative. Still, we believe that these oscillations bias the
determination of $h_{c}$, leading to the lower values of $V_{b}/h_{c}^{3}$
seen in Fig. 1$(d)$ for We in the range $0.006-0.03$ (where we empirically
observe such oscillations).

##  Appendix B Numerical methods

To solve Eqns. 1-3, we change coordinates to $(s,\tau)$, where
$s=z/(\frac{L(t)}{2})$ and $\tau=\log(L(t)/L_{0})=\log(1+2t/L_{0})$ (note this
is a different definition of $\tau$ than used in Appendix D), and $t$ is non-
dimensionalized with the advection timescale. Then the equations of motion
become

$\displaystyle\partial_{\tau}v-s\partial_{s}v+{\rm
We}^{-1}\partial_{s}p=-v\partial_{s}v$ $\displaystyle\partial_{\tau}\log
h-s\partial_{s}\log h+\frac{1}{2}\partial_{s}v=-v\partial_{s}\log h$
$\displaystyle
p=\frac{e^{\tau}}{h\sqrt{e^{2\tau}+\left(\frac{2}{L_{0}}\right)^{2}h_{s}^{2}}}-\left(\frac{2}{L_{0}}\right)^{2}\frac{e^{\tau}h_{ss}}{\left[e^{2\tau}+\left(\frac{2}{L_{0}}\right)^{2}h_{s}^{2}\right]^{3/2}}$
$\displaystyle h(\tau,\pm 1)=1\qquad v(\tau,\pm 1)=\pm 1$ (9)

We solve the equations of motion with the Dedalus pseudospectral framework
[42]. We represent $v$, $\log h$, $h_{s}$, and $p$ as Chebyshev series with
1024 modes. Nonlinear terms are calculated using the 3/2 dealiasing rule,
although the presence of more complicated nonlinear products introduce some
aliasing errors. We initialize with $p=\log h=1$, $h_{s}=0$, and
$v=\tanh\left[(2.2/\pi)\tan(\pi s/2)\right]$. We run simulations for $2/L_{0}$
ranging from $0.8$ to $2.0$ in increments of $0.5$, and for ${\rm
We}=10^{-4},$ $10^{-3}$, and 30 values ranging from $10^{-2.5}$ to $10^{0.5}$,
equally spaced in logarithm.

To timestep the equations, we use implicit-explicit timestepping methods from
[43]. In particular, we use the 2nd order, two-stage Runge–Kutta method for
the ${\rm We}=10^{-4}$ simulations, and the 3rd order, four-stage Runge–Kutta
method for all other simulations. All terms linear in the perturbation
variables are treated implicitly, and all nonlinear terms are treated
explicitly. As the bridge approaches breakup, the equations become more stiff.
To improve the stability of the timestepping scheme, we define a base state,
and evolve the (fully nonlinear) deviation of each variable away from this
base state. This allows us to implicitly timestep more linear terms in the
equations than evolving the variables directly. We set the base state to be
given by the problem variables every 100 timesteps. We initially take
timesteps of size $10^{-4}$, and evolve the system until breakup. To get
better time resolution near breakup, we then restart from the simulation state
between 150-200 timesteps before the breakup, taking timesteps that are half
as large. We continue to iteratively restart the simulations with smaller and
smaller timesteps until the timestep size is $6.25\times 10^{-8}$. For ${\rm
We}=10^{-4}$, we start with timesteps of size $2\times 10^{-4}$, and
iteratively restart the simulations until the timestep is $1.25\times
10^{-7}$.

##  Appendix C The three dynamical stages

### Quasi-static bridge evolution

In Fig. 6 we plot equilibrium bridge shapes along with simulated bridge shapes
in simulations with ${\rm We}=0.07$. The simulated bridges are very close to
the equilibrium bridges, until we approach $t_{u}$, the time at which the
equilibrium bridge becomes unstable. As ${\rm We}$ decreases, the quasi-static
dynamics follow the equilibrium shapes more closely.

Figure 6: Equilibrium bridge shapes (black lines) and simulated bridge shapes
(dotted yellow lines) for ${\rm We}=0.07$. We include bridges with different
initial aspect ratios, and at different times. Figure 7: $(a)$ The soap-film
bridge length at breakup $L_{f}$ (experiments) and the linear fit from which
we extract $L^{0}_{f}$ and $\Delta t_{b}$ in the main text. $(b)$ $\Delta
t_{b}/L_{0}$ as a function of $L_{0}$ as extracted from a linear
fit$L_{f}=L_{f}^{0}+2\sqrt{\text{We}}\Delta t_{b}$ in the simulations. $(c)$
$L_{0}^{f}$ as a function of $L_{u}(L_{0})$. Crosses show simulation results,
red dots show experimental results. The dashes line shows the relationship
$L_{0}^{f}=L_{u}$, which is very well satisfied in the simulations. Figure 8:
$(a)$ The satellite length at breakup $l_{b}$, as a function of
$\sqrt{\text{We}}$ and the linear fit used in the main text (experiments).
$(b)$ $l_{b}$ as a function of $h_{c}$ for experiments and simulations. $(c)$
$l^{0}_{b}/L^{0}_{f}$ as a function of $L_{0}$ as extracted from linear fits
in the simulations. ($d$) The ratio of speeds of the breakup region stretching
and that of the plates, $v_{b}/v_{p}$, as a function of $L_{0}$, extracted
from the slope in the linear fit of $l_{b}$ and $\Delta t_{b}$.

### Final bridge length $L_{f}$ and time to breakup

We show the bridge length at breakup $L_{f}$ as a function of
$\sqrt{\text{We}}\,L_{0}$ in Fig. 7 $(a)$, where each point represents a
single experiment and the lines show the linear fit. In Fig. 7 $(b)$ we show
how the breakup timescale $\Delta t_{b}/L_{0}$, extracted from the linear fit
$L_{f}=L^{0}_{f}+2\sqrt{\text{We}}\Delta t_{b}$ for each simulation, varies
with $L_{0}$. Fig. 7 $(c)$ shows $L_{0}^{f}$, extracted from this fit, as a
function of $L_{u}$, the unstable bridge length. In the simulations,
$L_{0}^{f}$ is very close to $L_{u}$. In the experiments, $L_{0}^{f}$ is
somewhat larger, likely due to slippage of the bridge contact line for low
${\rm We}$ experiments (Appendix F.1).

### The characteristic scale in the universal regime: $l_{b}$ and $h_{c}$

In Fig 8 we present the scaling of $l_{b}$, the lateral distance between
pinch-off points at the breakup time of the bridge. In panel $(a)$ we show the
ensemble averaged data from experiments as a function of We, along with a
linear fit to $l_{b}=l_{b}^{0}+c\sqrt{\text{We}}\,L_{0}$. It can be seen that
the different lines indeed have different slopes corresponding to the
different $L_{0}$. In panel $(b)$ we show that the data collapses when plotted
as a function of $h_{c}$ and that there is a linear dependence between $l_{b}$
and $h_{c}$. In this panel, the dots correspond to experimental results, and
the lines correspond to simulation results.

Panel $(c)$ shows the ratio $l_{b}^{0}/L_{f}^{0}$ as a function of $L_{0}$ for
the simulations, showing it is roughly constant. In panel $(d)$,
$v_{b}/v_{p}\equiv(l_{b}-l_{b}^{0})/(L_{f}-L_{f}^{0})=c/b$ is shown as a
function of $L_{0}$ for our range of simulations using the linear fits. The
ratio is very near to being constant, equal to $1.3$.

##  Appendix D Theoretical considerations

###  D.1 Equilibrium shapes

The equilibrium bridge shapes are completely characterized by two parameters,
which are typically chosen to be $L=2l/h_{0}$, where $2l$ is the length of the
bridge, and $S=V/V_{cyl}$ where $V$ is the volume of the bridge, and $V_{cyl}$
is the volume of a cylindrical bridge with length $L$ and height $h_{0}$. The
parameters $S$ and $L$ are not independent during the dynamics:
$S(t)=l(0)/l(t)=L_{0}/L(t)$. Note, that we always start from an initial
cylindrical bridge, i.e $S(0)=1$. We use the parameterization introduced in
[37] to determine the equilibrium shape for each $(L,S=L_{0}/L)$ pair (solving
the corresponding equations numerically using Mathematica). For each initial
aspect ratio $L_{0}$, we determine the critical bridge length $L=L_{u}$ once
these equations no longer have a solution.

###  D.2 Inertia driven double pinch-off in a symmetric set-up

Let us first rewrite equations (1)-(3) in a coordinate system $(s,\tau)$ where
the inertia from stretching of the bridge is more evident:
$z=\frac{L(t)}{2}s,\quad\tau=t$ so that
$\partial_{s}=\frac{L(t)}{2}\partial_{z},\quad\partial_{\tau}=\partial_{t}+s\sqrt{\text{We}}\partial_{z}$.
Using the modified velocity $u=v-s\sqrt{\text{We}}$, we arrive at

$\displaystyle
l(\tau)\partial_{\tau}u+u(\partial_{s}u+\sqrt{\text{We}})=-p_{s}$
$\displaystyle
p=\frac{l(\tau)}{h\sqrt{l(\tau)^{2}+h_{s}^{2}}}-\frac{l(\tau)h_{ss}}{(l(\tau)^{2}+h_{s}^{2})^{3/2}}$
$\displaystyle
l(\tau)\partial_{\tau}h+u\partial_{s}h=-\frac{h}{2}(\partial_{s}u+\sqrt{\text{We}})$
$\displaystyle h(\tau,\pm 1)=1\qquad$ $\displaystyle u(\tau,\pm 1)=0$ (10)

with $l(\tau)\equiv\frac{L(\tau)}{2}=\frac{L_{0}}{2}+\tau\sqrt{\text{We}}$.

We see that if $p_{s}\sim u\partial_{s}u$ while
$\partial_{s}u\gg\sqrt{\text{We}}$ (equivalently
$\partial_{s}v=l(t)\partial_{z}v\gg\sqrt{\text{We}}$), the inertial term would
no longer be negligible but the explicit dependence on We would still
disappear from the equations. Indeed, this is what happens at the end of the
second stage, as a fast outflow develops in the central region (note the
flattening of the free surface of the bridge) implying $\partial_{z}v\gg 1$
there. On the other hand, this condition is not satisfied close to the
boundaries: from equation (2) and the boundary conditions (3), the condition
$\partial_{s}u=-\sqrt{\text{We}}$ should be dynamically satisfied there
(reflecting that the surface is pinned at the boundaries so that the radial
velocity should remain zero there). Note that once the fluid inertia becomes
important, in the limit $\partial_{s}u\gg\sqrt{\text{We}}$, also $u\approx v$
so that at leading order we can use $u=v$ in the breakup region. Thus we do
not make a distinction between $u$ and $v$ in our discussion of the universal
solution in the main text.

Here we derive the universal breakup dynamics more carefully, taking into
account the length of the bridge $l(\tau)\approx l(\tau_{f})$ as an additional
parameter (which depends on We and $\rm L_{0}$) entering the dynamics, as seen
in equations (10). Similarly to the main text, we use the ansatz
$h=h_{c}H\left[\frac{t_{f}-t}{t_{c}},\frac{s}{s_{c}}\right]$ and
$u=u_{c}U\left[\frac{t_{f}-t}{t_{c}},\frac{s}{s_{c}}\right]$. Note that here
the anzatz is in the $s$ coordinate, which is normalized by the length of the
bridge, as apposed to $z$. Plugging it into equation (10) with $\text{We}=0$,
and requiring that all terms are of the same order in the limit $h_{c}\ll 1$,
we get that $h_{c}=ls_{c}=z_{c},$ $v_{c}=u_{c}=ls_{c}/t_{c}=z_{c}/t_{c}$ and
$t_{c}=h_{c}^{3/2}$. As compared to the main text, $l$ appears explicitly in
the dynamical equations. Then, the equations for $H,U$ become universal in the
limit $h_{c}\ll 1$, i.e. independent of $l$, only if one assumes
$ls_{c}=h_{c}$. This gives the relation we use in the main text $h_{c}=z_{c}$,
resulting in the scaling form (6) for $H,U\equiv V$, which then satisfy
equations (1)-(2), where $l$ no longer appears.

##  Appendix E Modeling

###  E.1 Compressibility, viscosity and gravitational effects

The compressibility of the air inside the bridge can be neglected. The largest
Mach number based on the plate velocity is $\text{Ma}=v_{p}/c_{s}\sim
10^{-2}$. The flow velocity induced by surface tension will be at most an
order of magnitude larger since $\text{We}\sim 0.1$ in those experiments and
$h_{c}\sim 0.1h_{0}$. Thus, the largest Mach number based on the flow velocity
will still be $\ll 1$. Furthermore, the pressure difference induced by surface
tension is at most of the order of $\sigma/h_{min}\sim 10^{2}\,{\rm N}/{\rm
m}^{2}$ (taking $h_{min}=0.05h_{0}\approx 10^{-4}\,{\rm m}$), while the
atmospheric pressure is $1\,\text{atm}=10^{5}\,{\rm N}/{\rm m}^{2}$, so that
the deviations in the density of the air are of the order $10^{-3}$.
Equivalently, we may compare the scaling of the pressure gradients in the
leading order $\propto\text{Ma}^{-2}$ to the stress induced by surface tension
$\propto\text{We}^{-1}$ in the momentum balance equation, giving
$\text{Ma}^{-2}\gg\text{We}^{-1}$, i.e., the flow is incompressible in the
leading order.

We measure the surface tension of our soap solution to be $\sigma=25\,{\rm
mN}/{\rm m}$. It is measured using the shape of a pendant-drop, using a matlab
code developed for this purpose in the Howard Stone lab. Viscous stresses
become comparable to surface tension stresses at the scale
$l_{\nu}=\rho\nu^{2}/\sigma=5\times 10^{-9}\,{\rm m}$ where $\nu$ is the
kinematic viscosity of air $\nu\approx 1.5\times 10^{-5}\,{\rm m}^{2}/{\rm
s}$. The maximal Ohnesorge number based on the minimal height $h_{c}\sim
1\,{\rm mm}$ in the experiment is given by $\text{Oh}=\sqrt{l_{\nu}/h_{c}}$
and is of the order of $\text{Oh}\sim 10^{-3}$. The Ohnesorge number at the
pinch-off points does eventually become $O(1)$, so that viscous effects become
important. This however does not influence the earlier breakup stages in the
central region of the bridge on which we focus here. In addition, the minimum
Reynolds number in the experiments is
$\text{Re}=\sqrt{\text{We}}/\text{Oh}\sim 10^{2}$ implying that viscous
effects are sub-leading both to inertia and surface tension during the breakup
and before pinch-off.

The maximal Bond number is $\text{Bo}=g\rho h^{2}/\sigma\approx 0.05$ so that
the effects of gravity are negligible (as evidenced by the axisymmetry of the
soap-film bridge up to breakup).

###  E.2 The soap-film surrounding the bridge

The inertia of the liquid in the soap film, compared to that of the air inside
the bridge, may be neglected in our experiments for most of the breakup
process. To compare the inertia of the two we assume that the acceleration of
the two fluids is comparable, meaning that we only need to compare their
masses. The thickness of the soap film is smaller than $500\,{\rm nm}$, so we
estimate $h_{s}\sim 1\,\mu\text{m}$. The density of the soap solution is
approximately that of water, $\rho_{s}\sim 10^{3}\,{\rm kg}/{\rm m}^{3}$. The
condition on the height of the bridge such that the soap solution inertia is
negligible is thus: $2\pi h_{min}h_{s}\rho_{s}\ll\pi h_{min}^{2}\rho_{a}$,
giving $h_{min}\gg 2\rho_{s}/\rho_{a}h_{s}\sim 10^{-3}\,\text{m}\approx
0.1h_{0}$. The smallest bubble heights occur in the central region before
breakup. These are smallest for the lowest We, reaching heights of
$0.05h_{0}$. Thus, our lowest We experiments may be somewhat affected by soap
film inertia in the final stages of breakup.

###  E.3 The one dimensional approximation

We model the dynamics by the equations of a slender jet, even when the aspect
ratio $L$ is not large. Initially, the justification for this approximation is
that deviations from a radially constant axial velocity $v_{z}(r,z,t)=v(z,t)$
and pressure $p(r,z,t)=p(z,t)$ are small: the initial pressure is uniform
(constant atmospheric pressure and constant mean curvature for equilibrium
shapes), and the boundary conditions for the velocity (which drives the
dynamics) are also uniform in the radial direction. Then, working in
cylindrical coordinates with $0\leq r\leq h(z,t)$, we can expand around a
radially constant solution for $v_{z}\approx
v_{z}^{0}(z,t)+\tilde{v}_{z}(r,z,t)$ and the pressure
$p(r,z,t)=p_{0}(z,t)+\tilde{p}(r,z,t)$, assuming deviations
$\tilde{v}_{z},\tilde{p}$ to be small in amplitude. The radial velocity is
then determined by incompressibility and is given in the leading order by
$v_{r}=-v_{z}^{0}r/2$, and in particular at the free surface
$v_{r}=-v_{z}^{0}h/2$. This is the same relation obtained within the slender
jet approximation, which means that $v_{z}^{0}$, $p_{0}(z,t)$ and $h(z,t)$
satisfy the equations used in the slender jet approximation [36] (i.e the
momentum equations for $v_{r},v_{z}$ are self consistent to leading order, so
that radial perturbations do not grow). In addition, when the bridge is
stretched radial perturbations become more and more suppressed, as it enters
into the regime of a slender jet.

###  E.4 Differences between experiments and modeling

There are several physical aspects in which the detailed dynamics in
experiments and simulations are distinct. One difference is that the
stretching protocol is not the same: while in simulations the edges are pulled
with a constant speed throughout, in experiments that speed invariably changes
in time, potentially with asymmetries between the two plates. Another
difference is in the boundary conditions. There is some slippage of the edges
of the bridge in experiments while those are pinned in simulations (Appendix
F.1).

##  Appendix F Experimental Considerations

###  F.1 Contact line slippage

Most of our experiments were performed rapidly enough so that the soap film
bridge effectively remained pinned at the boundaries. However, for the
smallest aspect ratio we used, $L_{0}\sim 1.1$, and especially for the slowest
experiments, $\text{We}\sim 0.002$, we did observe slippage of the soap film
at the boundaries. Indeed, we observe an $\sim 20\%$ slippage of the contact
line of the soap film bridge at the plates for the lowest We. We observe this
phenomenon for all the bridges with these parameters, an example is given in
Fig. 9 $(a)$ where the shape of bridges with three different We is shown. This
indicates that the timescales forstretching and slipping are similar in these
experiments, probably due to the long time it takes to reach $L_{u}$ for this
$L_{0}$ ($L_{u}$ being more than twice larger than $L_{0}$). The main effect
of slippage is to increase the actual normalized volume $L_{0}$ for these
experiments (since we use the height at the plates to normalize the volume),
pushing $L_{u}$ to higher values. We therefore discard the three lowest We
experiments for the inference of $L^{0}_{f}$. Despite this, we find the
difference between the expected $L_{u}$ and $L_{f}^{0}$ to be the largest for
this aspect ratio.

Figure 9: $(a)$ The soap film bridge shape at breakup for $L_{0}=1.1$. Note
that the bubble height at the boundaries is lower than $1$, implying there was
slippage at the boundaries. $(b)$ Asymmetry in the breakup shape for the
highest We for three different aspect ratios, $(L_{0},{\rm
We})\approx(1,0.48)$, $(1.7,0.35)$, and $(2.4,0.24)$. The asymmetry is
inherited from the asymmetry between the two plates.

###  F.2 Definition of We

To define the Weber number, we must determine the velocity of the plates for
each experiment. In the experiments the velocity changes with time, as the
speed with which the bridge is stretched increases from zero until it roughly
saturates. We present the corresponding velocity profiles (as extracted from
$l(t)$) in Fig. 10. To define We for each experiment we compute the time-
averaged square velocity, beginning from a threshold speed equal to
$0.15\,{\rm m}/{\rm s}$, so that the acceleration phase is not included, and
more weight is placed on the later stages (when the relevant dynamics are
expected to occur). In Fig. 10 $(d)$ we show how that threshold compares with
the slowest experiments. Introducing such a threshold is physically justified
for the purposes of the analysis in the main text since we expect the early
stages to correspond to the quasi-static phase of the dynamics, which is
irrelevant for the breakup dynamics in the later stages. However, it is also
true that while We plays a key role in the analysis in the main text, it is
not uniquely defined in the experiments (since the plate velocity does not
quite plateau). We have tried different definitions (such as using the
averaged velocity or the velocity when the bridge length increases by a
particular, large enough, increment) which did not produce a significant
difference for our results except for a slight multiplicative increase in We.
Note that the discrepancy between experiments and simulations could not be
resolved by such an increase, as the required shift in We would be too large,
corresponding to plate velocities which are larger than the maximal plate
velocity in the experiment (also, a single such correction could not produce a
match between experiments and simulations for both $L_{f}$ and $V_{b}/V$).
Finally, note the ratio of the plate velocity and the stretching velocity of
the breakup region $v_{b}/v_{p}$ as inferred for the experiments is
independent of such multiplicative factors in the definition of We.

Figure 10: $(a)$ Velocity profile for a particular motor rotation speed,
producing We-number in the range $1.6-2.5$. $(b)$ Histogram of the initial
aspect ratio of the soap-film bridge in the experiments. $(c)$ Velocity
profiles for the different experiments. $(d)$ Zoom in on small velocities and
the threshold used for the definition of We.

###  F.3 Symmetric breakup in experiments and role of acceleration

Two important features of our pulling protocol are that the pulling of the two
plates is roughly symmetric (but see below for comments on boundary
asymmetries) and that accelerations are small. These two features cause the
initial necking of the bridge (where over pressure initially builds up and the
instability develops) to occur roughly at the mid point between the two
plates. Generally, wherever the initial necking point forms, the stretching
can be considered to be locally symmetric around that point if its location
does not accelerate. Then, there is no preferred direction in the reference
frame of that point. If that is not the case, asymmetric features in the
height profile will have time to develop. The condition for symmetric breakup
is thus that the externally imposed acceleration is slower than that induced
by surface tension: $\sigma/(\rho h^{2}\dot{v})\gg 1$, where $h$ is a typical
height and $\dot{v}$ a typical acceleration scale related to the movement of
the boundaries. In our experiments, while there is always some acceleration of
the plates, that acceleration remains small (i.e., since we do not reach high
We), and we thus remain in the symmetric regime and do not observe breakup
features that are due to acceleration, as explored in [31].

For the highest We experiments we observe a systematic asymmetry in the
breakup: the pinch-off always first occurs on the left, as seen in Fig. 9
$(b)$ for a few examples with different We and $L_{0}$. This asymmetry seems
to originate from an asymmetry at the boundaries. Note however that this did
not seem to have a significant affect for the earlier breakup dynamics during
the universal dynamics, and in particular did not seem to alter the scaling of
the satellite bubble size for these We.

###  F.4 Realizability of the relevant regime for a liquid bridge

While satisfying $\text{We}\ll 1$ for a liquid bridge requires a smaller set-
up and slower speeds than those used in our experiments, it is still possible
to keep ${\rm Re}\gg 1$. Indeed, for a water bridge with plate velocities of
the order of $v_{p}\sim 10\,{\rm cm}/{\rm s}$ and boundary radius
$h_{0}=1\,{\rm mm}$ (such that it is below the capillary wavelength), one
finds $\text{We}\sim 10^{-1}$ and $\text{Re}\sim 10^{2}$. Ref. [27] also
includes bridges in this regime: using a larger set-up ($h_{0}\sim 10{\rm
cm}$) with slower speeds ($v_{p}\sim 1{\rm mm/s}$) such that ${\rm We}\sim
0.001$ and ${\rm Re}\sim 10^{2}$ (placed in a Plateau tank to achieve a small
Bond number).

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