# Atomisation of an acoustically levitated droplet: Experimental observations
of a myriad of complex phenomenon

S. K. Saroj1  R.M. Thaokar1<EMAIL_ADDRESS>1Department of Chemical
Engineering, IIT Bombay, Mumbai Maharashtra, 400076 India

###### Abstract

We report the dynamics of a droplet levitated in a single-axis acoustic
levitator. The deformation and atomization behavior of the droplet in the
acoustic field exhibits a myriad of complex phenomena, in sequences of steps.
These include the primary breakup of the droplet through stable levitation,
deformation, sheet formation, and equatorial atomization, followed by
secondary breakup which could be umbrella breakup, bag breakup, bubble breakup
or multistage breakup depending on the initial size of the droplet. The
visualization of the interfacial instabilities on the surface of the liquid
sheet using both side and top-view imaging is presented. An approximate size
distribution of the droplet after a complete breakup is also provided. Lastly,
an aggregation of the atomized smaller droplets is observed after the complete
atomization. The primary breakup of the droplet precedes with a stable
levitation of the droplet, when the acoustic force balances the downward
gravity force and the resulting ellipsoidal shape of the droplet is a
consequence of the balance of deforming acoustic force and the restoring
surface tension force. The acoustic force changes with the change in the shape
of the droplet leading to further deformation of the droplet, ultimately
resulting in a highly flattened droplet, with a thin liquid sheet at the edge
(equatorial region). The thinning of the sheet is caused by the differential
acceleration induced by the greater pressure difference between the poles and
the equator. As the sheet thickness reduces to the order of a few microns,
Faraday waves develop at the thinnest region (preceding to rim) which causes
the generation of tiny-sized droplets perpendicular to the sheet. The
corresponding, hole formation results in a perforated sheet that causes the
detachment of the annular rim which breaks due to Rayleigh Plateau (RP)
instability. The radial ligaments generated in the sheet possibly due to
Rayleigh Taylor (RT) instability break into droplets of different sizes. The
secondary breakup exhibits We dependency and includes umbrella, bag, bubble,
or multistage, ultimately resulting in complete atomization of the droplets.
Both the primary and the secondary breakup of the droplet admit interfacial
instabilities such as Faraday instability, Kelvin Helmholtz (KH) instability,
RT instability, and RP instability and are well described with visual
evidence.

##  1 Introduction

A liquid droplet can be levitated in the air using acoustic radiation force
generated by sound waves. The phenomenon, termed acoustic levitation, is one
of the best methods for contactless transportation, deformation, and breakup
of liquid droplets. The levitation of a droplet by sound waves finds several
important applications in the field of material science (Andrade et al.
(2020a)), biology (Akkoyun et al. (2021)), and analytical chemistry
(Bayazitoglu & Mitchell (1995)). For example, Andrade et al. (2020b) developed
a device based on acoustic fields for picking up Surface Mount Device (SMD)
elements from one position and transporting and releasing them at another
position without any contact with the solid object.

A liquid droplet can also undergo fragmentation due to a variety of reasons,
such as the gravity fall of naturally occurring raindrops (Villermaux & Bossa
(2009)), due to the effect of airflow (Kirar et al. (2022)) or due to
impaction with solid surfaces (Lohse & Villermaux (2020); Villermaux & Bossa
(2011)) etc. The atomization of the droplet has important applications such as
internal combustion engines (Lefebvre & McDonell (2017)), aircraft engines (Ju
et al. (2017)), cooling (Breitenbach et al. (2018)), agricultural spray (Law
(2001)), inkjet printing (El-Molla (2017)), pharmaceutical and medical devices
(Naidu et al. (2022)) etc.

When a droplet is placed in the acoustic waves, under specialized conditions,
it could exhibit both levitation as well as fragmentation. The deformation
dynamics of such a levitated droplet, which can eventually lead to breakup and
fragmentation, depends upon the relative interplay of surface tension of the
liquid-air interface and the acoustic pressure. A levitated droplet is known
to exhibit a variety of dynamical shape deformations depending on its size and
ambient conditions. Such a droplet is almost spherical at low sound
intensities and becomes gradually flattens as the sound intensity is
increased. An oblate spheroidal shape is formed at moderate sound intensities
(Andrade & Marzo (2019)). Hong et al. (2017) designed an acoustic levitation
technique by generating an acoustic vortex field, which levitates a wide range
of liquid or solid objects. This methodology has been suggested to control the
deformation of water droplets. Geng et al. (2014) reported experimental and
theoretical investigations of the vertical vibration of a droplet levitated in
an acoustic field. The vibration of the droplet is harmonic due to a
combination of both acoustic radiation and gravity forces. For an equal volume
of the droplet, the vibration frequency varies with the change in shape of the
levitated droplet. A good agreement between the theory and experiment for a
droplet smaller than 2.5 mm was observed.

The acoustically levitated and stretched droplet can eventually undergo
breakup when the inertial and pressure forces overcome the capillary forces.
Aoki & Hasegawa (2020) experimentally investigated the dynamics of
acoustically levitated droplets of water and ethanol. Both stable and unstable
conditions of the levitated droplet were predicted, through dependence on the
diameter of the droplet and the sound pressure. They suggested the following
relation for the acoustic pressure $P_{max}$ beyond which stable levitation is
not possible and a droplet can undergo breakup,

$P_{max}=\sqrt{\sigma\rho_{G}c^{2}\Big{(}\frac{3.2}{d}-\frac{1.3\pi}{\lambda}\Big{)}}$
(1)

where, $\sigma$, $\rho_{G}$, c, d and $\lambda$ are surface tension, density
of air, speed of sound, diameter of droplet, and wavelength of sound wave,
respectively. The typical values of the wavelength of sound in their work were
$\lambda=c/$f$=8.5$ mm for $c=340m/s$ and $f=40kHz$. Thus, the susceptibility
of a levitated droplet to atomization increases as the liquid’s surface
tension decreases and the intensity of sound pressure increases. Andrade &
Marzo (2019) in their numerical study, observed that a levitated water droplet
disintegrates in an acoustic field when the separation distance between the
emitter and reflector is less than resonance. They observed that the droplet
deformed into a pancake shape when the separation distance between the emitter
and reflector was higher than the resonance separation. More recently, the
atomization of the acoustically levitated droplet was observed by some
researchers (Andrade & Marzo (2019); Aoki & Hasegawa (2020); Naka & Hasegawa
(2020)). The atomization process includes flattening the droplet into a pan-
cake shape, developing capillary waves, and splitting the droplet into a large
number of smaller droplets (Andrade & Marzo (2019)), Aoki & Hasegawa (2020);
Naka & Hasegawa (2020). The rapid deformation of the droplet was suggested to
be due to the Kelvin–Helmholtz instability, which creates interfacial
deformation (Naka & Hasegawa (2020)). The measured velocity of the interface
before atomization and the critical velocity from the Kelvin–Helmholtz
instability were compared by Naka & Hasegawa (2020). They reported an
excellent match between experimental and theoretical velocity, whereby the
atomization was attributed to the Kelvin–Helmholtz instability at the
interfacial region of the droplet.

Acoustically levitated droplets have been used to investigate different types
of phenomena. For example, Kumar et al. (2010) studied the vaporization of a
nano-silica encapsulated water droplet by heating with $CO_{2}$ laser in an
acoustic levitator. They observed bowl and ring shape of the droplet during
evaporation in the acoustic field. Further, Basu et al. (2012) investigated
the atomization of acoustically levitated fuel droplets by external heating.
They reported that the onset of the KH instability at the droplet’s edge
induces droplet atomization. The small-scale atomization for diesel, kerosene,
and bio-diesel was attributed to their higher surface temperature. Ethanol
droplets were found to be more stable because of the smaller surface
temperature due to their higher vapor pressure and latent heat of
vaporization. In another study, it was observed that viscous droplets
levitated in acoustic fields break into smaller droplets due to cavitation of
a bubble inside the droplet (Zeng et al. (2018)), generated by localized
heating inside the droplet using a laser source. The breakup is because of the
radial acceleration of the droplet that arises due to the oscillation and
expansion of the bubble inside the droplet.

The few above-discussed studies on the breakup of acoustically levitated
droplets describe the deformation of the droplet into a thin liquid sheet
before the onset of atomization. However, the events leading to the
atomization of an acoustically levitated droplet are not very well
investigated in the literature. On the other hand, the breakup of a liquid
sheet has been extensively investigated for sheets produced by a liquid jet
impacting on a solid circular surface (Villermaux & Clanet (2002); Clanet &
Villermaux (2002); Bremond et al. (2007)), two co-linear jets Mulmule et al.
(2010), oblique impingement of two liquid jets (Dighe & Gadgil (2019)) and
droplet impacting on solid surfaces (Lohse & Villermaux (2020); Josserand &
Thoroddsen (2016); Villermaux & Bossa (2011)). The liquid jet impacting on a
solid circular surface generates a circular liquid sheet of varying thickness
in the radial direction. The liquid sheet eventually becomes unstable and
disintegrates at the edge (Villermaux & Clanet (2002); Clanet & Villermaux
(2002)). The liquid sheet breakup mechanism is classified in two regimes
depending on the Weber number $We_{L}=\frac{\rho_{L}U_{L}^{2}d_{j}}{\sigma}$,
where, $U_{L}$, $d_{j}$, and $\sigma$ are the jet velocity, jet diameter and
surface tension of the liquid, respectively. The liquid sheet is considered to
be in a smooth regime for $We_{L}<We_{Lc}$ and flapping regime for
$We_{L}>We_{Lc}$, $We_{Lc}$ is the critical Weber number of the order of
$10^{3}$. In the smooth regime, there is no surface instability and the liquid
velocity of the sheet remains constant up to the sheet edge where a rim is
formed. The sheet atomizes at the edge on account of the Rayleigh Plateau
instability of the rim. Cusp formation could also be seen, wherein, the
droplet ejection takes place when the centrifugal acceleration dominates over
the capillary action (Clanet & Villermaux (2002)) as the droplets move along
the cusp region. In the flapping regime, the KH instability is known to
develop because of the relative motion of the liquid wavefront in the air
which causes a flag-like motion. This flag-like motion generates radial
acceleration-deceleration at the edge of the sheet (rim), which triggers the
Rayleigh Taylor instability and results in the disintegration of the edge
(Villermaux & Clanet (2002)). The liquid sheet expansion and recoiling of a
droplet impacting a surface happens at constant pressure. Villermaux & Bossa
(2011) argued that the generation of the ligaments is independent of the
impacting velocity when the droplet impacts on the solid surface (Villermaux &
Bossa (2011)). The formation of ligaments from the sheet is attributed to a
Rayleigh-Taylor instability, experienced by the decelerating rim, which itself
tries to simultaneously undergo a Rayleigh-Plateau instability.

The effect of the acoustic field on the atomization characteristics of a
liquid sheet generated by two co-linear jets was investigated by Mulmule et
al. (2010). An acoustic speaker which was capable to deliver sound frequencies
from 65 to 18000 Hz was located normal to the sheet. Surface ripples were seen
to travel toward the edge of the sheet when the speaker was switched on. Hole
formation in the sheet was also observed that resulted in the reduction in the
sheet diameter. Both the dilatation and sinuous modes of the KH instability
are known to couple at the lowest order under acoustic forcing and the growth
rate of the dilatation mode is no longer negligible. In another similar study,
the variation of liquid sheet thickness was found to play an important role in
different instability characteristics. It was observed that while the
aerodynamic effect dominates for smaller acoustic frequencies the thinning
effect dominates at a higher frequency at a fixed Weber number (Dighe & Gadgil
(2019)). Fang et al. (2022) numerically investigated the instability of a
liquid sheet under the influence of the acoustic field. The acoustic field was
applied from both sides and perpendicular to the liquid sheet. They predicted
that the unstable region is affected by the surface tension, viscosity, and
density of the liquid and air as well as the amplitude of the acoustic field.
The instability was suggested to be due to the combined effect of KH waves and
parametric oscillation. The sub-harmonic oscillations were shown to change to
harmonic oscillations by an increase in the We due to the suppression of
parametric oscillation while the parametric instability was found to overcome
the KH instability at higher acoustic acceleration. The instability exhibited
sensitive dependence on the thickness variation (Jia et al. (2022)).

A liquid droplet can atomize by a mode of breakup, termed the bag breakup mode
in literature, and is observed in several situations, such as a droplet
falling freely in the air (Villermaux & Bossa (2009)), a droplet exposed to
continuous horizontal jet (Kulkarni & Sojka (2014)) and droplet exposed in
swirling airflow (Kirar et al. (2022)) without any equatorial atomization. A
freely falling droplet, for example, undergoes flattening followed by bag
formation because of a radial pressure gradient along the droplet surface. The
temporal development of the radius of the droplet and the bag is exponential
(Kulkarni & Sojka (2014)). Kirar et al. (2022) observed “retracting bag
breakup” for a droplet subjected to a swirling airflow. The interfacial
instability of the drop was suggested to be because of the RT mechanism.

While there have been several studies on acoustic droplet levitation,
deformation, and atomization, this work presents a comprehensive experimental
investigation into a variety of aspects of the fate of a liquid droplet
levitated at the node of a standing wave in an acoustic trap. Specifically,
the work addresses levitation, droplet stretching, droplet thinning, primary
atomization and various modes of secondary atomization of a liquid droplet
levitated in an acoustic trap. The physics of these phenomena is discussed
deriving from literature studies and our own studies, and several
instabilities including the Kelvin Helmoholtz, Faraday, Rayleigh Plateau and
Rayleigh Taylor instabilities, are observed and identified in experiments,
discussed and comparisons made with respective theories. The work demonstrates
a myriad of complex, nonlinear, coexisting phenomena that renders a single
droplet into countless atomized droplets.

##  2 Experimental details

###  2.1 Experimental setup

A picture of the experimental arrangement is shown in Figure 1. The
experimental setup included an acoustic levitation system, a fiber optics
light source (Nikon, model: C-FI230), a high-speed camera (Vision Research,
Phantom v12), and a computer. A light source was used for the illumination of
the droplet and a high-speed camera was used to capture the images at a fixed
time interval. The captured images were recorded on a computer. To capture
top-view images another high speed camera Phantom VEO 710 camera was used.
This camera had to be placed at 60 degrees to the vertical to enable proper
visualization.

The acoustic levitator was constructed in accordance with the guidelines given
by Marzo et al. (2017). It was made using two spherical cups of 4.5 cm, which
are vertically placed at a center-to-center distance of 9 cm. The acoustic
levitator consisted of 72 transducers, Arduino nano and a Dual H-Bridge motor
driver. Each spherical cap had 36 transducers. The main components of the
levitator are the transducers (CUI devices, 2223-CUSP-TR80-15-2500-TH-ND) that
transform the electrical input into acoustic waves. The operation frequency of
the transducer was equal to 40kHz. The corresponding wavelength of acoustic
wave at 25${}^{\circ}C$ equals 8.5 mm. Arduino Nano was used to generate a
high-speed square wave signal. The droplet was injected into the levitator
using a syringe. The images were recorded from 4000 to 140k fps depending on
the requirement of the experiment. The images were recorded with the camera
focused on a certain droplet region to know the type of interfacial
instability. The initial diameter of droplet is varied from 1.1 mm to 2.4 mm.
The captured images were analyzed using the Image J software. The maximum
pressure (pressure at anti-node) was estimated as 2400 Pa. For more details,
the reader is advised to see the appendix section A.4. The maximum pressure
calculated by the simulation agrees closely with the pressure given by Marzo
et al. (2017).

Figure 1:  Experimental setup.

###  2.2 Materials and method

The fluid properties of the droplet strongly influence the dynamics of an
acoustically levitated droplet. The liquid droplet considered in this study
was a mixture of 90% ethanol and 10% water. Before the experiment, the mixture
was stirred continuously in an ultrasonic bath for 30 minutes to ensure proper
ethanol-in-water mixing. The corresponding surface tension and density of this
mixture are 22 $mN/m$ and 830 $kg/m3$. The droplet was injected into the
levitator using a syringe. The droplet then undergoes deformation followed by
atomisation on the order of a millisecond. Therefore, the imaging was done at
a frame rate of 4000 fps to 140000 fps.

##  3 Results

We present the experimental results and observations in this section, followed
by a discussion on possible mechanisms for these observations in the next
section. The droplet gets levitated only for critical acoustic parameters.
Since the acoustic wavelength is $8.5mm$, the upper limit of the droplet size
that can levitate in this levitator is equal to 4.4 mm. We only consider such
levitated droplets in this study. The experimental observations include stable
and unstable levitated droplets depending upon the Weber number. The terms
”stable droplet” or ”unstable droplet” are used for a levitated droplet that
either resists (stable) or undergoes (unstable) atomization, respectively. Our
primary focus in this article is to understand the instabilities associated
with the atomisation of an acoustically levitated unstable droplet.

The radial expansion of a droplet in an acoustic field is classified into
different regimes, as illustrated in Figure 2. For better clarity, the
corresponding top and side view images are also shown in Figure 3. Droplet
flattening occurs when an injected droplet is levitated at the node of a
standing wave. The droplet first transforms from a spherical shape to a flat
disc shape (see 3 (a) and (b)) and is termed the stretching regime. This is
followed by the thinning regime that refers to the thinning of the edge region
(equatorial region of the droplet) ((see 3 (c)). Atomization at the edge
refers to the disintegration of the liquid sheet at the edge (see 3 (d)) Bag
formation occurs at the end of edge atomization during the deceleration of the
atomized sheet. The bag breakup occurs in 4 different ways: umbrella breakup,
bag breakup, bubble breakup, and multistage breakup. The mechanism of
atomization includes several interfacial instabilities in various breakup
regimes. In the following sub-sections, we detail the observations made at
different stages from the time a droplet is introduced in an acoustic trap to
the point where it undergoes complete shattering and disintegration.

Figure 2:  Different regimes of the droplet breakup.

###  3.1 Droplet Levitation in acoustic trap and critical We

A droplet when injected at the node of a standing wave in an acoustic
levitator does center-of-mass oscillations while simultaneously undergoing
deformation, to finally reach the equilibrium position (node of the acoustic
field). Experiments show that the oscillations decrease with an increase in
the size of the droplet. Once at the node, the droplet remains steady and
admits deformation at low We. In this case, the acoustic force is not
sufficient to overcome the surface tension force and therefore the droplet
does not undergo breakup. The oscillations of the droplet stop after a few
seconds of injection as the acoustic and surface tension forces completely
balance each other. The static equilibrium shape of the droplet depends on the
Weber number (We) which is defined as $We=\frac{\rho_{a}U^{2}d_{0}}{\sigma}$,
where $\rho_{a}$, U, $d_{0}$, and $\sigma$ are the density of air, average
acoustic velocity, initial diameter of the droplet and surface tension of the
liquid, respectively. Note that the velocity
$U=\frac{P_{0}}{\rho_{0}v_{0}}\frac{1}{\sqrt{2}}$ and is of the order of
$4.09$ m/s since $P_{0}$ is of the order of 2400 Pa, $\rho_{a}$ and $v_{0}$ of
the order 1.22 $kg/m^{3}$ and 340 $m/s$ respectively.

Figure 3:  Side view and top view images at different regimes of the droplet:
(a) initial droplet, (b) droplet flattened into a flat disc shape, (c)
thinning at the edge, and (d) atomisation at the edge). The side view images
are captured at 6200 fps and the top view images were captured at 14485 fps.

Figure 4 (a) shows the experimental data for the steady deformation of a
stable levitated droplet (without breakup) placed in an acoustic standing
wave. The droplet assumes an oblate spheroidal shape and the corresponding
variation of its deformation is represented by ($\frac{2R}{d_{0}}$) at
different We are presented in Figure 4 (b). Here $2R$ is the length of the
principal-major axis of the oblate spheroidal droplet (R termed as radius of
the oblate droplet in this work). The value of $2R$ and thereby the
deformation of the droplet increases with increasing We along with a
corresponding reduction of the length of the principal minor axis (termed
thickness $h$ here), such that $\frac{2R}{d_{0}}$ $\simeq$ $We^{1/2}$ (figure
4 (b)), indicating the prevalence of non-linearity. A similar observation has
also been reported in numerical calculations (Lee et al. (1991)). There exists
a critical We=1.36 (in our experiments) beyond which the droplet can no longer
attain steady deformation (termed stable droplet), but instead continues to
elongate (termed unstable droplet) ultimately leading to its breakup.

Figure 4:  Static deformation of the droplet in the standing wave: (a)
captured images at different We and (b) $\frac{2R}{d_{0}}$ vs We.

###  3.2 Droplet flattening

The droplet, beyond the critical We$>$1.36, changes from a spherical shape to
an oblate shape, and its radius $R$ continues to increase with time as the
droplet is squeezed at the poles. The flattening of the droplet results in a
highly flattened pan-cake like liquid sheet. Figure 5 (a) shows the
visualization images for different types of breakup modes depending on the We.
The corresponding evolution in diameter of the droplet (2R) as a function of
time at different We is presented in Figure 5 (b) which includes both the
regimes, i.e. stretching as well as thinning (before the initiation of
atomization). The stretching and thinning regimes are separated by a dotted
line. The variation of rim velocity ($U_{L}$), during radial expansion, as a
function of 2R is shown in Figure 6. The range of rim velocity, admitted in
the thinning regime, is approximately of the same order for all the We in the
stretching regime. The onset of the thinning regime can be easily seen in the
$U$ vs $t$ plot (figure 6) and the thinning is initiated earlier (shorter
times) at higher We.

A better understanding of the droplet stretching in the stretching regime is
seen in figure 5 (b) and (c) which shows the evolution of diameter (2R) as a
function of time for different We for the stretching (flattening) regime. The
stretching time reduces with an increase in the We (initial diameter of the
droplet). There is a good agreement between the experiment and deformation
predicted by the equation 13, the theory and mechanism of which are discussed
in the next section.

Note that the droplet is placed at the node of a standing sound wave, where
the pressure is zero and the associated velocity field in the air is maximum.
This velocity field of air, associated with the sound wave, alternates
sinusoidally in time and acts as the ambient air velocity field in which the
droplet is placed. When the droplet is placed at the node, it can be likened
to a problem of a droplet placed in an oscillatory uniform velocity field,
wherein the droplet disturbs the ambient velocity field. The resulting
oscillatory velocity field is such that when air goes around the droplet, it
creates stagnation points at the poles, generating positive pressure fields.
Importantly, at the equator, the air velocity is maximum, although
oscillatory, which creates a non-oscillatory low-pressure region at the
equator, consistent with the Bernoulli equation. It is this low pressure,
created at the equator of the droplet, that leads to the flattening of the
droplet. This is quantitatively demonstrated in the next section.

Figure 5:  (a) Side view visualization images at different stages, (b)
evolution of the diameter of the droplet and (c) corresponding rim velocity at
different time for various We. Note: the images have been recorded at 6200 fps
and the respective video are available at: (i) (umbrella breakup), (ii)  (Bag
breakup), (iii) (Bubble breakup) and (iv) (Multistage breakup). Figure 6:  Rim
propagation velocity ($U_{L}$) as a function of 2R at different We
corresponding to figures 5 (b) and (c).

###  3.3 Droplet Thinning

As the droplet stretches in the flattening regime, the pressure difference
between the pole and equator ($P_{p}-P_{E}$) keeps increasing (details can be
seen in appendix figure 33), and then shows a nonlinear increase, which
results in a sudden increase in the stretching rate ( with a corresponding
increase in radial velocity), and thereby resulting in a sudden reduction in
thickness at the edge of the liquid sheet. Shi (1995) made a similar
observation using boundary element calculations, claiming that the inception
of rim formation at the edge of the droplet, could lead to thinning of the
region just preceding the rim. This is akin to jet thinning in Newtonian and
non-Newtonian jets. For more clarity, we simulated the sound pressure
distribution around the droplet of different aspect ratios. The simulation
details are provided in Appendix A.5. The aspect ratio is chosen in accordance
with the experimental findings. It is clear that the compression pressure
increases at the poles while it decreases at the equator during flattening
(increase in $R/h$ ratio). The nonlinear increase in the equatorial velocity,
due to the flattening of the droplet, creates a corresponding nonlinear
decrease in the equatorial pressure, leading to the onset and sustenance of
the thinning of the droplet. The non-linearity is also reflected in the
increase in the stretching factor $f$ with We, in the model to be discussed
later.

Figure 7 shows the velocity of the liquid-air front at the equatorial region
during radial expansion. The velocity is determined by extracting digital
images recorded by a high-speed camera at 11000 frames per second for 1.82 ms
before the atomization commences. The droplet is positioned in such a way that
the edge region is completely captured during radial propagation of the
liquid-air wavefront. The velocity is shown for both the stretching and
thinning regimes. In the stretching regime, the velocity does not change
significantly (it increases gradually), whereas it is faster in the thinning
regime.

Figure 7:  Velocity of the radially expanding droplet before initiation of
breakup. Note: the images are recorded at 1100 fps with focusing on the
equatorial region of the droplet. The velocity is calculated for the 1.5 ms
before initiation of breakup which covers both stretching as well as thinning
regime.

Thus, the liquid sheet thins out near the equatorial region to replicate a
membrane. The inertial force due to radial liquid velocity $U_{L}$ is
stabilized by the surface tension force, which causes the accumulation of the
liquid at the edge resulting in the formation of the rim as shown in Figure 10
(a). The rate of change of momentum at the rim is resisted by the surface
tension force (Culick (1960); Villermaux & Clanet (2002)). A balance of the
inertial acceleration of the sheet and the interfacial tension,
$\rho_{L}U_{LTC}^{2}h=2\sigma$, leads to the Taylor Culick velocity
$U_{LTC}=\sqrt{\frac{2\sigma}{\rho h}}$. According to the Villermaux & Clanet
(2002), a liquid sheet propagates in the air without initiation of breakup
when $\frac{\rho_{L}U_{L}^{2}h}{\sigma}>2$. For a constant velocity of
stretching of a sheet, the sheet breaks when the thickness reduction is of the
order of $2\sigma/(\rho_{L}U_{L}^{2})$. For example, in a liquid sheet formed
by coaxial jets, the liquid velocity $U_{L}$ of the sheet remains constant,
and a steady state is typically considered. The thickness $h$ decreases with
$r$, the radial extent of the sheet, since mass conservation demands that
$h=Q/(2\pi rU_{L})$. In this case, the liquid sheet breaks when the thickness
h reaches that given by eqn 2.

$h=\frac{2\sigma}{\rho_{L}U_{L}^{2}}$ (2)

In the present case of droplet breakup in an acoustic field, the liquid
velocity increases exponentially (see Figure 7) in the thinning regime. The
inertial force $\rho_{L}U_{L}^{2}h$ can therefore decrease to a value of
$2\sigma$, the Taylor Cullick criteria of breakup, only if there is a
correspondingly faster decrease in the thickness of the sheet.

At the edge, the rim thickness is greater than in the preceding regions due to
the dominance of surface tension. As the rim and thereby the sheet accelerates
due to lower pressure at the equator, $U_{L}$ increases with time, it draws an
accelerating fluid from the region just behind the rim, in the equatorial
region. The acceleration results in thinning since the local liquid is
accelerated, and the mass balance then demands that the sheet reduces in
thickness. This is a reinforcing process, where the high velocity in the thin
region, further increases the radial extent of the thin region. While this may
appear akin to the jet thinning mechanism it should be noted that the
capillary pressure due to azimuthal curvature that plays an important role in
the jet thinning mechanism is absent in the sheet thinning process. The
mechanism involved in thinning of the liquid sheet at the equatorial region is
qualitatively discussed in Appendix section A.5.

The velocity and acceleration in the thinning regime are actually periodic, as
confirmed by experimental measurements obtained at 140k fps from top and
bottom view images. The standing wave has an acoustic frequency of 40k Hz.
Therefore, we captured the images at an fps higher than the frequency of the
standing wave. Figure 8 shows the dynamics of the radial expansion of the
liquid sheet in the thinning regime. Both the top and side view digital images
at time intervals of 7.14 $\mu$ s are shown in Figure 8. The rim velocity
($U_{L}$) and variation in the vertical amplitude with time in the thinning
regime are shown in Figure 9 (a) and Figure 9 (b), respectively. The velocity
$U_{L}$ in figure 9 (a) when compared with the velocity shown in Figure 7
indicates period averaged (40 kHz) radial propagation in the latter, and
therefore without any visible velocity fluctuations, especially in the
thinning regime. This is essentially due to the lower frame speed in those
experiments, 11000 fps, much smaller compared to the acoustic vibration
frequency. The linear increment in the amplitude occurs in the unstable region
(see Figure 9 (b)), commensurate with the net acceleration of the drop edge.
It should be noted here that $U_{L}$ in figure 9 does show periodic
deceleration of the drop edge.

Figure 8:  Instantaneous top and side view digital images recorded at 140000
frames per second in the thinning regime. The images are recorded focusing at
the equatorial region. Here, TV and SV indicate the top view and side view,
respectively. The corresponding top and side view video can be found at: (a)
(video: top view) and (b)  (video:side view). Figure 9:  Rim velocity (a) and
amplitude of vibrated liquid sheet (b) with time during thinning regime.
Velocity is calculated from the side view results presented in figure 8. Note:
velocity is measured for 0.7 ms before the initiation of breakup and amplitude
(vertical displacement of vibrating liquid sheet at thin region) is measured
for 0.22 ms before the initiation of breakup.

The radial deformation time t (stretching+thinning) decreases with time as
shown in table Table 1

###  3.4 Formation of Faraday waves at the thinned equatorial region before
rim breakup

The rim and thin region of the liquid sheet is indicated in Figure 10 (a). A
non-linear wave appears on the surface of the liquid sheet at the thinnest
portion (preceding to the rim) (see Figure 10 (b)). This non-linear pattern of
the wave on the surface of the liquid is reported in the literature for wave
generation due to the vertically oscillating liquid bath (Oza et al. (2014);
Tambasco et al. (2016); Bush (2015)). The wave pattern that has a strong
resemblance to the Faraday waves grows on the liquid sheet and induces a
capillary wave (due to vertical oscillation of the liquid sheet) (see Figure
10 (c)). The liquid sheet vibrates in the vertical direction which is shown in
side view images of Figure 8.

Figure 10:  (a) rim formation, (b) initiation of nonlinear instability, (c)
formation of capillary wave and (d) increasing of amplitude .

###  3.5 Breakup due to Faraday instability at the thinned equatorial region
before rim breakup

The vertical vibrations cause alternate radial acceleration and deceleration
of the rim resulting in azimuthal deformation of the rim shown in Figure 10
(d). The radial propagation of the liquid sheet with increasing velocity leads
to a continuous reduction in the thickness. These azimuthal deformations
propagate backward into the sheet, causing azimuthal perturbations in the
Faraday waves in the thin region of the liquid sheet. Our observation suggests
that a liquid sheet of a very small thickness, subject to an acoustic field
shows much enhanced vertical vibrations. The amplitude of the vibration
continues to increase linearly with time along with a reduction in the
thickness (see Figure 9 (b)). The Faraday waves now develop into a Faraday
instability on the liquid sheet’s surface, and a strong ejection of very tiny
droplets is seen perpendicular to the surface of the liquid sheet. The
ejection of these droplets from the liquid surface produces holes in the
liquid sheet (see Figure 11 (a)). The perforation of the liquid sheet causes
the ligaments to merge and break into droplets (Figure 11 (b)) thereby
weakening the attachment of the rim with the flattened droplet. The pictorial
representation of the ligament merging during perforation of the liquid sheet
at the equatorial region is shown in Figures 12. The images are presented at a
time interval of 35 $\mu$s. The ligaments are interconnected with each other
during the perforation of the sheet. The ligaments merge with each other
leading to a web of liquid in the perforated sheet. The merging and breakup of
the liquid ligaments then lead to the formation of spherical droplets.

###  3.6 Fragmentation at the rim of equatorial region

The liquid rim detaches from the liquid sheet and breaks into droplets (Figure
11 (c)). Unlike the droplets generated by the Faraday instability, which are
ejected upwards, perpendicular to the expanding sheet, the droplets generated
by rim breakup are ejected and scattered in the direction of sheet expansion.
The generation of tiny droplets by the Faraday instability and their ejection
in the vertical direction continues until the bag formation starts. The
measured size of the tiny-sized droplets, generated by the Faraday
instability, varies between 20 $\mu$m to 34 $\mu$m. The wavelength of the
deformed rim is around 330 $\mu$m and breaks into droplets with the size of
the order of 100 $\mu$m. At this stage, the generation of the capillary waves
of constant wavelength appears on the surface of the liquid sheet. The
capillary wavelength on the surface of the liquid sheet is 65 $\mu$m. A
detailed discussion on the droplet size distribution is presented later.

Figure 11:  Fragmentation of the liquid sheet: (a) top view (TV) and side view
(SV) images during perforation, (b) rim detachment and (c) rim breakup. The
corresponding video can be seen in the caption of figure 8.

Fragmentation at the equatorial region of the droplet continues during the
radial expansion of the decelerating liquid sheet as shown in Figure 13.
During this period, the droplets are ejected in both radial and perpendicular
to the liquid sheet. The liquid sheet, after undergoing a complete breakup of
the thinning region at the edge of the sheet near the equator, leads to the
cessation of further growth and marks the end of the fragmentation process.
The atomisation at the equatorial region strongly depends upon the We
(diameter of the droplet). Table 1 shows the equatorial breakup time
($t_{eb}$) as a function of We. Equatorial breakup time decreases with the
increase of We (an increase in diameter), indicating that the acoustic field
hastens the dynamics of the system.

Table 1:  The deformation time (stretching+thinning) (t), equatorial breakup time ($t_{eb}$) and secondary breakup ($t_{sb}$) as a function of We. Note: $t_{s}b$ represents the time interval between the end of equatorial atomization and the start of the secondary breakup. We | t (ms) |  $t_{eb}$ (ms) |  $t_{sb}$ (ms)
---|---|---|---
1.37 | 7.26 | 4.52 | 2.42
1.45 | 6.28 | 3.7 | 3.22
1.56 | 5.138 | 2.6 | 5.48
1.92 | 1.9 | 1.45 | 15

This marks the end of the primary breakup, we term the equatorial breakup of
the liquid sheet, whereafter the secondary breakup starts.

Figure 12:  The merging of the interconnected ligaments during hole creation
due to ejection of tiny sized droplets perpendicular to the liquid sheet at
equatorial region .

###  3.7 Bag formation and breakup of different kinds

The acoustic levitator has the capacity to levitate the droplet only half of
its wavelength i.e $\lambda/2=4.4mm$. The sheet diameter is larger than half
of the wavelength therefore, the pressure difference between the center and
edge of the liquid sheet causes the sheet to bend resulting in the formation
of the bag-like shape (see the last image in which the arrow indicates
downward).

Figure 13:  Instantaneous top and side view digital images during
fragmentation at the equatorial region which converted into bag like
structure.

While the droplet breakup at the equatorial region is similar for all $We$,
the secondary breakup (i.e. breakup of the bag-like structure) depends upon
the diameter of the initial diameter of the droplet, and thereby on the $We$.
The observed secondary breakups are classified as (a) umbrella breakup, (b)
bag breakup, (c) bubble breakup and (d) rim collapse and sheet breakup
(multistage breakup).

####  3.7.1 Umbrella breakup

Figure 14 shows the instantaneous digital images of the umbrella breakup of
the liquid sheet. The images are recorded at 48k frames per second and
presented at a time interval of 0.4 ms. Once a droplet is levitated in the
acoustic trap it undergoes stretching, thinning, and Faraday and equatorial
breakup, as well as fragmentation and retraction, as discussed in the earlier
section 3.5 and section 3.6 (The reader is advised to see the details in the
movie the link for which is provided in the supplementary material), the
thickness of the droplet starts to decrease again as seen in the first picture
in figure 14, and a new thinning region is again formed. The newly formed
thinned region shows capillary waves and edge breakup through a combination of
Faraday instability, correspondingly formed holes, Rayleigh Taylor, and
Rayleigh Plateau mechanisms constituting the secondary breakup mechanism. The
liquid droplet that has got transformed into a sheet now thins over the entire
region of the droplet. Thinning of the liquid sheet leads to an increase in
its frequency to reach the harmonic condition. The harmonic condition leads to
the Faraday instability developing on the entire surface of the liquid sheet.
The detailed calculation is discussed in section 5.5.

Figure 15 shows the development of different types of interfacial
instabilities on the liquid sheet during umbrella breakup mode. The capillary
wave pattern is developed when the frequency of the liquid sheet becomes equal
to half of the applied acoustic frequency due to thinning. The lattice mode
and checkerboard pattern are observed at the edge of the sheet during the
breakup. The lattice mode pattern was also observed by Vukasinovic et al.
(2007) at the interface of a vertically vibrating sessile droplet and has been
attributed to the Faraday instability. The checkerboard pattern is associated
with the coupling of the capillary wave pattern with the azimuthal wave. A
similar kind of Faraday wave pattern has also been observed elsewhere Shats et
al. (2014); Khan & Eslamian (2019).

The Faraday instability causes the generation and growth of the instability
from the equatorial to the center region of the liquid sheet. The Faraday
waves formed on the entire surface of the droplet become unstable, causing
droplet ejection from the entire liquid sheet and corresponding hole
formation. The holes expand and cause the merging of the liquid sheet and
result in ligaments that ultimately break due to the RP Instability.

The above is termed as umbrella breakup occurs since the sheet, after
fragmentation, and before the onset of secondary instability builds a
positive, concave upwards curvature. The liquid sheet completely breaks during
the formation of the upward curvature.

Figure 14:  Instantaneous top view digital images during umbrella breakup
regime. The corresponding video is available at  video:Umbrella breakup Figure
15:  The local variation of the Faraday pattern on the liquid sheet during
Umbrella breakup.

####  3.7.2 Bag breakup

Umbrella breakup occurs when the sheet undergoes further thinning and
admittance of Faraday instability subsequent to droplet levitation,
stretching, thinning, primary equatorial instability, and sheet retraction. On
the other hand, bag formation occurs when after the equatorial instability and
sheet retraction, the thickness at the equatorial region is not reduced to any
significant extent such that harmonic condition and, thereby, the Faraday
instability cannot set in. As a result, as the liquid sheet decelerates
(decreased rate of expansion of liquid sheet), droplets accumulate at the
edge, resulting in the formation of the rim, without any equatorial breakup.
The diameter of the liquid sheet can then increase without much further
equatorial breakup such that it is beyond the spatial range of the acoustic
pressure field in the trap (that varies up to 4.5 mm). Figure 16 shows the top
view of the digital images recorded during bag formation at 14485 frames per
second and displayed at 0.35 ms intervals. The unsupported sheet at the edges
bends downwards under gravity and results in the formation of a bag-like
structure. The development of the bag increases the surface area of the liquid
sheet and the pressure conditions inside the bag lead to significant
stretching and, thereby, thinning of the bag. The harmonic condition is
thereby achieved by a further reduction in the thickness of the liquid sheet
during bag growth. During this time, the Faraday instability is developed at
the surface of the thinned sheet. This causes the droplets to be ejected
perpendicular to the sheet, resulting in sheet perforation. The remaining
sheet merges and breaks into the larger-sized droplet due to RP instability.

Figure 16:  Instantaneous top view digital images during bag breakup regime.
The corresponding video is available at (video:bag breakup).

####  3.7.3 Bubble breakup

Subsequent to droplet levitation, stretching, thinning, and primary equatorial
instability followed by sheet retraction, bag formation can occur wherein the
bag thins and breaks by Faraday instability. If the thinning of the bag is
delayed, the bag (stretched liquid sheet) can close onto itself, resulting in
bubble formation. This bubble can then undergo bubble breakup. Figure 17 shows
the bubble breakup of a droplet injected in the acoustic levitator. The liquid
sheet transforms into a bag-like structure similar to the one discussed in
section 3.7.2. During bag formation, the thickness of the rim increases with
time due to the flow of liquid accumulating at the rim. During rim collision,
the bag transforms into a bubble shape shown in Figure 17. Due to the larger
thickness of this bubble, no instability is observed at the bubble surface.
The bubble then begins to expand into the radial direction with a simultaneous
reduction in the thickness of the liquid sheet. The instability emerges at the
thinnest region near the rim, as shown in Figure 17. As the thickness further
reduces to the magnitude that invokes the Faraday instability, the instability
propagates across the entire surface of the bubble resulting in droplet
ejection from the surface of the liquid sheet, and a hole is seen to form. The
perforated sheet transforms into ligaments and breaks due to RP instability.
The breakup of the bubble leads to a residual liquid sheet that again gets
levitated the rim deforms and breaks once again. The corresponding video is
provided in the caption of the figure 17.

Figure 17:  Instantaneous top view digital images during bubble breakup
regime. The corresponding video available at  (video:Bubble breakup).

####  3.7.4 Multi stage breakup

Figure 18 shows instantaneous images of the multistage sheet breakup. The
phenomenon prior to bubble formation (up to rim collision) is similar to the
section 3.7.3. Unlike the bubble breakup case, the bubble in the multi-stage
breakup does not burst. The rim begins to deform, which leads to equatorial
thinning of the sheet. The surface of this thinned-out bubble is locally
similar to a sheet and exhibits fragmentation through a combination of
Faraday, RT, and RP instabilities 3.5. The atomization of the sheet is
stopped, and a rim is formed once again at the edge of the sheet, such that
the sheet again expands and thins locally (see Figure 18) resulting in another
cycle of sheet breakup. This breakup continues until a rim forms and the sheet
collapses, resulting in ligament formation and breakdown as a result of RP
instability. The reader is suggested to refer to the movie provided in the
supplementary material for more information.

Figure 18:  Instantaneous top view digital images during multistage breakup
regime. The corresponding video is available at (video:multistage breakup)

##  4 Droplet size distribution

As seen from the experimental results presented in the previous section, a
variety of instabilities, Faraday, KH, RP and RT, lead to sheet (formed from
the original droplet by thinning) breakup. Since each instability is
associated with its own unstable wavelengths, together with non-linearities
leads to a very complex size distribution of droplets resulting from the
atomization. This section includes the size distribution of droplets generated
during different regimes of the breakup. The droplet size was measured using
Image J software. The estimated error is considered as $\pm 5\%$.

###  4.1 Droplet size distribution during equatorial breakup

Figure 19 shows the size distribution of the droplets generated in the radial
direction and normal direction during atomization at the equator of the
expanding liquid sheet. The total breakup time at the equatorial region
($t_{eb}$) depends on the We as discussed in an earlier section and for the
droplet breakup presented in figure 19, this is around 2.8 ms. The equatorial
breakup time ($t_{eb}$ is the time from the start ($t^{\prime}=0$) to the end
of the atomization process at the equatorial region ($t^{\prime}=t_{eb}$) and
therefore the size distribution of the generated droplets varies with time
between $t^{\prime}=0$ to $t^{\prime}$. The droplet breakup mechanism consists
of (i) Faraday breakup, (ii) RT breakup, and (iii) Capillary breakup (RP
instability). The droplet generation perpendicular to the sheet is due to the
Faraday instability. On the other hand, the vertical oscillation of the sheet
creates radial acceleration that causes RT instability. Figure 19 (a) shows
the droplet size distribution at $t^{\prime}$=0.25 ms. The droplet size
distribution here includes rim breakup (indicated by arrow), the capillary
breakup of merging ligaments of a perforated liquid sheet, and small-sized
droplets generated due to the Faraday instability. These are in fact well
captured in figure 12. The size of the droplets due to Faraday instability is
mostly in the range of 20 to 34 $\mu$m. The size of the droplets generated
from the rim breakup varies from 100 to 120 $\mu$m. The droplets generated due
to the merging of thin ligaments of perforated sheet vary from 40 to 80
$\mu$m. The contribution of the RT instability is not significant up to this
point. Figure 19 (b) shows the droplet size distribution generated by Faraday
instability perpendicular to the liquid sheet during the equatorial breakup.
The number of generated droplets measured is around 50. The droplet size
distribution is found to be in the range of 20 to 34 $\mu$m.

Figure 19 (c) shows the droplet size distribution at $t^{\prime}$=0.75 ms. For
the image at $t^{\prime}=0.75ms$, most of the droplets generated at
$t^{\prime}=0.25ms$, would have gone out of the frame such that the image
mostly has droplets that are generated between the time period
$t^{\prime}=0.25-0.75$ ms. As the rim separates from the thin liquid sheet,
the thin liquid sheet vibrates vertically and generates a radial acceleration,
resulting in the radial generation of droplets due to RT instability. The
mechanism of formation of radially generated droplets due to RT instability
will be discussed in the next section. The size of the droplets varies from 60
to 80 $\mu$m. The smaller-sized droplets (20 to 34 $\mu$m) in this
distribution correspond to the Faraday instability, which continues to be at
play.

Figure 19:  Droplet size distribution at the equatorial region at the
different instant of breakup time. The total breakup time is approximately 2.8
ms. The corresponding visualization video for the equatorial breakup can be
found at:  video top view: Droplets size equatorial. video side view: Faraday
droplets

The RT instability is suppressed as the liquid velocity decreases with time
(deceleration of the liquid sheet). In this regime ($t^{\prime}=0.75-1.5ms$)
the breakup takes place because of combined RT and capillary instability as
shown in Figure 19 (d) to yield droplets of size 100 to 200 $\mu$m. The
larger-sized droplet in Figure 19 (d) is because of the merging of the
droplets (see Figure 20).

The equatorial breakup continues with a weakening of the RT instability with
time (see 19 (e)). The process of merging droplets is significantly suppressed
here, whereby larger-sized droplets ($>$ 200 $\mu$m) are not formed
($t^{\prime}=1.5-2.0ms$).

The capillary breakup takes over the RT instability between $t^{\prime}$= 2 ms
and $t^{\prime}$= 2.65 ms which can be clearly seen in Figure 19 (f). The
deceleration of the sheet and relative acceleration leads to the suppression
of RT instability. Moreover, the thinning region reduces in extent. The
prevailing vertical generation of the droplet causes perforation and
separation of the edge rim of the liquid sheet to create liquid threads and
ligaments which break into droplets by capillary breakup Figure 21. The
Faraday instability too seems to have reduced at this point. The reader is
suggested to refer to the movie provided in the supplementary material for
more information.

Figure 20:  Merging of two droplets in radial direction. Figure 21:  Droplet
generation at the late stage of the equatorial breakup (deceleration phase)
during smaller radial velocity

###  4.2 Droplet size distribution of very tiny sized droplet ejected
perpendicular to liquid sheet and larger sized during bag breakup

Figure 22 (a) shows the droplet size distribution of tiny droplets produced
from the Faraday instability during bag formation. Droplet size distribution
is measured over 10 consecutive images captured at 48000 fps (measured from
figure 28). The sizes of the droplets are found to vary in the range of 20 to
34 $\mu$m, similar to those formed during equatorial breakup thereby
confirming their formation by the Faraday instability. Figure 22 (b) shows the
droplet size distribution of the generated droplets during bag breakup after
the sheet perforation. The size of the droplets varies between 100 to 200
$\mu$ m, confirming that the droplets break due to RP instability (rather than
RT instability). Much larger droplets to the tune of 0.3mm to 0.7mm are formed
by the liquid rim breakup as well as the merging of ligaments unperforated
liquid sheets, which collapse and form ligaments. These droplets have a larger
diameter of 0.3 mm to 0.7 mm. This is clearly demonstrated in the video link
provided in the figure caption. This phenomenon occurs not only for complete
bag breakup, but also for other regimes such as umbrella, bubble, and multi-
stage. Therefore, it can be concluded that the large droplets are always a
result of liquid rim breakup and the breakup of ligaments formed by the
merging of the liquid sheet.

Figure 22:  Droplet size distribution of droplet ejected during bag
development: (a) tiny droplets (faraday breakup) and (b) larger droplets
(after sheet perforation due to RP instability). The corresponding
visualisation video can be found at video:Bag Faraday.

##  5 Discussion

The droplet breakup in swirl flow or gravitational fall as well as breakup of
sheets formed by co-axial jets, have been investigated in numerous studies.
Similarly, droplet oscillation, levitation, and break up in an acoustic
levitator have been addressed to some extent in the literature. In this
section, we invoke the results of these studies to understand the mechanism of
droplet levitation, stretching, thinning, equatorial breakup, bag formation,
interfacial instabilities, and droplet size distributions in our experimental
observations detailed in sections 3 and 4 on acoustically levitated droplets.

###  5.1 Mechanism of Droplet Levitation, deformation and critical We

In appendix A.2 we show that the total acoustic radiation force on the drop,
which when balanced by the weight of the droplet, yields the droplet position
in the trap below the node. The calculations show that a 2 mm droplet is
levitated around 300 microns below the center node position of the standing
wave. Given that the node-to-node distance is around 4.5 mm, the droplet can
be considered to be located almost exactly at the center of the node of the
standing wave.

The initial shape of the droplet is assumed to be spherical. The spherical
droplet changes to an oblate cylindrical shape in the standing wave.
Therefore, h can be expressed as

$h\sim\frac{d_{0}^{3}}{6R^{2}}$ (3)

where $d_{0}$ is the diameter of the undeformed droplet. The final shape of
the droplet is due to the inertial pressure of the gas and the surface tension
force.

$2\pi R(2\sigma)=\rho_{a}U^{2}\times 2\pi Rh$ (4)

By eliminating $h$ in the above equation

$\frac{2R}{d_{0}}=\frac{1}{\sqrt{3}}We^{1/2}$ (5)

Thus, the theoretical results from equation 5 indicate that the size of the
deformed droplet $\frac{2R}{d_{0}}$ $\simeq$ $We^{1/2}$ in agreement with
experiments.

The droplet then breaks at a critical We = 1.36. Danilov & Mironov (1992)
argue that the restoring force due to the curvature of the elipsoidal droplet
at the equator cannot balance the deforming negative pressure at the equator,
whereby the droplet cannot attain a steady shape. They predict a critical We
of around 1.7.

###  5.2 Mechanism of Flattening

For a droplet levitated at the node of a stationary sound wave, the pressure
field is approximated by A.1 as provided in the appendix. With the assumption
of inviscid, incompressible fluid flow, the current model determines droplet
deformation in accordance with the mechanisms in references Villermaux & Bossa
(2009); Kulkarni & Sojka (2014); Kirar et al. (2022). Thus for a droplet at
the node of a standing wave in an acoustic field, the droplet is subjected to
a time-varying pressure and velocity field, which in turn yields a non-zero
time averaged pressure field, on account of the Bernoulli equation, as given
by,

$P_{a}(r,z)=P_{a}(0)-\rho\frac{f^{2}U^{2}}{8d_{0}^{2}}r^{2}-\rho\frac{f^{2}U^{2}}{2d_{0}^{2}}z^{2}$
(6)

Where, $r=\sqrt{x^{2}+y^{2}}$ is the radial coordinate and
$P_{a}=\frac{\rho_{a}U^{2}}{2}$ is the stagnation pressure at z = 0 (see
Figure 23). The air velocity,
$U=\frac{P_{0}}{\rho_{0}v_{0}}\frac{1}{\sqrt{2}}$, the details about the
calculation of U are given in the A.1. The stretch factor is highly shape-
dependent and is expected to get nonlinear with $We$. Here, as in other
literature too, we use $f$ as a fitting parameter.

The Navier-Stokes equation and continuity equation have been solved to predict
the deformation of the droplet in the stretching regime. The experimental
observation is illustrated in Figure 2 for We = 1.45.

The Navier-Stokes equation and continuity equation in radial coordinates are
given as,

$\rho_{l}\Bigg{(}\frac{\partial u_{r}}{\partial t}+u_{r}\frac{\partial
u_{r}}{\partial r}\Bigg{)}=\frac{-\partial p}{\partial
r}+\mu_{L}\Bigg{[}\frac{1}{r}\frac{\partial}{\partial
r}\Bigg{(}r\frac{\partial u}{\partial r}\Bigg{)}-\frac{u_{r}}{r^{2}}\Bigg{]}$
(7)

$r\frac{\partial h}{\partial t}+u_{r}\frac{\partial(ru_{r}h)}{\partial r}=0$
(8)

The change in the height of the droplet during radial expansion is written as,
$h=\frac{d_{0}^{3}}{6R^{2}}$ and the velocity field inside the droplet is
written as,

$u(r,t)=\frac{r}{R}\frac{dR}{dt}$ (9)

The deformation of the droplet is due to the pressure difference between the
liquid and surrounding pressure. By neglecting the tangential component, the
normal component of the stress is balanced as,

$\frac{2\sigma_{l}}{h}=P_{L}(R)-P_{a}(R)$ (10)

$\frac{2\sigma_{l}}{h}$ is the capillary pressure at at the liquid-air
interface. $P_{L}(R)$ and $P_{a}(R)=P_{a}(R,0)$ are the inside (liquid) and
outside pressures (surrounding) respectively. By assuming the origin of a
cylindrical coordinate system at the center of the droplet (z=0), the pressure
at the edge of the droplet, $r=R$ is given below.

$P_{L}(R)=P_{a}(0)-\rho\frac{f^{2}U^{2}}{8d_{0}^{2}}R^{2}+\frac{2\sigma_{L}}{h}$
(11)

Defining $\phi=\frac{2R}{d_{0}}$, T = $t/\tau$,
$\tau=\frac{d_{0}}{U}\sqrt{\frac{\rho_{L}}{\rho_{a}}}$. Note as discussed in
the appendix, $P_{a}(0)=P_{a}(0,h)$. Thus the evolution of the diameter $R$ of
the deformed droplet can be given as,

$\frac{d^{2}\phi}{dT^{2}}-\Bigg{(}\frac{f^{2}}{4}-\frac{24}{We}\Bigg{)}=0$
(12)

The solution of the above equation by using the boundary condition at
$\phi(0)=1$ and $\phi^{\prime}(0)=0$ is given as,

$\phi(T)=\frac{1}{2}exp\bigg{(}-\frac{\alpha
T}{2}\bigg{)}\bigg{(}1+exp\big{(}\alpha T\big{)}\bigg{)}$ (13)

where, $\alpha=\frac{\sqrt{-96+f^{2}We}}{\sqrt{We}}$

Figure 23:  Sketch of the mechanism of droplet flattening.

The solid line and symbol in figure 24 (a) represent the evolution of 2R with
t obtained from the equation 13 and experiments, respectively. Figure 24 (b)
shows a good data collapse between experiment and theory and the data
collapses is seen to be better for higher Weber number. The difference between
the experiments and theory at smaller We may be attributed to the oscillations
of the smaller-sized droplet in the standing wave. It is important to note
that the stretching factor (f) varies with droplet size (or We) as listed in
the appendix.

It is useful to see if equation 12, which describes stretching of the droplet
with time, as a function of $We$, can be used to estimate the critical $We$.
At low We, the stretching factor is found to asymptote to a value of $f=9.2$.
The critical We beyond which a droplet is always stable, and shows stretching
can now be estimated using equation 12, assuming a steady state, such that
$We_{c}=96/f^{2}$. This yields $We_{c}\sim 1.14$ in close agreement with the
experimental value of around 1.36 reported in the results section.

Figure 24:  The comparison of experimental and theoretical results in the
stretching regime: (a) 2R vs t (b) $\phi$ vs $\alpha$*T .

###  5.3 Mechanism of thinning

The droplet placed in the acoustic standing wave expands radially with a
reduction of the height (thickness of the droplet) of the droplet. This
thickness is approximately uniform along the radial extent during the
stretching regime. The radial velocity of the expanding liquid is already
presented in section 3. The radial velocity can be expressed in terms of a
non-dimensional liquid Weber number defined as $W_{eL}$ =
$\frac{\rho_{L}U_{L}^{2}d_{o}}{\sigma}$. It was discussed in section 3.3, that
the stretching regime is followed by the thinning regime, where towards the
equatorial edge, a thin sheet is formed at the edge of the stretched droplet.
The radial expansion in this regime is highly nonlinear. It is then attempted
to explain both the stretching and the thinning regime with a single empirical
expression. Figure 25 shows the variation of the $W_{eL}$ with respect to
scaled time and the radial diameter during liquid sheet expansion. The
$W_{eL}$ changes slowly in the stretching regime (see Figure 25(a) and (b)). A
critical $W_{eL}$ is obtained as 2 which can be clearly observed in zoomed
view inserted in Figure25(a), beyond which the onset of stretching can be
observed. Thus the non-dimensional variation of the radial extent of the sheet
with time can be explained by a single empirical expression,
$Log(We_{L}/0.6)=0.08\;exp[10\;Log(We^{3.5}*T)]$.

Numerical calculations were conducted by inserting a droplet at the node with
an aspect ratio chosen in accordance with the results of the experiment. The
results indicated that in the stretching regime, the pressure difference
between the pole and the equator varies as $P_{P}-P_{E}$ $\simeq$ $R^{1/2}$
and $P_{P}-P_{E}$ $\simeq$ $R^{11/5}$ (see in appendix figure 33 (h)). This
illustrates that the thinning of the equatorial region is triggered by the
sudden rise in the $P_{P}-P_{E}$.

Figure 25:  (a) $W_{eL}$ vs $We^{3.5}*T$ and (b) $Log(W_{eL})$ vs
$Log(We^{3.5}*T)$.

###  5.4 Mechanism of Primary breakup: atomization at equatorial region

####  5.4.1 Fragmentation at the thinned equatorial region before rim breakup:
Faraday instability

The Faraday waves are nonlinear standing waves that develop on the surface of
a vibrating liquid when the frequency increases above a certain threshold.
These waves are known to exhibit a variety of patterns and ripples on the
surface of the vibrating liquid. The response frequency of these waves could
be sub-harmonic or harmonic with respect to the imposed acoustic frequency.
The levitator used in the present study has an imposed acoustic frequency of
40k Hz.

The frequency of the waves on a liquid surface can be expressed as (Kumar &
Tuckerman (1994); Bush (2015); Khan & Eslamian (2019)).

$\omega_{0}^{2}=(gk+\frac{\sigma k^{3}}{\rho_{L}})Tanh(kh)$ (14)

where the first term corresponds to gravity waves and the second to capillary
waves. The smaller wavelengths seen in the present study imply that the
$\frac{\sigma k^{3}}{\rho_{L}}$ $>>$ gk, thereby rendering the gravitational
effects negligible. The calculated value of the $\omega_{o}=127.1krad/s$,
correspondingly $f_{o}=20.23kHz$ for the experimental $\lambda=65\mu m$
($k=\frac{2\pi}{\lambda}$). The ratio of the $\frac{\omega_{0}}{\omega}\sim
0.51$, which shows the liquid sheet frequency is half of the applied frequency
$\omega$ and is thus in the sub-harmonic region. This is in good agreement
with the experimentally observed oscillation frequency of the sheet, measured
at 140,000 fps, which yields a value of $20$ kHz. It is known that a liquid
sheet shows parametric resonance only when the thickness of the liquid sheet
reduces to such an extent that $\omega_{0}$ is half the forcing frequency. For
higher thicknesses, $\omega_{0}>\omega/2$, where $\omega$ is the frequency of
the sound waves, which explains why only the thinned part of the stretched
droplet undergoes Faraday instability.

####  5.4.2 KH Instability

The Kelvin-Helmholtz type inter-facial waves develop when a thin liquid sheet
radially expands in the air. The dispersion relation for the KH instability of
a moving liquid sheet of constant thickness was first investigated by Squire
(1953). This dispersion relation is valid only for the long wave approximation
i.e. $kh\ll 1$ developed two possible modes of KH waves symmetric (varicose)
and asymmetric (sinuous mode). The dispersion relation for a sinuous mode of
the liquid sheet with constant thickness h moving with a velocity $U_{L}$,
which is excited by vertical vibration, is written as (Bremond et al. (2007))

$\frac{\omega}{kU}=\frac{1\pm(\alpha coth(kh/2))^{1/2}[-1+\frac{\sigma
k}{\rho_{l}U_{L}^{2}}(1+\alpha coth(kh/2)]^{1/2}}{1+\alpha coth(kh/2)}$ (15)

Where, $k=\frac{2\pi}{\lambda}$ is the wave number, $\lambda$ is the
wavelength and $\alpha=\frac{\rho_{a}}{\rho_{l}}$ is the ratio of densities.

The equation 15 is valid for the constant thickness which is not true for real
situations because thickness decreases with R as confirmed by the observation.
Therefore, Bremond et al Bremond et al. (2007) modified the above equation for
variable thickness $h=\frac{d_{0}^{2}}{8R}$. The dispersion relation in non-
dimensional form is given as

$\tilde{\omega}=\frac{\tilde{k}^{2}}{\tilde{k}+16\alpha\tilde{R}}\Biggl{(}1\pm\Bigg{[}\frac{16\tilde{R}}{We_{l}}\bigg{(}1+\frac{16\alpha\tilde{R}}{\tilde{k}}\bigg{)}-\frac{16\alpha\tilde{R}}{\tilde{k}}\Bigg{]}^{1/2}\Biggl{)}$
(16)

Where, $\tilde{k}=kd_{0}$, $\tilde{R}=R/d_{0}$,
$We_{l}=\frac{\rho_{L}U_{L}^{2}d_{0}}{\sigma}$, $\sigma$ is the surface
tension and $d_{0}$ is the initial diameter of the droplet. The pulsation is
equal to the real part of the above equation as the frequency of the acoustic
field is equal to the sinuous frequency. Therefore, the real part is expressed
as

$\tilde{k}(\tilde{R})=\frac{\tilde{\omega}_{0}}{2}\Bigg{[}1+\Bigg{(}1+\frac{64\alpha\tilde{R}}{\tilde{\omega_{0}}}\Bigg{)}^{1/2}\Bigg{]}$
(17)

Here, $\tilde{w}_{0}=2\pi fd_{0}/U_{L}$ is the frequency of the acoustic
field. For $f=40000Hz$, $d_{0}=$ 2 mm, $U_{L}=2.5m/s$, $\rho_{L}=830kg/m^{3}$,
$\rho_{a}=1.22kg/m^{3}$, $\tilde{R}=1.37$, the wavelength comes out to be
$\lambda=63.16\mu m$. Our experimental results show that the wavelength on the
surface of the liquid sheet in the thin region is equal to 65 $\mu m$ which
shows good agreement with the wavelength calculated by the equation 17. This
confirms that the waves on the surface of the liquid sheet are sinuous modes
of KH waves. Now we need to verify if this wave number is stable or unstable
at this imposed frequency.

The imposed oscillation frequency is sinuous mode with amplification and
damped. Thus at a critical radius beyond which the sheet is completely stable
and the amplitude of oscillation will be constant (Bremond et al. (2007)). We
have assumed that the initial diameter of the droplet is equal to the critical
radius and cut off frequency has been calculated by the equation

$\tilde{\omega}_{c}=\alpha
We_{L}\Bigg{(}1-\frac{16\tilde{R}_{c}}{We_{L}}\Bigg{)}^{1/2}$ (18)

The cut-off frequency comes out equal to $f_{0}=102Hz$ ($We_{L}$ = 350). The
system is unstable for the frequency smaller than $f_{0}$ which is very small
compared to the imposed frequency (40k Hz). Therefore, the waves on the
surface are not unstable wave and oscillate with constant amplitude and
wavelength. Therefore, it is confirmed that the instability on the surface is
not due to the sinuous mode of the KH instability.

The vertical vibration is not visible for the imaging conducted at smaller fps
compared to the imposed frequency. This is because the vertical vibration is
driven by the applied frequency.

Figure 26:  (a) Radial acceleration of the rim with time during propagating in
the air (thinning regime) before initiation of breakup which is calculated
from Figure 9 (a), and (b) RT instability growth rate as a function of wave
number.

####  5.4.3 Fragmentation at the rim of the equatorial region: Rayleigh Taylor
instability

The Faraday waves appear only at the thinned region of the sheet preceding the
rim of the liquid sheet and eventually when the Faraday instability sets in,
it results in the ejection of tiny-sized droplets as presented in the section
3.5. During this process, the thinned part of the liquid sheet vibrates
vertically with increasing amplitude. This fluctuation of velocity generates
radial oscillatory acceleration that also leads to instantaneous deceleration
of the sheet. The calculated acceleration is given in Figure 26.

Figure 27:  Instantaneous top view digital images during breakup at the
equatorial region. The images are shown at the time interval $\Delta t=69\mu
s$. The width of the image is equal to 5.62 mm. The corresponding video can be
seen at  video:RT instability

The droplet generation mechanism at the end of the sheet, is due to the
formation of holes due to droplet ejection by Faraday instability and the
corresponding stream-wise stretching of the perforated thin liquid sheet,
which results in the formation of ligaments. These ligaments then break into
droplets Figure 27 (see video provided in the caption) by a Rayleigh Plateau
instability. The formation of ligaments and their subsequent breakup is
further exacerbated by the Rayleigh Taylor instability. The ligament
generation in the radial direction is facilitated by the radial acceleration
and deceleration in the perforated sheet that can lead to the Rayleigh Taylor
instability (Jarrahbashi & Sirignano (2014)). The dispersion relation for the
RT instability and its wavelength can be determined by (Jarrahbashi &
Sirignano (2014); Sharma et al. (2021)).

$\omega_{RT}=\bigg{(}\frac{a(\rho_{L}-\rho_{a})k_{RT}-k_{RT}^{3}\sigma}{\rho_{L}+\rho_{a}}\bigg{)}^{1/2}$
(19)

where a is the acceleration, and $\omega_{RT}$ is the dimensional growth rate.
The corresponding most unstable wavelength is determined as

$\lambda_{RT}=2\pi\sqrt{\frac{3\sigma}{a(\rho_{L}-\rho_{a})}}$ (20)

The radial acceleration of the oscillating rim during radially propagating in
the air is calculated from the variation in velocity Figure 9 (a) and
presented in Figure 26. The maximum radial acceleration measured on the rim is
29000 $m/s^{2}$ and the corresponding azimuthal wavelength as measured in
experiments is 0.338 mm. The calculated wavelength by the equation 20 is equal
to 0.329 mm, indicating a fairly good agreement with the experiments.

The breakup of the thinned sheet at the equatorial region of the droplet,
results in delayed breakup of the rim. This is essential because even before
the rim breaks, the Rayleigh Taylor and Faraday instabilities already start
generating a significant number of droplets in the highly perforated sheet
preceding the rim. Aided by these perforations, the rim ultimately partly
separates and undergoes breakup by Rayleigh Plateau instability.

It is useful to mention here that even before the sheet breakup starts, the
azimuthal deformation observed in the rim before breakup is not observed prior
to the vertical oscillation (see Figure 10 (a)) of the thinned sheet. The
thinned sheet preceding the rim, when sufficiently thin, oscillates in a
vertical direction while radially propagating in the air (in the thinning
regime), which results in the azimuthal deformation on the rim (see Figure 10
(c) and (d)) on account of the RT instability. The azimuthal wavelength of the
rim compares well with the most unstable wavelength of the RT instability.
These azimuthal undulations infact propagate backward and create radially
propagating azimuthally distributed undulations.

As the sheet keeps ejecting droplets, by Faraday instability in a direction
perpendicular to the plane of the sheet, and droplets generated by RT and RP
instabilities in the direction of radial expansion of the sheet, the liquid
sheet subsequently breaks into droplets of larger diameter with a decrease in
the liquid velocity; the merging of lobes leads to the accumulation of drops
that re-forms the rim as the surface tension force overcomes the inertia
force.

###  5.5 Mechanism of secondary breakup

This section includes the mechanism behind the secondary breakup (after the
equatorial breakup). The secondary breakup includes umbrella breakup, bag
breakup, bubble breakup, and multi-stage breakup. The mechanism of each will
be discussed in the following sections.

####  5.5.1 Mechanism of umbrella breakup

The observation of the umbrella breakup is reported in section 3.7.1. The
umbrella breakup occurs for 1.36 $\leq$ We $\leq$ 1.45 i.e. for smaller
droplets. A typical droplet, after undergoing equatorial breakup, and rim
retraction, bends upwards to form a concave shape stretched droplet sheet,
resembling an umbrella. With increasing We the curvature of the umbrella
increases. Once the umbrella is formed and the umbrella-shaped droplet
stretches further to again get into the thinning regime, equatorial breakup
and KH as well as Faraday waves obeying the frequency of the vibrating sheet
to be half that of the applied frequency, start to appear on the surface of
the liquid sheet.

To confirm this, we measured the wave velocity by extracting the images frame-
by-frame during wave propagation on the surface of the liquid sheet, which was
found to be of the order of 2.5 $m/s$. By assuming that this velocity is equal
to the liquid velocity, the corresponding thickness is determined by the
Taylor-Culick criteria to be equal to 8.5 $\mu$m. The wavelength, 65 $\mu$m,
and the corresponding frequency of oscillation, 14 agree well with the
calculated oscillation frequency ($\omega_{0}$) of 127.1K, which is half of
the imposed acoustic frequency (251.2K). This is a clear indication of sub-
harmonic condition. Thus the wave appears on the entire surface of the liquid
sheet when the liquid sheet thickness reduces ($U_{L}$ increases) such that
the frequency of the liquid sheet becomes equal to half of the applied
acoustic frequency. It should be noted that the capillary wavelength can be
estimated as $\frac{U_{L}}{f}=62.5\mu$m, which demonstrates a surprisingly
good agreement with the measured wavelength. Once, the capillary wave appears
on the surface, the instability pattern changes at different locations of the
liquid sheet as shown in figure 15. This may be because of the uneven thinning
(locally) of the liquid sheet that causes the development of different modes
of Faraday instability. The Faraday instability leads to perforation of the
stretched umbrella-like liquid sheet leading to the merging of the ligaments
and their breakup due to RP instability.

For better clarity, the capillary wave propagation on the surface of the
liquid surface in the sub-harmonic condition can be seen in the video:
capillary wave formation and instability. This top-view video was recorded by
focusing on the liquid sheet surface at 140000 fps.

####  5.5.2 Mechanism of Bag breakup

The bag formation commences during the deceleration of the liquid sheet
towards the end of the equatorial breakup process (see Figure 13). In this
case, when the $We_{L}$ decreases and the surface tension dominates over the
inertia force. Therefore the ligaments and detached liquid rim (due to
capillary breakup) do not break into droplets and remain attached to the edge
of the liquid sheet. This results in an increase of the sheet thickness (see
the video given in the caption of figure 19) and prevents breakup by umbrella
mode (Faraday instabilty). Since the diameter of the stretched liquid sheet is
beyond the levitation capacity of the levitator, the edges of this stretched
liquid sheet curl downwards in the vertical direction. The pressure
distribution on the liquid sheet is such that it is maximum at the pole and
minimum at the edge of the sheet (see the appendix section in figure A.5).
This leads to the liquid sheet bending like a bag during its simultaneous
upward movement due to acoustic radiation pressure.

Figure 28:  Instantaneous side view digital images of droplet ejection during
propagation of instability in bag breakup regime. The images are presented at
a time interval of 20.8 $\mu$s. The corresponding video available
atvideo:Bagfaraday.

The extent of upward movement (z) during bag development and subsequent
ejection of the tiny-sized droplets, perpendicular to the liquid sheet due to
Faraday instability, are shown in figure 28. This occurs as the thickness of
the liquid sheet decreases to meet the prerequisite for sub-harmonic
condition. For additional information, the reader is advised see the video
given in the caption of figure 28. Figure 29 shows the distance traveled (z)
in the upward direction and the corresponding velocity and acceleration of the
liquid sheet during bag growth. It is seen that $z$ increases exponentially
with time before the initiation of a hole in the sheet. During this period,
the time-averaged velocity increases for a period of 1 ms with increased
acceleration and subsequent onset of deceleration where after, it starts
decreasing. The periodic acceleration and deceleration of the liquid sheet in
the vertical direction during bag formation ( Figure 29 (b)) suggests the
possibility of RT instability perpendicular to the surface of the liquid
sheet. The RT wavelength estimated corresponding to the maximum
acceleration/deceleration of the bag (12500 $m/s^{2}$) is $\lambda_{RT}$ =
0.502 mm. The measured wavelength of the deformed liquid rim during upward
propagation of the bag is equal to 0.72 mm (by top view visualization) and 0.6
mm (by side view visualization). The theoretical and measured values agree
reasonably well, suggesting that the RT instability may be responsible for the
deformation of the bag rim.

Figure 29:  The change of height (z), velocity ($v_{z}$) and acceleration
($a_{z}$) during bag formation (upward movement of the liquid sheet). The
vertical movement is calculated by focusing at the center of the liquid sheet.
Here, $t_{bag}$ denotes the time between the end of the equatorial breakup and
the start of the secondary breakup.

The velocity decrease in figure 29 appears to be after holes form due to the
ejection of tiny-sized droplets caused by the Faraday instability of the
thinned region of the bag. This reduction of the thickness plays an important
role in secondary breakup and depends upon the We number. The sub-harmonic
condition is only set up when the liquid sheet frequency becomes on the order
of half of the imposed acoustic frequency. As soon as the thickness reduces to
reach the sub-harmonic region the wave appears on the surface and droplet
ejection starts as the instability setup. A video of Faraday instability (
Faraday instability) on the surface of the bag is captured at a frame rate of
140k fps. The ejection of the droplets due to Faraday instability creates a
hole (see the first image of Figure 30). A few such holes are formed, which
expand radially with the formation of the liquid rim at the edge. The hole rim
thickness increases during hole expansion. The expansion of the holes creates
interconnected ligaments which become unstable due to RP instability and break
into the droplets. This process continues until the complete sheet breakup of
the liquid sheet. This video provides a nice illustration of how the liquid
sheet has changed into thick ligaments during perforation, which then breaks
into the larger size droplet: ( liquid sheet to ligaments). This video has
been recorded at 72 000 fps from top view. To calculate the rate of expansion
of the hole with time, we measured the diameter of the hole with time focusing
with 3 holes. The change in the diameter of the hole with time is presented in
Figure 30 (b).

Figure 30 (b) shows the expansion of the hole diameter ($d_{h}$) with time for
3 such holes with typical velocities of the order of $0.4-0.5$m/s. The hole
expansion at a constant velocity of a perforated liquid sheet is also reported
by Liu et al. (2022); Fraser et al. (1962); Taylor (1959); Culick (1960).
Contrary to this, some researchers have observed exponential growth Debrégeas
et al. (1995); Roth et al. (2005); Savva & Bush (2009). Our results also show
an onset of exponential growth $d_{h}$ with time (see Figure 30 (b) 2 and 3).
In this case the exponential velocity increases with time. The expansion
diameter $d_{h}\sim exp(t/\tau)$ is identical to the observation of Debrégeas
et al. (1995).

Figure 30:  Hole formation and ligaments breakup during bag breakup regime:
(a) visualization of top view images and (d) hole expansion diameter with
time. Here, $d_{h}$ is the hole diameter and $t_{h}$ is the time starting from
the hole formation.

####  5.5.3 Mechanism of Bubble breakup and Multistage breakup

The observation of the bubble breakup is reported in section 3.7.3. The bubble
breakup occurs for 1.56 $\leq$ We $\leq$ 1.92. The flattened droplet undergoes
thinning followed by atomization at the edge. The liquid sheet starts to bend
downwards leading to bag formation or bubble formation depending upon the
initial size of the droplet (We). When the droplet size is increased such that
during bag formation, its thickness does not reduce to an extent to admit sub-
harmonic conditions, the liquid sheet takes a near bubble (see Figure 17) like
structure. The bubble further expands due to the pressure differences between
the inside and outside of the bubble resulting in a further decrease in its
thickness. As the thickness reduces to attain harmonic conditions, non-linear
Faraday waves appear on the surface of the liquid as shown in Figure 31. These
non-linear waves which are quite different from bag and umbrella breakup can
be observed by focusing on the surface of the bubble. The measured wavelength
is equal to 65 $\mu$m confirming it is a Faraday wave. This results in hole
formation due to ejection of droplets due to Faraday instability, ligament
formation due to hole rearrangement, and the breakup of ligaments due to RP
instability as discussed in the section 5.5.2. The rim can then dramatically
close back and the bubble can show one more round of bubble expansion, finally
leading to complete atomization by a combination of Faraday and RP
instabilities.

Figure 31:  Instantaneous images of the growth of Faraday instability on the
surface of the bubble and how that instability eventually resulted in the
creation of a hole. The corresponding video is available at video: Bubble
instability

The multistage breakup takes place for larger droplets with We $>$ 1.92. In
this case, bag formation commences during equatorial breakup. The duration of
an equatorial breakup is very small for a multistage breakup. Thus a very
small volume of the droplet is lost due to the ejection of the liquid at the
edge. The rim thickness increases as the liquid flowing from inside the
droplet accumulates at the edge. As a consequence, instead of closing like a
bubble, the bag’s rim begins to expand in a radial direction. The breakup
occurs in stages similar to the equatorial breakup. The Faraday wavelength is
equal to 65 $\mu$m at the thinnest region indicating harmonic condition. The
rim deformation in the azimuthal direction and its wavelength is equal to 0.34
mm. This rim deformation is triggered through RT instability, which is similar
to equatorial breakup. The complete explanation is shown in the given
video:multistage. This video is presented after bag formation.

##  6 Conclusions

An experimental investigation of the breakup dynamics of a single droplet
levitated in the acoustic field is carried out for different sizes of the
droplet (thereby changing the We). Both top and side view imaging are
performed at a significantly higher frame rate (4000 fps to 140k fps) to
understand the mechanism underlying the breakup process and visualize the
associated interfacial instabilities. The droplet deforms with an increase in
We without any atomization for We $\leq$ 1.36. For We $>$ 1.36, the droplet
first transforms into a cylindrical liquid sheet during the stretching regime
followed by the thinning at the edge. During the stretching period, a slow
increase in velocity is observed, while the thinning regime exhibits an abrupt
increment. Our theoretical model agrees well with the experimental results for
the stretching regime. The radial expansion in thinning regime is highly non-
linear and an empirical relationship based on $W_{eL}$, We, and nondimensional
time scale (T) that explains both the stretching as well as thinning regimes
is proposed. The thickness reduces at a faster rate in the thinning regime. As
the thickness reduces to the order of the harmonic condition, the sheet
vibrates in the vertical direction. A non-linear (Faraday) wave appears on the
surface of the thin liquid sheet. The Faraday instability causes the ejection
of smaller droplets perpendicular to the sheet. This droplet ejection creates
the hole resulting in perforation of the liquid sheet. Then, the liquid rim is
detached from the liquid sheet and breaks due to RP instability. The capillary
(sinuous) wave generates due to the vertical vibration of the liquid sheet.
The vertical vibration induces radial acceleration and deceleration towards
the liquid-air interface (rim) resulting in the formation of the radial
ligaments because of the development of RT instability. The sizes of the
droplets, generated during the process, are related to the different types of
interfacial instabilities that develop (Faraday, RT and RP instability) during
equatorial breakup. The results of the experimental measurement show that
although the pattern of the Faraday waves changes throughout all regimes,
their wavelength remains constant. This suggests a parametric instability of
the imposed wavelength on the sheet, $\lambda\sim U/f$. The bag formation
commences during the atomization at the equatorial region. The liquid sheet
diameter is larger than the levitation capacity of the acoustic levitator.
Therefore, the liquid sheet bends downwards at the edges due to gravity. The
radial pressure distribution along the sheet diameter causes the bag
formation. The secondary breakup of the liquid sheet depends on the different
types of the bag formed based on the We. The umbrella breakup for 1.36 $\leq$
We $\leq$ 1.45, bag breakup for 1.45 $<$ We $\leq$ 1.56, bubble breakup for
1.53 $<$ We $\leq$ 1.92 and multi-stage breakup for We $>$ 1.92 are observed
experimentally. The mechanism of the breakup of the liquid sheet at different
modes is discussed in detail. Experimentally measured wavelength is compared
with calculated from the dispersion relation of the corresponding instability.

The work presents a myriad of complex phenomena that present significant
challenges in theoretical predictions of the various involved instabilities,
further exacerbated by the transient nature of the process. The work, however,
demonstrates a variety of phenomena that could be harnessed judiciously while
designing a large-scale atomizer based on acoustic fields.

## Appendix A

###  A.1 Basics of standing sound wave

Consider an incident sound wave of frequency $f=40kHz$, such that $\omega=2\pi
f$. The wavelength of sound is given by the distance the sound covers in 1
cycle i.e $1/f$ time. Thus $\lambda=2\pi/k=c_{o}/f$, where $c_{o}$ is the
velocity of sound, that is $c_{o}=\omega/k$. For the present case,
$\lambda=$8.5 mm. The trap has an end-to-end distance $L=$ 90 mm, the number
of the nodes where the droplet can be levitated (in the absence of gravity)
can be obtained by $z_{n}=(n\lambda/2)+(\lambda/4)$. The number of the nodes
(n) for the length L=$z_{n}$ is equal to $\sim$ 20\. This is exactly equal to
our calculation from simulation using Comsol multiphysics software.

The actual conditions in the tiny-lev levitator are a little different due to
multiple emitters and receptors. The incident pressure wave associated with a
standing sound wave is given by (Andrade et al. (2018)),

$P_{1}=P_{0}cos(kz)cos(\omega t)$ (21)
$\rho_{1}=P_{0}/c_{o}^{2}cos(kz)cos(\omega t)$ (22)

such that $z_{n}=L=(n\lambda/2)+(\lambda/4)$

$k=\frac{\pi}{2L}(2n+1)$ (23)

. In a standing wave, the nodes occur at $z_{n}=(n\lambda/2)+(\lambda/4)$. For
a single node, the node occurs at $z_{n}=L_{1}=3\lambda/4$. The associated
velocity with the standing sound wave that satisfies $\rho_{o}\frac{\partial
u_{1}}{\partial t}=-\nabla P_{1}$, is given by,

$u_{1}=\frac{P_{0}}{\rho_{0}v_{0}}sin(kz)sin(\omega
t)=\frac{P_{0}}{\rho_{0}v_{0}}sin(\omega t))$ (24)

The RMS velocity associated with this pressure $<u_{1}^{2}>$

$U^{2}=\langle
u_{1}^{2}\rangle=\bigg{(}\frac{P_{0}}{\rho_{0}v_{0}}\bigg{)}^{2}\frac{\int_{0}^{T=\frac{2\pi}{\omega}}sin^{2}(\omega
t)\,dt}{\frac{2\pi}{\omega}}$ (25)
$U=\frac{P_{0}}{\rho_{0}v_{0}}\frac{1}{\sqrt{2}}$ (26)

###  A.2 Derivation of Net Radiation Pressure

The Bernoulli equation associated with the sound wave second order gives,

$\nabla<P_{2}>=-<\rho_{1}\frac{\partial v_{1}}{\partial
t}>-\rho_{0}<v_{1}.\nabla v_{1}>$ (27)

The time average radiation pressure can be given as,

$\langle P_{2}\rangle=\frac{\langle
P_{1}^{2}\rangle}{2\rho_{0}v_{0}^{2}}-\frac{\rho_{0}\langle
u_{1}^{2}\rangle}{2}$ (28)

We get

$<P_{2}>=\frac{P_{o}^{2}}{4\rho_{0}c_{0}^{2}}\left(\cos^{2}{kz}-\sin^{2}{kz}\right)$
(29)

The associated Gorkov potential is given by,

$U_{Gorkov}=\frac{P_{o}^{2}}{4\rho_{0}c_{0}^{2}}\left(\frac{\cos^{2}{kz}}{3}-\frac{\sin^{2}{kz}}{2}\right)$
(30)

Integrating over a sphere, one gets,

$F_{rad}{\bf e_{z}}=\frac{5\pi
R^{3}kP_{o}^{2}}{6\rho_{0}c_{0}^{2}}\sin{2kz}\,{\bf e_{z}}$ (31)

The droplet cannot be levitated at the exact node of the standing wave when
gravity is present; however, it can be levitated just below the node point
depending on the gravity force.

The force of gravity on a levitated sphere can be expressed as

$F_{g}=-\rho_{L}g\frac{4\pi R^{3}}{3}\,{\bf e_{z}}$ (32)

The z value comes out to be -0.3 mm after equating the equation 31 and
equation 32, indicating that the droplet is levitated 0.3 mm below the node
point.

(33)

###  A.3 Pressure around the droplet in the presence of a droplet

$U^{2}=U_{x}^{2}+U_{y}^{2}+U_{z}^{2}$ (34)

$U_{x}=-(\gamma/2)$ x, $U_{y}=-(\gamma/2)$ y, $U_{z}=-\gamma$ z, where,
$\gamma=f\frac{U}{d_{0}}$

$P_{a}(0,z)\approx P_{a}(0)=P_{a}(r,z)+\frac{\rho_{0}U^{2}}{2}$ (35)

$P_{a}(0)=P_{a}(r,z)+\frac{\rho_{a}\gamma^{2}}{2}(U_{x}^{2}+U_{y}^{2}+U_{z}^{2})$
(36)

$P_{a}(0)=P_{a}(r,z)+\frac{\rho_{a}f^{2}U^{2}}{2d_{0}^{2}}(z^{2}+\frac{r^{2}}{4})$
(37)
$P_{a}(r,z)=P_{a}(0)-f^{2}\frac{\rho_{a}U^{2}}{2d_{0}^{2}}z^{2}-f^{2}\frac{\rho_{a}U^{2}}{8d_{0}^{2}}r^{2}$
(38)

###  A.4 Sound pressure distribution inside the acoustic levitator

COMSOL Multiphysics (COMSOL Inc.) software is used to simulate the pressure
field distribution in the acoustic field. The Pressure Acoustics, Frequency
Domain model is selected.

The following governing equation has been solve to get the pressure acoustic
field distribution.

$\nabla\cdot\biggl{(}-\frac{1}{\rho}(\nabla
p_{t}-q_{d})\biggl{)}-\frac{k_{eq}^{2}}{\rho}p_{t}=Q_{m}$ (39)

$k_{eq}^{2}=\biggl{(}\frac{\omega}{c}\biggl{)}^{2}-k_{z}^{2}$ (40)

Where $Q_{m}$ and q are mono pole and dipole sources. $\omega=2\pi f$, $f$ =
40k Hz. The geometry for the computational domain has been chosen according to
the experimental setup. The two spherical cup of diameter 4.5 cm have been
used at the top and bottom as acoustic field sources. The center distance
between to cup is 9 cm.

Figure 32:  Acoustic pressure field distribution inside the acoustic
levitator.

Figure 32 shows the Acoustic pressure field distribution inside the acoustic
levitator. The peak amplitude (pressure at anti node) is equal to 2400 Pa. The
number of nodes are equal to 20.

###  A.5 Pressure re-distribution due to the presence of a particle in an
acoustic field

The presence of a particle (liquid droplet in this case), significantly and
non-trivially alters the ambient undisturbed pressure distribution due to the
standing sound wave. We have used the COMSOL multiphysics software to get the
acoustic radiation pressure distribution around a droplet placed in a standing
wave acoustic field. Our experimental observation suggests that the droplet
shape changes with time in the standing wave. Therefore, different aspect
ratios of the deformed droplet were chosen according, to the experimental
results, which cover the different regimes of dynamics of droplets in the
acoustic field. The density, surface tension, and sound speed of the droplet
are 830 $kg/m3$, 22.5 $mN/m$, and 1540 $m/s$, respectively. The obtained
simulation results are presented in Figure 33. Figure 33 (a) shows the
distribution of the radiation acoustic pressure without any droplet. As the
droplet is placed in a standing wave, the sound waves are scattered around the
droplet such that the maximum pressure is at the pole region ($P_{P}$) and the
minimum pressure is around the equatorial region ($P_{E}$) (Figure 33 (b)).
The corresponding change in the pressure difference between the Pole and
Equator ($P_{P}-P_{E}$) for different aspect ratio is presented in Figure 33
(g), where P and E indicate the Pole and Equator, respectively, as marked in
Figure 33 (b). The figure shows an increase in $P_{P}-P_{E}$ with an increase
in $R$ and corresponding reduction in $h$, during the radial expansion of the
droplet such that $P_{P}-P_{E}$ $\simeq$ $R^{1/2}$ during stretching regime
and $P_{P}-P_{E}$ $\simeq$ $R^{11/5}$ during thinning regime (see Figure 33
(f)) .

The nondimensional acoustic pressure on the surface of the droplet is given by
Shi & Apfel (1996) in the form of acoustic velocity potential. The
$U=-\nabla\phi$ and first order acoustic pressure
$P=\rho_{a}\frac{\partial\phi}{\partial t}$. The velocity potential is
nondimensionalized by the velocity potential of incident wave $\phi_{0}$.

$P_{r}=\frac{1}{K^{2}}(n.\nabla\phi)^{2}-\frac{1}{2}\phi^{2}$ (41)

Where, $K=\frac{kd_{0}}{2}$, k is the acoustic wave number and n is the unit
normal vector on the surface of the droplet.

As the droplet placed in the standing acoustic field, the acoustic wave
scattered around the droplet and it experiences the non uniform radiation
stress which causes the deformation of the droplet against the surface tension
force. This is also reflected in our simulation results as shown in Figure 33
for without (a) and with (b) droplets in the standing acoustic field. The
deformed shape further modifies the scattered acoustic field and radiation
stress. This process continues until the transient energy is damped out
whereafter it reaches a steady deformation. That indicates that in order to
maintain equilibrium, the surface tension force is increased to the order of
the radiation force by adjusting the curvature of the droplet.

The first term on the right-hand side of the equation 41 denotes the suction
force in the outer normal direction and the second term denotes the squeezing
force in the opposite direction. The suction force increased as the droplet
becomes oblate shape under the effect of the acoustic field which causes the
further flattening of the droplet. And at the same time, the squeezing force
on the droplet is also increased (Shi & Apfel (1996)) which is similar to our
simulation results. Once the droplet gets flattened, the radiation pressure
becomes equal to the pressure inside the liquid sheet (see Figure 33 (e) and
Figure 33 (f)). The equilibrium can only be achieved if the surface tension,
suction stress, and internal pressure balance each other. The surface tension
force is proportional to the curvature ($\frac{1}{h}$) of the flattened
droplet. The suction stress is proportional to the $\frac{P_{0}^{2}R}{h}$ (Lee
et al. (1991)). For the smaller value of the h, the internal pressure can be
ignored because at this time the radiation pressure and pressure inside the
liquid sheet become equal (see Figure 33 (d). A similar argument is also
predicted by Lee et al. (1991). Therefore, the dynamics at the rim are the
competition between the surface tension and the suction stress. As the
$P_{0}^{2}R$ becomes larger during flattening, the surface tension force is no
longer able to balance the suction force. The increased velocity at the
equatorial region of the flattened droplet creates a higher suction. At this
time, there is an insufficient increase of the squeezing force because of
equal pressure. Therefore, the suction stress is completely dominated over the
surface tension which pulls the liquid resulting in thinning at the equatorial
region (Lee et al. (1991)).

Figure 33:  Acoustic pressure field variation during deformation of the droplet levitated on the node of standing wave: (a) without drop (b) 2R = 2.2 mm and h = 0.8 mm, (c) 2R = 2.5 mm and h = 0.54 mm, (d) 2R = 3 mm and h = 0.38 mm, (e) 2R = 4 mm and h = 0.2 mm and (f) 2R = 5 mm and h = 0.14 mm, (g) $P_{P}-P_{E}$ vs R and Log($P_{P}-P_{E}$) vs Log R. Where $P_{E}$ and $P_{P}$ are the acoustic pressure at the equator and pole region, respectively.  Table 2: . Stretching factor (f) for different We. We | f
---|---
1.37 | 9.2
1.45 | 9.28
1.56 | 9.89
1.92 | 9.93
2.4 | 12

## References

  * Akkoyun et al. (2021) Akkoyun, Fatih, Gucluer, Sinan & Ozcelik, Adem 2021 Potential of the acoustic micromanipulation technologies for biomedical research. Biomicrofluidics 15 (6), 061301.
  * Andrade & Marzo (2019) Andrade, Marco AB & Marzo, Asier 2019 Numerical and experimental investigation of the stability of a drop in a single-axis acoustic levitator. Physics of Fluids 31 (11), 117101.
  * Andrade et al. (2020a) Andrade, Marco AB, Marzo, Asier & Adamowski, Julio C 2020a Acoustic levitation in mid-air: Recent advances, challenges, and future perspectives. Applied Physics Letters 116 (25), 250501.
  * Andrade et al. (2018) Andrade, Marco AB, Pérez, Nicolás & Adamowski, Julio C 2018 Review of progress in acoustic levitation. Brazilian Journal of Physics 48, 190–213.
  * Andrade et al. (2020b) Andrade, Marco AB, Ramos, Tiago S, Adamowski, Julio C & Marzo, Asier 2020b Contactless pick-and-place of millimetric objects using inverted near-field acoustic levitation. Applied Physics Letters 116 (5), 054104.
  * Aoki & Hasegawa (2020) Aoki, K & Hasegawa, K 2020 Acoustically induced breakup of levitated droplets. AIP Advances 10 (5), 055115\.
  * Basu et al. (2012) Basu, Saptarshi, Saha, Abhishek & Kumar, Ranganathan 2012 Thermally induced secondary atomization of droplet in an acoustic field. Applied Physics Letters 100 (5), 054101.
  * Bayazitoglu & Mitchell (1995) Bayazitoglu, Yildiz & Mitchell, Garrick F 1995 Experiments in acoustic levitation-surface tension measurements of deformed droplets. Journal of thermophysics and heat transfer 9 (4), 694–701.
  * Breitenbach et al. (2018) Breitenbach, Jan, Roisman, Ilia V & Tropea, Cameron 2018 From drop impact physics to spray cooling models: a critical review. Experiments in Fluids 59, 1–21.
  * Bremond et al. (2007) Bremond, Nicolas, Clanet, Christophe & Villermaux, Emmanuel 2007 Atomization of undulating liquid sheets. Journal of Fluid Mechanics 585, 421–456.
  * Bush (2015) Bush, John WM 2015 Pilot-wave hydrodynamics. Annual Review of Fluid Mechanics 47, 269–292.
  * Clanet & Villermaux (2002) Clanet, Christophe & Villermaux, Emmanuel 2002 Life of a smooth liquid sheet. Journal of fluid mechanics 462, 307–340.
  * Culick (1960) Culick, Fred EC 1960 Comments on a ruptured soap film. Journal of applied physics 31 (6), 1128–1129.
  * Danilov & Mironov (1992) Danilov, SD & Mironov, MA 1992 Breakup of a droplet in a high-intensity sound field. The Journal of the Acoustical Society of America 92 (5), 2747–2755.
  * Debrégeas et al. (1995) Debrégeas, G, Martin, Pascal & Brochard-Wyart, F 1995 Viscous bursting of suspended films. Physical review letters 75 (21), 3886.
  * Dighe & Gadgil (2019) Dighe, Sandip & Gadgil, Hrishikesh 2019 Atomization of acoustically forced liquid sheets. Journal of Fluid Mechanics 880, 653–683.
  * El-Molla (2017) El-Molla, Sara 2017 Inkjet printing & spray deposition techniques for flexible electronic applications. PhD thesis, Dissertation, München, Technische Universität München, 2017.
  * Fang et al. (2022) Fang, Zi-dan, Wang, Tie-han, Jia, Bo-qi, Yang, Li-jun, Fu, Qing-fei, Sun, Da-kun & Sun, Xiao-feng 2022 Linear instability of a liquid sheet in a transverse standing acoustic field. Physics of Fluids 34 (6), 063310.
  * Fraser et al. (1962) Fraser, RP, Eisenklam, Paul, Dombrowski, Norman & Hasson, David 1962 Drop formation from rapidly moving liquid sheets. AIChE Journal 8 (5), 672–680.
  * Geng et al. (2014) Geng, DL, Xie, WJ, Yan, N & Wei, B 2014 Vertical vibration and shape oscillation of acoustically levitated water drops. Applied Physics Letters 105 (10), 104101.
  * Hong et al. (2017) Hong, ZY, Yin, JF, Zhai, W, Yan, N, Wang, WL, Zhang, J & Drinkwater, Bruce W 2017 Dynamics of levitated objects in acoustic vortex fields. Scientific reports 7 (1), 1–7.
  * Jarrahbashi & Sirignano (2014) Jarrahbashi, D & Sirignano, WA 2014 Vorticity dynamics for transient high-pressure liquid injection. Physics of Fluids 26 (10), 73.
  * Jia et al. (2022) Jia, Boqi, Wang, Tiehan, Fu, Qing-fei & Yang, Li-jun 2022 Experimental study on the stability and breakup of a planar liquid sheet under a standing wave acoustic field. Physical Review Fluids 7 (12), 124004.
  * Josserand & Thoroddsen (2016) Josserand, Christophe & Thoroddsen, Sigurdur T 2016 Drop impact on a solid surface. Annual review of fluid mechanics 48, 365–391.
  * Ju et al. (2017) Ju, Dehao, Sun, Xiaochuan, Jia, Xiaoxu, Huang, Zhong, Qiao, Xinqi, Han, Dong & Huang, Zhen 2017 Experimental investigation of the atomization behavior of ethanol and kerosene in acoustic fields. Fuel 202, 613–619.
  * Khan & Eslamian (2019) Khan, Talha & Eslamian, Morteza 2019 Experimental analysis of one-dimensional faraday waves on a liquid layer subjected to horizontal vibrations. Physics of Fluids 31 (8), 082106.
  * Kirar et al. (2022) Kirar, Pavan Kumar, Soni, Surendra Kumar, Kolhe, Pankaj S & Sahu, Kirti Chandra 2022 An experimental investigation of droplet morphology in swirl flow. Journal of Fluid Mechanics 938, A6.
  * Kulkarni & Sojka (2014) Kulkarni, Varun & Sojka, Paul E 2014 Bag breakup of low viscosity drops in the presence of a continuous air jet. Physics of Fluids 26 (7), 072103.
  * Kumar & Tuckerman (1994) Kumar, Krishna & Tuckerman, Laurette S 1994 Parametric instability of the interface between two fluids. Journal of Fluid Mechanics 279, 49–68.
  * Kumar et al. (2010) Kumar, Ranganathan, Tijerino, Erick, Saha, Abhishek & Basu, Saptarshi 2010 Structural morphology of acoustically levitated and heated nanosilica droplet. Applied Physics Letters 97 (12), 123106.
  * Law (2001) Law, S Edward 2001 Agricultural electrostatic spray application: a review of significant research and development during the 20th century. Journal of Electrostatics 51, 25–42.
  * Lee et al. (1991) Lee, CP, Anilkumar, AV & Wang, TG 1991 Static shape and instability of an acoustically levitated liquid drop. Physics of Fluids A: Fluid Dynamics 3 (11), 2497–2515.
  * Lefebvre & McDonell (2017) Lefebvre, Arthur H & McDonell, Vincent G 2017 Atomization and sprays. CRC press.
  * Liu et al. (2022) Liu, Yang, Zhang, Huiqiang, Li, Yanfei & Zhao, Zijie 2022 Numerical investigation on the hole characteristics and perforation mechanism of conical liquid sheet. Thermal Science and Engineering Progress 29, 101225.
  * Lohse & Villermaux (2020) Lohse, Detlef & Villermaux, Emmanuel 2020 Double threshold behavior for breakup of liquid sheets. Proceedings of the National Academy of Sciences 117 (32), 18912–18914.
  * Marzo et al. (2017) Marzo, Asier, Barnes, Adrian & Drinkwater, Bruce W 2017 Tinylev: A multi-emitter single-axis acoustic levitator. Review of Scientific Instruments 88 (8), 085105.
  * Mulmule et al. (2010) Mulmule, Aditya S, Tirumkudulu, Mahesh S & Ramamurthi, K 2010 Instability of a moving liquid sheet in the presence of acoustic forcing. Physics of Fluids 22 (2), 022101.
  * Naidu et al. (2022) Naidu, Haripriya, Kahraman, Ozan & Feng, Hao 2022 Novel applications of ultrasonic atomization in the manufacturing of fine chemicals, pharmaceuticals, and medical devices. Ultrasonics Sonochemistry 86, 105984.
  * Naka & Hasegawa (2020) Naka, M & Hasegawa, K 2020 Breakup characteristics of levitated droplets in a resonant acoustic field. Physics of Fluids 32 (12), 124109.
  * Oza et al. (2014) Oza, Anand U, Harris, Daniel M, Rosales, Rodolfo R & Bush, John WM 2014 Pilot-wave dynamics in a rotating frame: on the emergence of orbital quantization. Journal of fluid mechanics 744, 404–429.
  * Roth et al. (2005) Roth, CB, Deh, B, Nickel, BG & Dutcher, JR 2005 Evidence of convective constraint release during hole growth in freely standing polystyrene films at low temperatures. Physical Review E 72 (2), 021802.
  * Savva & Bush (2009) Savva, Nikos & Bush, John WM 2009 Viscous sheet retraction. Journal of Fluid Mechanics 626, 211–240.
  * Sharma et al. (2021) Sharma, Shubham, Singh, Awanish Pratap, Rao, S Srinivas, Kumar, Aloke & Basu, Saptarshi 2021 Shock induced aerobreakup of a droplet. Journal of Fluid Mechanics 929, A27.
  * Shats et al. (2014) Shats, Michael, Francois, Nicolas, Xia, Hua & Punzmann, Horst 2014 Turbulence driven by faraday surface waves. In International Journal of Modern Physics: Conference Series, , vol. 34, p. 1460379. World Scientific.
  * Shi (1995) Shi, William Tao 1995 Dynamics and instabilities of acoustically stressed interfaces. Yale University.
  * Shi & Apfel (1996) Shi, W Tao & Apfel, Robert E 1996 Deformation and position of acoustically levitated liquid drops. The Journal of the Acoustical Society of America 99 (4), 1977–1984.
  * Squire (1953) Squire, HB 1953 Investigation of the instability of a moving liquid film. British Journal of Applied Physics 4 (6), 167\.
  * Tambasco et al. (2016) Tambasco, Lucas D, Harris, Daniel M, Oza, Anand U, Rosales, Rodolfo R & Bush, John WM 2016 The onset of chaos in orbital pilot-wave dynamics. Chaos: An Interdisciplinary Journal of Nonlinear Science 26 (10), 103107.
  * Taylor (1959) Taylor, Geoffrey Ingram 1959 The dynamics of thin sheets of fluid. iii. disintegration of fluid sheets. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 253 (1274), 313–321.
  * Villermaux & Bossa (2009) Villermaux, Emmanuel & Bossa, Benjamin 2009 Single-drop fragmentation determines size distribution of raindrops. Nature physics 5 (9), 697–702.
  * Villermaux & Bossa (2011) Villermaux, Emmanuel & Bossa, Benjamin 2011 Drop fragmentation on impact. Journal of Fluid Mechanics 668, 412–435.
  * Villermaux & Clanet (2002) Villermaux, Emmanuel & Clanet, Christophe 2002 Life of a flapping liquid sheet. Journal of fluid mechanics 462, 341–363.
  * Vukasinovic et al. (2007) Vukasinovic, Bojan, Smith, Marc K & Glezer, ARI 2007 Dynamics of a sessile drop in forced vibration. Journal of Fluid Mechanics 587, 395–423.
  * Zeng et al. (2018) Zeng, Qingyun, Gonzalez-Avila, Silvestre Roberto, Ten Voorde, Sophie & Ohl, Claus-Dieter 2018 Jetting of viscous droplets from cavitation-induced rayleigh–taylor instability. Journal of Fluid Mechanics 846, 916–943.