# Effect of ambient gas on cavity formation for sphere impacts on liquids

###### Abstract

Formation of a splash crown and a cavity following the impact of a sphere on a
body of liquid is a classical problem. In the related problem of a droplet
splashing on a flat surface, it has been established that the properties of
the surrounding gas can influence the splashing threshold. At lower impact
speeds, this is due mainly to the influence of gas kinetic effects, since the
height of the gas lubrication film which is displaced during dynamic wetting
is often comparable to the mean free path of the gas. At higher Weber and
Reynolds numbers, on the other hand, inertial effects dominate and the density
of the gas becomes important in determining whether a splash occurs. In this
article, sphere impacts on a liquid body are investigated in a rarefied
atmosphere using high-speed photography. It is found that the threshold entry
speed for cavity formation is influenced by the density of the surrounding
gas, whereas changing the mean free path of the gas has no effect. We
attribute this phenomenon to the gas slowing the sealing of the thin crown
sheet behind the sphere. This assertion is supported with experimental
measurements of the liquid sheet thickness. In the range of parameters
considered, the splash crown influences the movement of the contact line, an
effect not previously observed.

Hollis Williams1, James Sprittles2, Juan C. Padrino3, and Petr Denissenko1

1School of Engineering, University of Warwick, Coventry CV4 7AL, UK

2Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK

3School of Engineering, Newcastle University, Newcastle upon Tyne, NE1 7RU, UK

Contact email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>

Keywords: boundary layers, contact line dynamics, flow instability, wetting

##  1 Introduction

A sphere plunging through the free surface of a body of liquid may splash and
form a cavity which trails behind it. Air-water entry of projectiles has been
studied since the nineteenth century [1, 2]. Early studies, based mostly on
the work of von Kármán and Wagner [3, 4], assumed that one can neglect the
effects of liquid viscosity and surface tension due to large Reynolds and
Weber numbers. These studies also neglect the presence of surrounding gas due
to its low density. Subsequent work has shown that this picture is
oversimplified and that the ambient gas influences dynamics of the cavity
following a liquid-solid impact [5, 6, 7]. Furthermore, Duez et al.
established that both viscosity and surface tension of the liquid play a role
in cavity formation, finding also that wettability of the impacting body is an
important factor in determining the threshold speed $U^{*}$ for air
entrainment [8]. Later, Marston et al. found that a falling sphere entraps a
small amount of air at the bottom of the sphere due to the air pocket which
forms because of the lubrication pressure in the gas layer between the sphere
and the liquid surface. This phenomenon is due to the air pocket which forms
because of the lubrication pressure in the gas layer between the sphere and
the liquid surface prior to impact. As the liquid surface deforms, a thin
sheet of air is produced which contracts to a bubble at the south pole of the
sphere [9]. Numerical simulations have found that cavity formation is
connected with pinning of the contact line around the sphere and that
viscosity of the gas in the film between the sphere and the liquid plays a
role during early impact [10, 11].

There have been many studies of other variants of the falling sphere problem,
although none have focussed directly on the influence of the surrounding gas
on cavity formation [12]. Aristoff et al. considered buoyant low-density
spheres, finding expressions for the pinch-off time of a cavity and the volume
of air entrained by the sphere [13]. Hurd et al. studied the water entry
characteristics of deformable elastomeric spheres, finding that the
oscillations of these spheres during impact results in new types of nested
cavities [14]. Watson et al. examined spheres with heterogeneous wetting
properties, finding that spheres which are partly hydrophilic and partly
hydrophobic always have asymmetric cavities and drift away from straight-line
trajectories [15]. Marston et al. observed cavity formation for heated sphere
impacts, finding that there is an inverted Leidenfrost effect when the sphere
temperature is much larger than the boiling point of the liquid, which either
produces a cavity with smooth walls or a double cavity structure [16]. Mansoor
et al. also studied superhydrophobic spheres in detail and used a splash-guard
mechanism to eliminate the phenomenon known as surface seal [17]. Related
studies have considered projectiles with varying aspect ratios and impacts on
a two-phase fluid [18, 19, 20].

The process of splash curtain formation in the sphere impact problem shares
some similarities with splashing of a liquid droplet, but is believed to be
driven by different physical mechanisms [21, 22]. Nevertheless, the
similarities which exist might lead one to wonder whether the dynamics of the
splash curtain can be influenced by the properties of the surrounding gas,
given that it is now well-established that the properties of the surrounding
gas play a significant role in influencing droplet splashing. This line of
investigation was initiated by Xu et al. who observed that splashing of a
droplet when it impacts against a flat smooth surface can be completely
suppressed by reducing the pressure of the surrounding gas [23]. Xu studied
the dependence of droplet splashing on the roughness and texture of the
surface, confirming that there is a different type of splashing caused by
surface roughness (called prompt splashing) which must be distinguished from
corona splashing on a smooth surface due to the presence of the surrounding
air [24]. Benkreira and Khan demonstrated that air entrainment in the related
problem of a dip coating flow can also be suppressed under reduced pressures
and attributed this effect to the mean free path of the gas [25].

The regimes considered by these authors are typically for low impact speeds,
where gas kinetic effects are important and the mean free path plays a
significant role. This occurs because the maximum speed at which the liquid
can wet the solid surface is controlled by the speed at which the wetting gas
lubrication film in front of the moving contact line is displaced. The height
of these films is typically extremely thin (of the same order as the mean free
path of the gas) and as a consequence both droplet splashes and dip coating
flows can be successfully described by models which incorporate kinetic
effects [26, 27, 28]. As an example, the model of [26] uses kinetic theory in
the gas film as described by the Boltzmann equation and combines this with
regular hydrodynamics in the liquid phase as described by the Navier-Stokes
equations.

On the other hand, in the Gordillo-Riboux model for droplet splashing, the
threshold speed for splashing is determined using the fact that splashing
occurs due to a vertical lift force on the edge of the liquid sheet from the
surrounding gas. This lift force has two contributions: the lubrication force
$\sim K_{l}\mu_{g}V_{t}$ and the suction force $\sim
K_{u}\rho_{g}V_{t}^{2}H_{t}$, where $\mu_{g}$ is the viscosity of the gas,
$\rho_{g}$ is the density of the gas, $V_{t}$ is the initial velocity of the
ejected lamella and $H_{t}$ is the initial height of the lamella. $K_{l}$ and
$K_{u}$ are coefficients which are derived from detailed calculations of the
lift force in the region located between the lamella and the substrate (these
calculations show that $K_{l}$ is approximately equal to a sum of two terms
which both depend on the mean free path of the gas) [27]. The lubrication
force captures the viscous contributions to the lift force on the lamella edge
and the suction force captures the inertial forces. Note that viscosity is
actually a mean free path effect, since changes in gas pressure do not change
the viscosity of the gas. During the characteristic impact time, viscous
effects are confined to thin boundary layers with a typical width much lower
than the radius of the droplet $R$. Since the gas Reynolds number
$\text{Re}_{g}$ based on $H_{t}$ and $V_{t}$ as the characteristic scales is
$\sim\mathcal{O}(10)$, both the viscous and inertial contributions to the lift
force must be considered. The model suggests as expected that at higher Weber
and Reynolds numbers the dominant forces in the droplet splashing problem for
a smooth dry surface are inertial in nature [27].

Experimental evidence for this was found by Burzynski et al., who found that
gas entrapmment is not the mechanism which is responsible for splashing at
high Weber and Reynolds numbers and that splashing is influenced primarily by
the density, not the mean free path, of the surrounding gas [29]. This was
done by considering a flywheel experiment and different gases at atmospheric
pressure, with splashing outcomes analysed by measuring the size, velocity and
angle of ejected secondary droplets. The splashing outcome was also determined
from the total volume ejected, where the theoretical expression for the volume
is calculated using the Gordillo-Riboux theory [27]. Guo et al. conducted
numerical simulations of droplet impingement and splashing on dry and wet
surfaces at very high impact speeds, finding that splashing on a dry surface
can be suppressed by lowering the ambient gas density and that the properties
of the ambient gas do not significantly influence splashing on a wet surface
[30]. The simulations used a dynamic contact line model to define the boundary
condition at the moving contact line and found that increasing the density of
the gas increases the number of secondary droplets ejected from the lamella.

The Gordillo-Riboux model has since been expanded by Pierzyna et al. using a
data-driven threshold model which re-defines the threshold for droplet
splashing on a dry smooth surface by collating a large number of experimental
sources with different conditions and analysing the data with an uncertainty
qualification analysis combined with machine learning [31]. The Gordillo-
Riboux model can be considered as the special case where the parameter $\beta$
for determining splashing is a constant (in the general case, each impact
parameter has a different dependence on $\beta$). This more detailed threshold
model incorporating a large number of diverse experimental results observed a
linear dependence of the splashing threshold $\beta$ on impact speed $V$,
surface tension $\sigma$ and gas density $\rho_{g}$ and an inversely
proportional relation between $\beta$ and the liquid viscosity $\mu_{l}$. This
new model exhibits only weak dependency on the radius of the droplet $R$,
liquid density $\rho_{l}$, gas viscosity $\mu_{g}$ and the mean free path of
the gas $\lambda_{g}$. These dependent variables are eliminated by a recursive
feature elimination process to leave the four impact conditions mentioned
above. The eliminated variables are only irrelevant in their sense that they
are irrelevant beyond the expression for $\beta$, so the Gordillo-Riboux
captures all the influence of the eliminated variables on the splashing
parameter, such that they are needed for any further corrections.

In this paper, we use high-speed photography to investigate the influence of
the surrounding gas on cavity formation following the impact of a smooth dry
wettable sphere on the free surface of a liquid body. We begin in Section 2 by
varying properties of both the solid and the liquid, finding agreement with
existing experimental data on threshold speeds for cavity formation. We then
investigate the influence of the surrounding gas on cavity formation by using
air, argon, and helium at different pressures hence varying gas density and
the mean free path, obtaining results which diverge from the existing
theoretical models. In particular, we observe that cavity formation following
a sphere impact on a liquid surface can be completely suppressed by reducing
the pressure of the gas. This is the first time that the transition from a
cavity to a no cavity event has been effected only by changing the properties
of the ambient gas. We find that the most important parameter for influencing
cavity formation is the density of the ambient gas and that the mean free path
of the gas has negligible influence. In Section 3, we provide a theoretical
explanation for this effect based on the phenomenon by which the ambient gas
influences the sealing of the crown sheet behind the sphere. We finish in
Section 4 with conclusions and suggestions for future work.

##  2 Core Experimental Results

Spheres were released from an electromagnet above a container of silicone oil
located inside a depressurised chamber. The chamber was filled with air,
argon, or helium at pressures from 0.05 bar to atmospheric. Silicone oil was
used instead of water to reduce the threshold entry speed for cavity formation
and to prevent damage to the vacuum pump by water vapour. The vapour pressures
of the oils used are all less than 0.002 bar, well below the pressures used in
experiments, so that the influence of oil evaporation can be neglected.

The spheres were made of untreated chrome steel with density of $7.8$ g/cm3
and static contact angle $\theta\approx 20-25^{\circ}$ which makes them
oleophilic. The mean square roughness of the spheres was measured using a
Bruker Contour profilometer at $0.3\,\mu$m, well below the thickness of the
viscous boundary layer, reaching $\mathcal{O}(100\,\mu\text{m})$ at the sphere
equator. Typical image sequences acquired when a sphere of diameter 15 mm
plunges into the oil at atmospheric pressure, at 0.53 bar, and at 0.5 bar are
shown in Fig. 1, demonstrating that reduced pressures can suppress cavity
formation. The impact of the sphere into the liquid was captured against an
LED backlight with a Photron FASTCAM SA-X2 high-speed camera at a frame rate
of 12 500 fps. To provide consistent surface wettability, the sphere was
washed after each experiment with dilute ethanol, and then washed twice with
deionised water.

Figure 1: Chronophotography of a 15 mm sphere plunging into 1.74 cP silicone
oil at $V=2$ ms-1 at times $1.6$, $3.2$, $4.8$, and $6.4$ ms after touchdown.
The pressures are (a) $1$ bar, (b) $0.53$ bar, and (c) $0.5$ bar,
respectively. Videos #1, #2, and #3 in the supplementary material

.

A series of experiments was conducted using spheres with diameters $2$, $8$,
and $15$ mm; silicone oils with viscosities $\mu_{L}=0.87$, $1.74$, and $2.61$
cP; and entry speeds $V$ from 1 to 3.5 ms-1. The oil density varies little
with viscosity, being $\rho=0.9$, $0.87$, and $0.9$ g cm-3, respectively, and
the surface tensions are measured to be in the range $\sigma$ between $15$ and
$20$ mN/m. Three different gases are used for the surrounding atmosphere
(helium, air and argon) which have molecular masses $M=4$, $29$, and $39$
Daltons, respectively. Since the mean free path of the gas $\lambda_{g}$
depends on the molecular mass, this allows us to separately isolate the
effects of changing the density and the mean free path of the gas. Results are
shown as points in Fig. 2 in the parameter space of the capillary number
$\text{Ca}=\mu_{\rm L}V/\sigma$ and the normalised gas density
$\rho/\rho_{\text{atm}}$. No-cavity events are plotted with blue markers and
the impacts where a cavity was formed are shown in red.

Our main qualitative result is that, in a range of parameters, the gas density
influences whether a cavity forms during an impact on the free surface of a
liquid. The molecular mass of the gas and the pressure seem to have no effect
provided the density is kept constant, as confirmed by Fig. 3 where the mean
free path is plotted at the horizontal axis, which is in stark contrast to
experiments on droplet splashing [23]. We note that the speed plateaus in a
region of large gas densities exceeding 0.6 atmospheric density. The short
range of densities over which the threshold speed varies suggests that the
curve should instead be considered as a transition between two regimes: one
where the role of the ambient gas is important and one where the influence of
the gas is negligible. In all cases, the transition between cavity and no-
cavity regimes occurs in a threshold manner and is prone to perturbations in
experimental conditions such as rotations of the falling sphere. This is
confirmed by occasional observations of no cavity cases at unusually high
entry speed such at the highest no-cavity point in Fig. 2. At gas pressures
below 0.05 bar, when the dynamic pressure of the liquid becomes close to the
gas pressure, the oil boils upon impact and hence this range is not
considered.

Figure 2: Diagram of cavity/no cavity regimes at varying gas density ratio
$\rho/\rho_{\text{atm}}$ and capillary number Ca. Three parameters are varied
as indicated in the legend: sphere diameter, molecular mass of the gas, and
dynamic viscosity of the liquid. Cavity/no cavity pairs with the same sphere
size, oil viscosity, and gas type are connected by a line to indicate
uncertainty of the cavity/no cavity threshold (red, blue, and green lines are
used for the 15, 8, and 2 mm balls, respectively). Black diamonds indicate
regimes shown in Fig. 1. The red triangle and the black dashed line correspond
to the experimental data and the theoretical prediction from [8]. Figure 3:
Diagram of cavity/no cavity regimes similar to that in Fig. 2 with the inverse
mean free path $\ell_{\text{atm}}/\ell$ at the horizontal axis. Note that,
although a trend related to the change in gas density is present, a number of
experiments with He prevents us from plotting an unequivocal divider between
splash and no splash regimes.

##  3 Experimental Observations

Similarly to our results, the approximate independence of threshold speed
$U^{*}$ on the sphere size for wettable spheres was previously reported by
[32] over a similar range of diameters and the inversely proportional
dependence of threshold velocity on liquid viscosity was reported in [8], i.e.
for a given gas the transition is characterised by the capillary number. In
contrast, the dependence of the splashing regime on the density of the
surrounding gas observed in Fig. 2 contradicts the experimental results of
Duez et al. [8], who _verified that the diameter of the impacting sphere does
not influence the threshold, nor does the gas pressure (varied between 0.1 and
1 atm)_ , a range of pressures in which we observe an approximate doubling of
the critical speed.

From our experiments, we distinguish three main scenarios for sphere impact:

(a) Flow-detached cavity: the flow detaches from the sphere around the
equator, forming a vertically growing crown as shown in Fig. 1a;

(b) Flow-attached cavity: the flow remains attached, i.e. wetting the sphere,
forming a crown which starts closing behind the sphere but whose contact line
does not reach the axis near the ball surface due to being slowed down by the
gas (Fig. 1b);

(c) No cavity: the flow remains attached, forming a crown whose base closes at
the axis behind the ball before it is fully submerged below the unperturbed
liquid surface (Fig. 1c).

To understand the effect of varying the ambient pressure, compare snapshots of
a no-cavity event at the same gas density but with two different gases, air
and helium, shown in Fig. 4 in the sphere’s frame of reference. Observe that
the speed with which the liquid sheet rises up around the sphere is similar
i.e. molecular weight / mean free path appears to have no effect on crown
formation (although appearance of the crown and the ejection of droplets
differs). Next consider time sequences at higher air densities, one slightly
below the cavity/no-cavity threshold, and one slightly above, shown in Fig. 5.
By comparing the right-hand side of Fig. 5 with Fig. 4, observe that the speed
of the crown propagation towards the pole is lower at higher gas density. When
the gas density becomes too high (left-hand panels of Fig. 5), polar
propagation of the crown slows down far enough for the contact line to not
join at the apex and it instead goes below the liquid surface so that the
cavity forms.

Figure 4: Testing the influence of gas molecular mass (and thus mean free
path). Comparison of the impact in air at $p=140$ mbar (left panels of each
image) and in helium at $p=950$ mbar (right panels). Both cases correspond to
the same gas density of 0.14 atmospheric. The sphere has diameter 15 mm and
impact speed $V=2.06$ ms-1. Time separation between frames is 1 ms. Videos #4
and #5 in the supplementary material.

Figure 5: Testing the influence of gas density. Comparison of the impact in
air at the density of 0.525 atmospheric (left panels) and in air at the
density of 0.5 atmospheric (right panels). The sphere has diameter 15 mm and
impact speed $V=2.06$ ms-1. Time separation between frames is 1 ms. Parameters
are chosen close to the threshold so that the increase in the gas density by
5% leads to cavity formation. Videos #2 and #3 in the supplementary material.

.

We now proceed to sketch the dynamics of the splash crown shown in Fig. 11 and
explain how it is altered by the ambient gas. The influence of liquid
evaporation on droplet splashing at reduced pressure was previously ruled out
[33, 34]. Similarly, liquid evaporation only influences contact line motion
for sufficiently slow processes involving volatile liquids, so can be
discounted [35]. The fact that the threshold speed only depends on the gas
density suggests that the effect is related to the dynamic pressure of the
gas, i.e. to inertial effects. Furthermore, independence of the threshold
speed on the sphere radius suggests that the effect might relate to
deceleration of the crown sheet itself which, in turn, slows down propagation
of the contact line.

Figure 1 suggests that the difference between cavity/no-cavity appears above
the unperturbed liquid level and above the equator of the sphere. This was
confirmed by filming the experiment from beneath the free surface to measure
the growth rate of the wetted circle below the equator, similar to experiments
by [20]. As seen in Fig. 6, evolution of the wetted region of the sphere with
and without cavity formation is similar at the lower hemisphere, hence we
focus on the upper hemisphere and the stage when the crown appears. The black
dashed line indicates the free-fall trajectory which would be expected for the
displaced liquid from geometric considerations, indicating that there is no
influence on the motion of the contact line from trapped air in the pre-impact
layer between the solid and the liquid and that cavity formation is not due to
gas entrapment. We therefore eliminate the possibility that the observed shift
of the threshold entry speed is related to the gas trapped in the layer
between solid and liquid and instead argue that the difference is due to the
modification of the dynamics of the splash crown above the equator by the
surrounding gas.

Figure 6: Growth rate for wetted part of sphere against time. The dashed black
line shows the radius of the cross-section of the sphere at the level of the
liquid. Red markers correspond to the experiment where cavity was formed (air
at 1000 mbar), blue markers to the experiment with no cavity (air at 150
mbar). The impact speed is $V\approx 2.3$ ms-1 in both cases.

The advance of the root of the crown (and hence the contact line) is where the
gas plays its role, separating the cavity and no-cavity regimes. By comparison
of Fig. 5 and Fig. 4, we observe that the liquid-solid angle at the root of
the crown is close to 90∘ at low gas density and decreases at higher gas
densities. Here, one should distinguish between the advancing (apparent)
contact angle at the liquid-solid boundary and the angle which the crown sheet
makes with the surface of the sphere, which is defined by the ratio between
the speed of the crown root and the polar velocity of liquid feeding into the
crown. It follows from the equality of the arc length and the tangent length
for sufficiently small arc angles that when the speed of the crown root and
the polar velocity are the same, the angle is 90∘. If this angle falls below
$90^{\circ}$, the speed of the advancing contact line decreases. Near the
cavity/no cavity threshold, the ambient gas slows the speed of the contact
line by up to 50%, resulting in a 50% decrease in the threshold entry speed as
seen in Fig. 2.

Slowing down of the crown is shown in detail in Fig. 7, where we plot the
spatiotemporal image illustrating propagation of the crown towards the pole.
The image was created by stacking the semi-circular strips adjacent to the
sphere at sequential time instants. The dashed sinusoidal arch represents
advance of the horizontal liquid surface along the sphere and its shape is
dictated by basic geometric considerations. The root of the crown is seen to
decelerate and is either overtaken by the liquid surface or collapses at the
pole. In the first case, the cavity forms and in the latter case, the liquid
surface closes with no cavity as a result.

Figure 7: Spatiotemporal image of the curtain propagating towards the pole in
the experiments shown in Fig. 5 of the main text (left panel is air at $p=525$
mbar and right panel is air at $p=500$ mbar). The dashed line indicates
position of the unperturbed liquid surface. At lower gas density (right
panel), the crown deceleration is not sufficient to prevent it from collapsing
at the pole. At higher gas density (left panel), deceleration is higher and
the liquid surface overtakes the crown, forming a cavity.

A natural question one might ask is whether the gas flow around the sphere as
it falls towards the liquid surface has an influence on the splash crown. To
check this, in Fig. 8 we show a laser sheet visualisation with smoke as the
seed material and air at atmospheric pressure. The resulting flow
visualisation is as expected, with the presence of a wake behind the falling
sphere which then flows out of the sealing crown and forms a pair of vortices
either side, so we also rule this out as the reason for cavity formation.

We see from Fig. 2 that a cavity forms at sufficiently high impact speeds,
even when the gas density is very low. We suggest that this plateau at low gas
densities is due to the fact that in this regime the surrounding gas does not
slow the crown and that there are instead instabilities or rotational effects
such that a cavity almost always forms for sufficiently fast impacts on the
liquid surface. In Fig. 9, the case where the crown collapses at the pole
unimpeded is shown. However, the contact line propagation becomes unstable and
the flow separates at some point, creating an asymmetric cavity.

Figure 8: Oil vapour and lasersheet visualisation of the gas flow behind the
falling sphere (left) and of the vortex formed behind the crown. $V\approx
1.6$ ms-1, air at atmospheric pressure. The bottom of the image coincides with
the unperturbed liquid level. Video #8 in the supplementary videos.

Figure 9: Asymmetric splash at low gas density at higher impact speed:
Formation of cavities on one side while the most of circumference is cavity-
free. $V\approx 2.8$ ms-1, Helium, $p=138$ mbar left, side view and $p=200$
mbar right, top view. Videos #6 and #7 in the supplementary videos.

In order to provide understanding and aid the development of a model, the
thickness of the liquid crown was measured by adding blue dye to the oil. The
thickness of the liquid sheet is measured by the ratio of red channel in the
colour image to the total brightness. Levels are calibrated using a wedge-
shaped cuvette with known thickness. In Fig. 10, visible images of the splash
are shown together with the corresponding thickness maps. Taking into account
the fact that at each location we observe two layers of the crown, its
thickness can be evaluated at the level of 50-100 $\mu$m.

Figure 10: Sheet thickness measured by the blue-coloured oil: Contour levels
of the liquid sheet thickness and the corresponding optical images.
Atmospheric pressure, 13 cm drop height. Levels are plotted with the 2-fold
increments in the liquid thickness. Observe the thickness corresponding to the
major part of the curtain corresponds to the interval 80-160 micron so that,
accounting for the two parts of the curtain captured, the thickness of the
curtain is half this value, approximately 50-100 $\mu$m.

##  4 Discussion

Summarising the above experimental observations, we propose the following
sequence of events on impact:

(1) The sphere displaces liquid which, rising along the sphere, forms a
collar. Viscous effects are negligible in the formation of the bulk of the
collar as the liquid boundary layer thickness $\delta\sim\sqrt{\mu/\rho_{\rm
liq}\cdot R/V_{0}}\sim 100~{}{\rm\mu\text{m}}$, whilst the thickness of the
bulk of the collar is of the order of millimetres as seen in Fig. 4. Here, we
assume that the velocity of the liquid in the collar is of the same order of
magnitude as the impact velocity.

(2a) Due to surface tension, a rim appears at the leading edge of the collar,
similar to that observed for vertical cylinders [20]. At too high an impact
speed, the flow separates from the sphere at the equator and the cavity forms
without the crown propagating along the surface of the sphere as in Fig. 1a.

(2b) At lower velocities, the rim moves past the equator but starts to depart
the surface due to centrifugal force, and the crown forms as shown in Fig. 11.
In the absence of gas and surface curvature, the velocity of propagation of
the wetting front $V_{\rm front}$ would be defined by the balance between the
pressure created by the concave surface formed due to the sphere wettability
and the viscous stress in the vicinity of the front [11, 23]:

$\frac{\sigma}{r_{\rm front}}\sim\frac{\mu V_{\rm front}}{r_{\rm
front}}\Longrightarrow V_{\rm front}\sim\frac{\sigma}{\mu}$ (1)

which corresponds to $\text{Ca}_{\text{front}}\sim 1$ as reported in [8]. In
our case with 1.74 cP oil, $V_{\rm front}\sim 10$ ms-1 which agrees with the
speed measured from videos and also agrees with the observed threshold impact
velocity of 2-3 m/s independent of the sphere radius.

(2c) Below the transition speed, the crown crown closes at the pole before the
sphere is submerged and no cavity is observed.

Figure 11: Schematic in the frame of reference moving with the root of the
crown. In the absence of the ambient gas, propagation of the front is defined
by the balance between capillary pressure under the concave liquid surface and
viscous forces. When the ambient gas density is high enough, it slows the
crown (dashed line), which changes the shape of the curved interface and slows
down propagation of the liquid front.

To confirm the plausibility of relating the cavity formation to deceleration
of the crown by the surrounding gas, we present an order of magnitude estimate
of the stresses involved in the problem at $V_{0}=2$ $\text{ms}^{-1}$ (drop
height of 20 cm) and $D=15$ mm. The dominant term is the ambient pressure
$P_{0}>5000$ Pa. This pressure provides continuity of the liquid so that the
liquid displaced by the sphere moves up along the solid surface. The next term
is dynamic pressure of the liquid at $P_{\rm liq}\sim\rho_{\rm
liq}V_{0}^{2}/2\sim 2000$ Pa. Besides this, we have dynamic pressure of the
gas $P_{\rm dyn\,gas}$. To evaluate this term, we suppose the crown speed
relative to the gas $V_{\rm gas}$ to be given by the speed of the crown of
$2V_{0}$ estimated from the experiment in Fig. 5 and add another factor of
$V_{0}$ for the circulatory flow which forms behind the falling sphere to find
that the dynamic gas pressure at which the cavity/no-cavity transition occurs
is given by

$P_{\rm dyn\,gas}\sim\rho_{\rm gas}V_{\rm gas}^{2}/2\sim\rho_{\rm
gas}(3V_{0})^{2}/2\sim 10~{}\text{Pa}$ (2)

The radius of curvature of the advancing front $r_{\rm front}$ shown in Fig.
11 is defined by the balance of capillary force and centrifugal terms which
accelerate fluid into the crown:

$\frac{\sigma}{r_{\rm front}}\sim r_{\rm front}\rho_{\rm
liq}\frac{V_{0}^{2}}{R}\Longrightarrow r_{\rm front}\sim\sqrt{\frac{\sigma
R}{\rho_{\rm liq}V_{0}^{2}}}.$ (3)

Reassuringly, for sphere diameters from 2 to 15 mm, this estimate for $r_{\rm
front}$ varies from 70 to 200 micron, which does not contradict experimental
observations. The corresponding capillary pressure varies from 150 to 400 Pa.
The thickness of the crown $h_{\rm crown}$ is measured experimentally (Fig.
10), resulting in a radius-independent value of 50-100 micron for the 16 mm, 8
mm, and 4 mm spheres.

The deceleration $a_{\rm stop}$ required for the crown moving at the speed of
$2V_{0}$ to completely stop at the pole, passing the distance of approximately
$R$, is

$a_{\rm stop}\sim\frac{2V_{0}^{2}}{R}.$ (4)

The deceleration of the crown due to the dynamic pressure of the gas is

$a_{\rm crown}\sim\frac{P_{\rm dyn\,gas}}{\rho_{\rm liq}\,h_{\rm crown}}.$ (5)

The exact shape of the liquid surface adjacent to the advancing wetting front
depends on the angle which the root of the crown makes with the surface of the
sphere and affects the speed of propagation of the wetting front. A sharper
angle increases the curvature, increasing the capillary term in (1) and hence
the speed of the front, whereas angles above 90∘ lower the curvature and the
speed of the front. In this way, by slowing the crown, the ambient gas slows
the wetting front (1), preventing it from reaching the pole before the falling
sphere submerges below the surface. Combining (2)-(4), we have

$\frac{a_{\rm crown}}{a_{\rm stop}}\sim\frac{\rho_{\rm gas}V_{\rm
gas}^{2}}{2\rho_{\rm liq}h_{\rm crown}}\frac{R}{2V_{0}^{2}}\sim
2\,\frac{\rho_{\rm gas}}{\rho_{\rm liq}}\frac{R}{h_{\rm crown}}$ (6)

At the set of parameters when the cavity/no-cavity transition occurs (Fig. 5),
$a_{\rm crown}/a_{\rm stop}$ varies between $1$ and $2$, depending on the
sphere radius $R$. This supports our hypothesis that the presence of the
ambient gas shifts the threshold speed for cavity formation, manifesting in
the lowering of the threshold $\rm Ca$ between 0.2 and 0.6 atmospheric density
in Fig. 2. We emphasise that (6) is an order of magnitude estimate valid
within the considered range of parameters and not a scaling.

##  5 Conclusions

We have investigated the transition between regimes in which a cavity forms or
fails to form during sphere impacts on a free liquid surface. We have found
that the threshold speed for cavity formation is significantly influenced by
the ambient gas, where the gas density is the defining parameter and the mean
free path in the gas has negligible influence. The diagram of cavity/no-cavity
regimes (Fig. 2) shows two limiting cases: one at low gas density (below 0.2
atmospheric air density) and one at high gas density (above 0.6 atmospheric).
At low gas densities, cavity/no cavity transition occurs due to instability in
the contact line propagation and appears not to be affected by the surrounding
gas. As the gas density increases, the dynamic pressure of the gas slows the
propagation of the crown towards the pole of the sphere, reducing the
threshold speed.

In this work, we have focussed on smooth oleophilic spheres, but it would
certainly be an interesting avenue for further work to study the influence
which surface wettability/roughness can have on the effect which we have
discovered [36]. As explained in Section 1, the extension of the results on
the influence of the surrounding gas for droplet splashing on a flat smooth
surface in [23] to droplet splashes on rough and textured surfaces is somewhat
involved, so we would expect the extension of our work to rough and textured
spheres to be similarly difficult. Splashes on smooth surfaces experience a
type of splashing which is caused by the presence of the atmospheric gas,
whereas splashes on rough or textured surfaces, with the different dynamics of
contact line propagation, the effect of the ambient gas may differ. The effect
of corona deceleration can be completely isolated by pulling a high vacuum
[24].

Studies of the influence of gas density on the sphere impact problem can be
developed in many directions. A recent study has found that the gas density is
also the most relevant quantity influencing the ejection of the lamella during
droplet splashing at high Re and We, so we anticipate that our result may
stimulate some models which attempt to unify the splashing mechanism which is
found in the droplet splash and sphere impact problems in these regimes [29,
30]. This is, however, a complicated problem and there are many other factors
which affect the sphere splashing mechanism (including instabilities of the
flow and contact line) which will need to be studied further in experiments.
Further experimental work can also be conducted to investigate the exact shape
of the liquid surface at the root of the crown [37].

Our results will also complement the development of multi-scale computational
simulation of air-water entry of spheres since the existing models have
focused only on the effect of the viscosity, rather than the density, of the
gas [11]. We also anticipate that our findings will stimulate theoretical
models which modify the classical Wagner model for impacts with the presence
of ambient gas. The existing models of this type only consider pre-impact or
early-impact air-water interactions where viscous forces in the gas deform the
liquid, whereas we have demonstrated that the gas continues to play a role
long after the pre-impact stage [38]. In terms of applications, it is
interesting that the splash of a projectile which falls and hits a liquid
surface can be suppressed by altering the properties of the ambient gas above
the liquid, implying that it is possible for a projectile to slip through a
surface unobserved more easily in a rarefied atmosphere. This complements the
existing observation that sphere splashing can be suppressed by using a smooth
rather than a rough sphere [36].

## Acknowledgements

This research has been financially supported by EPSRC Grants No. EP/N016602/1,
No. EP/S029966/1, No. EP/P031684/1, and No. EP/P020887/1.

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