Eect of ambient gas on cavity formation for sphere impacts on liquids Hollis Williams1 James Sprittles2 Juan C. Padrino3 and Petr Denissenko1 1School of Engineering University of Warwick Coventry CV4 7AL UK

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

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: 1Hollis.Williams@warwick.ac.uk, P.Denissenko@warwick.ac.uk

2J.E.Sprittles@warwick.ac.uk, 3padr0006@umn.edu

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

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.

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 K´arm´an 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. consid-

ered buoyant low-density spheres, ﬁnding expressions for the pinch-oﬀ time of a cavity and the volume

arXiv:2210.15369v1 [physics.flu-dyn] 27 Oct 2022

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 splash-

ing) 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

∼KlµgVtand the suction force ∼KuρgV2

tHt, where µgis the viscosity of the gas, ρgis the density

of the gas, Vtis the initial velocity of the ejected lamella and Htis the initial height of the lamella.

Kland Kuare 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 Klis 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 Regbased on Htand Vtas the characteristic scales is ∼ 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 βfor determining splashing is a constant (in the general case, each

impact parameter has a diﬀerent dependence on β). This more detailed threshold model incorporating a

large number of diverse experimental results observed a linear dependence of the splashing threshold β

on impact speed V, surface tension σand gas density ρgand an inversely proportional relation between

βand the liquid viscosity µl. This new model exhibits only weak dependency on the radius of the droplet

R, liquid density ρl, gas viscosity µgand the mean free path of the gas λg. 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 β, 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 depres-

surised 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 for-

mation 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/cm3and static contact angle

θ≈20 −25◦which makes them oleophilic. The mean square roughness of the spheres was measured

using a Bruker Contour proﬁlometer at 0.3µm, well below the thickness of the viscous boundary layer,

reaching O(100 µm) 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.

A series of experiments was conducted using spheres with diameters 2, 8, and 15 mm; silicone oils

with viscosities µL= 0.87, 1.74, and 2.61 cP; and entry speeds Vfrom 1 to 3.5 ms−1. The oil density

varies little with viscosity, being ρ= 0.9, 0.87, and 0.9 g cm−3, respectively, and the surface tensions

are measured to be in the range σbetween 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 λgdepends 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 Ca = µLV /σ and the normalised gas

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摘要：

EectofambientgasoncavityformationforsphereimpactsonliquidsHollisWilliams1,JamesSprittles2,JuanC.Padrino3,andPetrDenissenko11SchoolofEngineering,UniversityofWarwick,CoventryCV47AL,UK2MathematicsInstitute,UniversityofWarwick,Coventry,CV47AL,UK3SchoolofEngineering,NewcastleUniversity,NewcastleuponTyne...

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Eect of ambient gas on cavity formation for sphere impacts on liquids Hollis Williams1 James Sprittles2 Juan C. Padrino3 and Petr Denissenko1 1School of Engineering University of Warwick Coventry CV4 7AL UK.pdf

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Eect of ambient gas on cavity formation for sphere impacts on liquids Hollis Williams1 James Sprittles2 Juan C. Padrino3 and Petr Denissenko1 1School of Engineering University of Warwick Coventry CV4 7AL UK

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