A numerical study of a hollow water droplet falling in air

Mounika Balla, Manoj Kumar Tripathi†, and Kirti Chandra Sahu∗

Department of Chemical Engineering, Indian Institute of Technology Hyderabad, Sangareddy 502 285, Telangana, India†Indian Institute of Science Education and Research Bhopal 462 066, Madhya Pradesh, India

We numerically study the dynamics of a hollow water droplet falling in the air under the action ofgravity. The focus of our study is to investigate the effects of the difference in radii (‘thickness’) ofthe hollow droplet, gravity and surface tension at the air-water interface on shape oscillations and thebreakup dynamics of the hollow droplet. We found that the oscillations of the inner interface (innerair bubble) are mostly periodic, while the outer interface undergoes irregular oscillations due to theinteraction with the surrounding air. Increasing the ‘thickness’ of the hollow droplet decreases theamplitude of oscillations which further decays with time for high surface tension. It is observed thatfor a fixed value of the ‘thickness’ and low surface tension, the hollow droplet undergoes transitionfrom the oscillatory regime to the dripping regime as it falls. The velocity contours are used toexplain the behaviour observed in the present study. The deformation and shape oscillations of thehollow droplet are also compared with those observed in the case of a normal droplet of equal liquidvolume falling in air.

∗ ksahu@iith.ac.in

2

I. INTRODUCTION

Fundamental understanding of the dynamics of compound droplets can be useful in the fabrication of fusion targetpellets and the development of materials processing techniques [1, 2]. The collision of compound droplets on solidsurfaces and free surfaces has also been studied by several researchers (see, for instance, Refs. [3–5]) due to itsrelevance in many engineering applications [6]. In the present work, a compound water droplet with air as its innercore is termed as a “hollow” droplet.

Hollow droplets can also be observed in natural phenomena [7]. Raindrops reach earth in a remarkable range ofshapes and sizes because of the complex interaction between droplets and atmosphere [8]. Thus, the possibility ofobserving hollow raindrops can not be discarded. The first mention of this idea goes back to the late 1800s by E. J.Lowe. In the atmosphere, a hollow raindrop may be formed in two ways. (i) From a liquid sheet in storms: smallspherical raindrops falling from the clouds coalesce to increase their sizes. A big raindrop undergoes deformation andflattens out to a ‘pancake’ like shape. As it continues to fall, its shape widens and becomes thinner, which in turnallows the surrounding air to enter inside, and thereby changing its shape to an upward-turned bag. In some situations(such as in a rainstorm), the upward-turned bag may close from the top to form a hollow raindrop. (ii) From a crystalof frozen fluid inside a cloud droplet that melts under the heat of condensation to release inert gas. This resultantinert gas serves as the core of a hollow droplet [7]. A normal liquid droplet of radius R is stable if the inside pressureexceeds the outside pressure by 2σ/R, where σ is the interfacial tension. As the inside pressure becomes very large asR → 0, water droplets should not form in the clouds from the saturated vapour. Thus, several researchers (see, forinstance, Ref. [9]) have attempted to answer this paradox and investigated why the clouds contain liquid droplets ina saturated environment. In this regard, Aston [10] showed that the idea of hollow droplets in clouds can be used toexplain this paradox. Thus, a question that arises is how different is the dynamics of a hollow droplet as comparedto a normal droplet of the same liquid volume.

The main objective of the present work is to study the shape oscillations and breakup phenomena observed in caseof a hollow water droplet falling in quiescent air. It is to be noted that in the atmosphere, the hydrodynamics ofraindrops is complex and influenced by several parameters, such as temperature, humidity gradients, air current, etc.,whose effects have been neglected in the present study. Thus, the present work can be considered as a first attemptto study the hydrodynamics of a hollow raindrop. In literature, the fluid dynamics of hollow droplets has receivedfar less attention as compared to normal droplet (a blob of heavier fluid falling in a lighter medium) and a bubble (ablob of lighter fluid rising in a heavier medium) [11–14]. A brief overview of some of the important contributions tothe dynamics of the normal droplet/bubble is provided below.

In the creeping flow limit, Hadamard [15] and Rybczynski [16] were the first to derive the expression of the terminalvelocity and the flow field inside and outside of a small bubble/drop. Since then drops/bubbles have been investigatedanalytically, numerically and experimentally by several authors focusing on breakup dynamics [17–19], path instability[11, 20–23] and shape oscillations [24–26]. In the creeping flow regime, Koh & Leal [24, 25] found that a slightlynonspherical droplet comes back to a spherical shape, whereas a largely distorted droplet continues to deform withtime. Agrawal et al. [26] extended this to the inertia dominated regime and showed that an initial nonspherical waterdrop falling in air undergoes prominent shape oscillations, whose amplitude increases with the increase in inertia andat later times the droplet shape becomes asymmetrical due to nonlinear effects. Balla et al. [27] have investigated theeffect of inertia and surface tension on the dynamics of a falling tilted liquid droplet in the air. They found that athigh aspect ratios and sufficiently large Reynolds numbers the droplet undergoes fragmentation. Recently Deka et al.[28] studied the formation and shape oscillations of a non-spherical droplet released from a nozzle. The above reviewis indeed a small section of a large volume of study conducted on normal droplets and bubbles.

The focus of the present study is to numerically investigate the dynamics of a hollow droplet, which has not beenconsidered so far to the best of our knowledge. A volume-of-fluid (VoF) based Navier-Stokes solver [29, 30] has beenused, and the effects of the ‘thickness’ of the hollow droplet, the Galilei number, Ga (i.e. the ratio of the gravitationalforce to the viscous force) and the Eotvos number, Eo (i.e. the ratio of the gravitational force to the surface tensionforce) have been investigated. The dynamics of a hollow droplet has also been compared with that of a normal dropletof equal liquid volume. We found that the inner interface of the hollow droplet (inner air bubble) undergoes periodicshape oscillations about its spherical shape with a time-period of about half of that of a normal droplet of equalvolume. On the other hand, the deformation of the outer interface of the hollow droplet is found to be irregular. Fora low value of Ga, the amplitude of these oscillations decreases with time, as observed by Koh & Leal [24, 25] in thecase of normal droplets. Increasing the ‘thickness’ of the hollow droplet decreases the amplitude of oscillations for lowvalues of Eo (high surface tension). However, for high values of Eo (small surface tension), the droplet deformationleads to a ‘spike’ like structure at the bottom when the thickness of the hollow droplet is small. The behaviour of thehollow droplet observed in the present study is explained by analysing the velocity contours. The rest of the paper isorganised as follows. The problem formulation and the numerical method used in this study are discussed in SectionII. The results are presented in Section III, and concluding remarks are given in Section IV.

3

II. FORMULATION

The hydrodynamics of a hollow liquid droplet falling in the air under the action of gravity (g) is investigatedvia three-dimensional numerical simulations of the incompressible Navier-Stokes and the continuity equations. Theschematic diagram is shown in Fig. 1. Here, a2,0 and b2,0 represent the initial inner radii along the horizontal andvertical directions, respectively; ε is the ‘thickness’ of the hollow droplet, such that the corresponding initial outerradii are a1,0 = a2,0 + ε and b1,0 = b2,0 + ε. Thus, by specifying a2,0 and ε, which are varied in the present study, andby keeping the volume of water in the hollow droplet equals to that of a normal spherical droplet of radius, Req, onecan calculate b2,0 as

b2,0 =R3

eq − ε(a2,0 + ε)2

ε2 + 2a2,0ε. (1)

In order to compare the dynamics of a hollow droplet with that of a normal droplet of equivalent radius Req, theinitial aspect ratio of the normal droplet, (a0/b0) is kept the same as that of the outer interface of hollow droplet,(a1,0/b1,0), and the constant volume constraint, i.e., 4πR3

eq/3 = 4πa02b0/3 is used. Here, a0 and b0 are the horizontal

and vertical radii of the equivalent normal droplet.

(a) (b)

x zyyi

Hollow dropa2,0

b2,0

Air

FIG. 1. (a) Schematic diagram of a hollow liquid droplet falling in air. A computational domain of size H(75Req)×W (75Req)×L(75Req) is used and the droplet is initially placed at yi = 70Req. Req is the equivalent spherical radius of a normal droplet forthe same volume of liquid in the hollow droplet. (b) The zoomed view of the hollow droplet; ε, a2,0 and b2,0 are the ‘thickness’,the initial inner radii along the horizontal and vertical directions, respectively.

A Cartesian coordinate system (x, y, z) is used to model the flow dynamics, such that gravity (g) acts in thenegative y-direction as shown in Fig. 1. Initially, both the droplet and surrounding air are stationary. To minimisethe boundary effect, a sufficiently big computational domain of size 75Req×75Req×75Req is considered in the presentstudy. Free-slip and no-penetration conditions are imposed at all the boundaries of the computational domain.

The governing equations are given by

∇ · u = 0, (2)

ρ

[∂u

∂t+ u · ∇u

]= −∇p+∇ ·

[µ(∇u +∇uT )

]+ σδ(x− xf )κn− ρgj. (3)

Here, u = (u, v, w) denotes the velocity field, where u, v and w represent the velocity components in the x, y and zdirections, respectively; p is the pressure field; t denotes time; j denotes the unit vector along the vertical direction;δ(x− xf ) is the delta distribution function (denoted by δ hereafter) whose value is zero everywhere except at theinterface, where x = xf ; κ = ∇ ·n is the interfacial curvature; n is the outward-pointing unit normal to the interface.

The interface separating the air and liquid phases is tracked by solving an advection equation for the volume fractionof the liquid phase, c (such that, c = 0 and 1 for the air and liquid phases, respectively). This is given by

∂c

∂t+ u · ∇c = 0. (4)

4

The density, ρ and the viscosity, µ are assumed to depend on c as

ρ = (1− c)ρa + cρl, (5)

µ = (1− c)µa + cµl, (6)

where ρl, µl and ρa, µa are the density and dynamic viscosity of water and air, respectively. At time t > 0, a1, a2,and b1, b2 represent the outer and the inner radii of the hollow droplet in the horizontal and the vertical directions,respectively. Similarly, a and b are the instantaneous radii of a normal droplet in horizontal and vertical directions,respectively.

The following scaling is used to non-dimensionalise the above governing equations:

(a1, a2, b1, b2, a, b, ε) = Req

(a1, a2, b1, b2, a, b, ε

), (x, y, z) = Req (x, y, z) , t = Req/V t, u = V u, p = ρaV

2p,

µ = µaµ, ρ = ρaρ, δ = δ/Req, (7)

where V (≡√gReq) is the velocity scale and the tildes designate dimensionless quantities. After dropping tildes from

all the nondimensional variables, the governing dimensionless equations are given by

∇ · u = 0, (8)

∂u

∂t+ u · ∇u = −∇p+

1

Ga∇ ·[µ(∇u +∇uT )

]+ δ∇ · nEo

n− ρj,

where the dimensionless density and dynamic viscosity are given by

ρ = (1− c) + cρr, (9)

µ = (1− c) + cµr. (10)

The various dimensionless numbers appearing in Eqs. (8)-(10) are the Galilei number, (Ga(≡ ρag1/2Req

3/2/µa)),

the Eotvos number, (Eo(≡ ρagReq2/σ)), the density ratio (ρr(≡ ρl/ρa)) and the viscosity ratio (µr(≡ µl/µa)). The

Galilei and Eotvos numbers characterised the relative importance of the gravitational force over the viscous force andthe surface tension force, respectively. As we investigate a hollow water droplet falling in the air, in the present study,the values of ρr and µr are fixed at 998 and 55, respectively.

(a) (b)

t = 0.0

3.4

FIG. 2. Typical grid used in the present study. (a) A hollow droplet with ε = 0.38 and (b) an equivalent normal droplet att = 0 and t = 3.4. The view is along the x-axis. The smallest grid size is ∆ = 0.018. The other dimensionless parameters area1,0/b1,0 = a0/b0 = 1.175, Re = 10 and Eo = 0.005.

A volume-of-fluid (VoF) based Navier-Stokes solver [29, 30] has been used in the present study. A height-functionbased balanced-force, continuum-surface-force formulation has been incorporated for the inclusion of the surface force

5

term in the Navier-Stokes equations [31]. The numerical method used in the present study is similar to the one usedby Tripathi et al. [27]. This solver has been validated extensively in our previous studies [32, 33]. Thus to avoidrepetition, we refer the reader to Refs. [11, 26, 27] for the detailed description of the numerical method and validationof the present solver. However, we have conducted a grid convergence test to ensure the accuracy of the numericalmethod used in the present study.

In Figs. 2(a) and (b), a hollow droplet with ε = 0.38 and a2 = 0.8 and an equivalent normal droplet, along with thecomputational grid used, are shown at t = 0 and 3.4. The rest of the parameters are Re = 10 and Eo = 0.005. In thepresent study, the grid refinement has been performed by dividing the computational domain into three regions: (i)the inner region which is a spherical region of dimensionless radius 2, where the grid size is the smallest (∆ = 0.018),(ii) the intermediate region that consists of a rectangular parallelepiped of dimensionless height equals to 7, and widthand breadth equal to 4, where the grid size is 0.07 and (iii) the outer region, where an adaptive grid refinement isimplemented that provides an intermediate grid size (∆ = 0.07) on regions with velocity gradient. In the interfacebetween these regions, there is a gradual change in the grid size. Note that the grid set-up of the inner and theintermediate regions moves with the drop, while in the outer region, the grid remains fixed. A grid convergence testhas been conducted by performing the simulations using three different grids having the smallest grid size ∆ = 0.036,0.018 and 0.009, and it is found that the results obtained using different grids are indistinguishable. In view of thisgrid convergence test, ∆ = 0.018 is used to generate the rest of the results presented in this study.

III. RESULTS AND DISCUSSION

We begin the presentation of our results by investigating the effect of varying the ‘thickness’ of the hollow droplet,ε for Ga = 10 and Eo = 0.005. The temporal variations of the aspect ratios of the hollow droplet, Ari = ai/biwith ε = 0.3 and a2,0 = 0.8 are presented in Fig. 3(a), where i = 1 and 2 represent the outer and inner radii,respectively. The corresponding temporal variation of the aspect ratio (a/b) of an equivalent normal droplet is shownby the dashed line in Fig. 3(a). It can be seen that at t = 0, the values of Ar1 of the hollow droplet and (a/b) ofthe normal droplet are equal (≈ 0.776) and thus they are initially prolate. It can be seen in Fig. 3(a) that both theinner and the outer radii of the hollow droplet undergo prolate-spherical-oblate shape oscillations, like in the case of anormal droplet. However, the time-periods of oscillations of the inner and outer radii of the hollow droplet are muchsmaller (about half) as compared to the time-period of the equivalent normal droplet. Close inspection of Fig. 3(a)also reveals that the inner interface of the hollow droplet exhibits symmetrical oscillations about a spherical shape(Ari = 1). On the other hand, the outer interface of the hollow droplet is slightly aperiodic. It can be seen in Fig.3(a) that the amplitudes of the oscillations of the inner and outer interfaces of the hollow droplet decrease rapidlywith time. A similar dynamics was observed in the case of normal droplets by Koh & Leal [24, 25] in the creepingflow regime. In contrast, the amplitude and the time-period of shape oscillations of the equivalent normal droplet arealmost constant for this set of parameters considered, and the time-period is found to be the same as the theoreticallyobtained time-period under the inviscid approximation [34]:

Tp = 2π/

√n(n− 1)(n+ 2)

1

ρrEo≈ 4.96, (11)

with n = 2 (the fundamental mode).The temporal evolutions of the shape of the hollow droplet with ε = 0.3 and a2,0 = 0.8, and the equivalent normal

droplet are depicted in Fig. 3(b). It can be seen that both are prolate at t = 0, as they start to fall. For t > 0 both thedroplets undergo prolate-spherical-oblate shape oscillations. For the hollow droplet, it can be seen that the ‘thickness’is constant (ε = 0.3) at t = 0, but as time progresses the ‘thickness’ becomes non-uniform. At t = 3.9 and 5.1, it canbe seen that the ‘thickness’ at the top and bottom parts of the hollow droplet is smaller than that in its equatorialplane. Close inspection of Fig. 3(a) indicates that the inner and the outer interfaces of the hollow droplet becomeout of phase, as can be seen during t = 4.4− 5.8. For example at t = 4.7, the outer interface is oblate while the innerinterface is prolate. At later times, as the amplitude of shape oscillations decreases (see Fig. 3(a)), the gravity startsto dominate by draining the liquid in the downward direction. Due to this, a ‘spike’ like structure is clearly visible atthe bottom part of the hollow droplet at t = 8.0. At this stage, liquid from the top part moves to the bottom of thehollow droplet. Thus, the ‘thickness’ of the hollow droplet becomes even smaller at the top part and becomes biggerat the bottom part of the hollow droplet. The surface tension can not hold this accumulated liquid at the bottom ofthe hollow droplet. Thus, the ‘spike’ elongates until it gets detached from the main droplet as a satellite droplet. Thebreakup can be due to the Rayleigh-Plateau type instability. After this again the hollow drop may undergo shapeoscillations or break up if the inner and outer interfaces of the hollow droplet rupture. However, our solver is notfacilitated with the breakup model to capture this phenomenon. This droplet behaviour observed in the case of thehollow droplet at the later times is not predominant in the case of the normal droplet as the shape oscillations occur

6

(a)

0 2 4 6 8

t

0.6

0.8

1

1.2

1.4

1.6

Ari

outer (i = 1)

inner (i = 2)

normal drop

(b)

t = 0

3.9

5.1

6.4

8.0

70.00

yCG

62.45

57.09

49.70

38.29

70.00

yCG

62.40

57.06

49.64

38.23

FIG. 3. (a) Temporal variations of the inner and outer aspect ratios of a hollow droplet, Ari = ai/bi with ε = 0.3 and a2,0 = 0.8;i = 1 (outer interface) and 2 (inner interface). The result corresponds to the equivalent normal droplet is shown by dashedline. (b) The shapes of the hollow and the normal droplets at different times. The view is along the x-axis. The rest of theparameters are Ga = 10 and Eo = 0.005.

at a constant amplitude (see Fig. 3(b)). As the volume of the hollow and normal droplets is kept constant, we foundthat they fall almost at the same velocity, which is evident from the same values of yCG (approximately) of the hollowand the equivalent normal droplets at each time as mentioned in Fig. 3(b).

The temporal variations Ari of the hollow droplet with ε = 0.38 and a2,0 = 0.8 are presented in Fig. 4(a). Theresult of an equivalent normal droplet is shown by the dashed line in Fig. 4(a). As we have fixed the value of a2,0 andvolume of the hollow droplet, and increased its thickness (ε), the droplet becomes oblate at t = 0. In this case, att = 0, the value of Ar1 of the hollow droplet and (a/b) of the normal droplet is ≈ 1.176. It can be seen that even foroblate hollow droplet, the inner aspect ratio undergoes symmetrical oscillations about Ar,i = 1, but the deformation ofthe outer aspect ratio is complex having a double periodic type behaviour. In this case too (ε = 0.38), the time-periodof the oscillations of the inner radius is about half of that of the equivalent normal droplet (≈ 4.95). However, acomparison of results presented in Fig. 4(a) with those in Fig. 3(a) reveals that the amplitude of oscillations of thehollow and normal droplets with ε = 0.38 is much smaller than that with ε = 0.3.

The temporal evolutions of the shape of the hollow droplet with ε = 0.38 and a2,0 = 0.8, and the equivalent normaldroplet are depicted in Fig. 4(b). As the amplitude of oscillations is less than that observed in the case of a hollowdroplet with ε = 0.3, unlike in Fig. 3(b), it can be seen that the hollow droplet undergoes a stable shape deformation(constant thickness) at early times (till t = 4.6), after which at t ≥ 5.7 in Fig. 4(b), we see some non-uniformityin the thickness of the hollow droplet. The ‘spike’ like structure, as seen in the hollow drop with ε = 0.3 presentedin Fig. 3(b), does not appear for ε = 0.38. The equivalent normal droplet undergoes oblate-spherical-oblate shapeoscillations, but with some surface instabilities as seen in Fig. 4(b). We have not fully understood these instabilitiesyet and this needs further investigation which is a part of our future study. However, to understand the physicalmechanism of the shape oscillations, we have plotted the contours of u in Figs. 6 (a) and (b) for ε = 0.3 and ε = 0.38,respectively.

7

(a)

0 2 4 6 8

t

0.6

0.8

1

1.2

1.4

1.6

Ari

outer (i = 1)

inner (i = 2)

normal drop

(b)

t = 0

3.4

4.6

5.7

8.0

70.00

64.25

59.47

53.89

38.25

70.00

64.23

59.45

53.85

38.21

yCG yCG

FIG. 4. (a) Temporal variations of Ari for a hollow droplet with ε = 0.38 and a2,0 = 0.8; i = 1 (outer interface) and 2 (innerinterface). The result of the equivalent normal droplet is shown by dashed line. (b) The shapes of the hollow and the normaldroplets at different times. The view is along the x-axis. The rest of the parameters are Ga = 10 and Eo = 0.005.

The influence of the ‘thickness’ of the hollow droplet is summarised in Fig. 5, where the temporal variations of theaspect ratio of the inner bubble, Ar2 are shown for different values of ε until the interface is ruptured. The volumeof liquid for all values of ε is kept constant. ε = 0.348 represents a hollow drop consisting of two perfectly sphericalinner and outer interfaces. For ε < 0.348 and ε > 0.348, we get initially prolate (see Fig. 3) and initially oblate (seeFig. 4) hollow droplets. It can be seen in Fig. 5 that for ε = 0.348 the hollow droplet remains mostly spherical andfragments as the outer interface deforms and touches the inner interface of the hollow droplet. The amplitude andtime-period of oscillations (before rupture) of the inner interface are minimum at ε = 0.348. Decreasing ε below andabove ε = 0.348 increases amplitude and time-period of oscillations of the inner interface.

To understand the velocity field around the drop, the contours of u velocity for hollow and normal droplets areshown in Fig. 6(a) and (b) for ε = 0.3 and 0.38, respectively. In Fig. 6(a), it can be seen that the hollow droplet (withε = 0.3) and the equivalent normal droplet develop negative and positive u velocity fields around them indicating thatthe droplets are being stretched in the radial direction leading to the deformation. Similar contours are also observedfor the w component of the velocity vector (not shown). It is observed that the intensity of u velocity field for thehollow droplet is higher than that of the equivalent normal droplet. Close inspection also reveals that the region withthe velocity gradient increases as the time progresses for both the hollow and normal droplets. At early times (saytill t = 5.1), the regions with velocity gradient look symmetrical. However, at later times (t = 6.4 and 8 for this setof parameters), it can be seen that these regions become slightly asymmetrical and two more vortices appear at thetop interface of the droplets. This is a sign of vortex shedding, which is responsible for the nonlinear behaviour atlater times. A similar phenomenon was observed by Agrawal et al. [26] in the case of a normal nonspherical droplet.

Fig. 6(b) shows the u velocity field for the hollow droplet with ε = 0.38 and the equivalent normal droplet. Thevelocity fields for ε = 0.38 look qualitatively similar to those observed for ε = 0.3. However, with a closer inspection,we can find that, unlike ε = 0.3, the intensity of the velocity fields of the hollow and normal droplets becomes almostthe same. This is expected, as increasing the value of ε decreases the amplitude of oscillations (see Fig. 4(a)). Anotherinteresting dynamics is seen at t = 4.6 and 5.7, where asymmetry in the u velocity contours are observed in the wake

8

0 3 6 9 12 15

t

0.5

1

1.5

2

2.5

Ar2

0.250.30.3480.380.4

ε

break-up

break-up

break-up

FIG. 5. Temporal variations of Ar2 of a hollow droplet with a2,0 = 0.8 for different values of ε. The breakup time for eachvalue of ε is marked. The rest of the parameters are Ga = 10 and Eo = 0.005.

region of the normal droplet. This is due to the appearance of the surface instability in the case of the equivalentnormal droplet at t = 4.6 (Fig. 6(b)).

(a) (b)

t = 3.9

5.1

8.0

6.4

-0.5 0.0 0.5

t = 3.4

4.6

5.7

8.0

FIG. 6. The contours of u for the hollow droplet (the left side of each panel) and the equivalent normal droplet (the right sideof each panel). (a) ε = 0.3 and (b) ε = 0.38. The rest of the parameters are Ga = 10 and Eo = 0.005.

Next, we investigate the effect of Eo on the deformation of a hollow droplet. Figs. 7(a) and (b) present the temporalvariations of Ari of a hollow droplet with ε = 0.38 and a2,0 = 0.8 for Eo = 0.004 and 0.01, respectively. The value ofGa taken is 10. For a fixed value of Ga, increasing the value of Eo signifies a decrease in the surface tension. It can beseen from Eq. (11) that increasing Eo increases the time-period of oscillations of a normal droplet. The values of the

9

time-period of oscillations for Eo = 0.004 and 0.01 obtained using Eq. (11) are 4.44 and 7.02, respectively. It can beseen in Figs. 7(a) and (b) that for Eo = 0.004 and 0.01, the inner interface of the hollow droplet undergoes periodicoblate-prolate-oblate oscillations. The time-periods of the oscillations of the inner interface of the hollow droplet forEo = 0.004 and 0.01 are 2.05 and 3.3, respectively, which are about half of the corresponding theoretically obtainedtime-periods for a normal droplet. The outer interface, on the other hand, does not exhibit any periodic oscillations.The shapes of the hollow droplet for Eo = 0.004 and 0.01 are directed in Fig. 8. It can be seen that for higher valueof Eo (Eo = 0.01) the surface wave becomes prominent at the inner interface and a ‘spike’ like structure emergesfrom the bottom of the hollow droplet.

(a)

0 2 4 6 8

t

0.6

0.8

1

1.2

1.4

Ari

outer (i = 1)

inner (i = 2)

(b)

0 2 4 6 8

t

0.6

0.8

1

1.2

1.4

Ari

outer (i = 1)

inner (i = 2)

FIG. 7. Temporal variations of Ari for ε = 0.38 and Ga = 10. (a) Eo = 0.004 and (b) Eo = 0.01.

(a) (b)Eo = 0.004 0.01

t = 4.0

5.0

6.0

7.0

t = 3.25

4.85

6.35

8.0

FIG. 8. The shapes of the hollow droplet at different times for (a) Eo = 0.004 and (b) Eo = 0.01. The rest of the parametersare the same as those in Fig. 7. The view is along the x-axis.

Finally, the effect of the Galilei number, Ga on the inner and outer aspect ratios, Ari of the hollow droplet withε = 0.38 and Eo = 0.01 is presented in Figs. 9(a) and (b) for Ga = 100 and 500, respectively. The dynamics can becompared with the results for Ga = 10 in Fig. 3(a). It can be seen that for a higher values of Ga (Ga ≥ 100), the

10

inner interface undergoes oscillations, while the breakup occurs at the outer interface at early time. Figs. 10(a) and(b) present the shapes of the hollow droplet at different times for Ga = 100 and 500, respectively. It can be seen thatthe deformation becomes very complex at high values of Ga.

(a)

0 2 4 6 8

t

0.6

0.8

1

1.2

1.4

Ari

outer (i = 1)

inner (i = 2)

(b)

0 2 4 6 8

t

0.6

0.8

1

1.2

1.4

Ari

outer (i = 1)

inner (i = 2)

FIG. 9. The temporal variations of Ari of a hollow droplet with ε = 0.38 and Eo = 0.01: (a) Ga = 100 and (b) Ga = 500.

(a) (b)

t = 3

5

7

Ga = 100 500

FIG. 10. The shapes of the hollow droplet at different times for (a) Ga = 100 and (b) Ga = 500. The rest of the parametersare the same as those in Fig. 9. The view is along the x-axis.

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IV. CONCLUDING REMARKS

The hydrodynamics of a hollow water droplet falling in the air is investigated via three-dimensional numericalsimulations. An open-source finite volume Navier-Stokes solver [29, 30] based volume-of-fluid method is used to trackthe interface. A grid convergence test is conducted to ensure the accuracy of the numerical method used. The focusof the present study is to investigate the effects of ‘thickness’ of the hollow droplet, the Galilei number (Ga), theEotvos number (Eo) on the dynamics of the hollow droplet. The dynamics of the hollow droplet is also comparedagainst that observed in the case of an equivalent normal droplet falling in air. It is found that the droplet undergoesoblate-prolate oscillations in its downward path. The oscillations of the inner interface of the hollow droplet are foundto be periodic with a time-period of about half of the time-period of an equivalent normal droplet. On the other hand,the deformation of the outer interface of the hollow droplet is irregular. The amplitude of oscillations of the inner andouter interfaces of the hollow droplet decreases with time for low values of the Galilei number. Our study for differentvalues of Eo reveals that for low Eo (high surface tension), increasing the ‘thickness’ of the hollow droplet decreasesthe amplitude of oscillations, but for high Eo (low surface tension), the deformation leads to a ‘spike’ like structureat the bottom when the thickness of the hollow droplet is small. Thus, we can conclude that with increasing Eo, thehollow droplet undergoes a transition from the oscillatory regime to the dripping regime as it falls. The velocity fieldis also analysed to study the deformation dynamics observed in the present work.

ACKNOWLEDGMENT

KS thanks Science and Engineering Research Board (SERB), India for providing financial support through thegrant number, MTR/2017/000029.

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