Bubble pinch-off and scaling during liquid drop impact on liquid poolBahni Ray, Gautam Biswas, and Ashutosh Sharma Citation: Phys. Fluids 24, 082108 (2012); doi: 10.1063/1.4746793 View online: http://dx.doi.org/10.1063/1.4746793 View Table of Contents: http://pof.aip.org/resource/1/PHFLE6/v24/i8 Published by the American Institute of Physics. Related ArticlesEffects of streamwise rotation on the dynamics of a droplet Phys. Fluids 24, 082107 (2012) On self-similarity in the drop-filament corner region formed during pinch-off of viscoelastic fluid threads Phys. Fluids 24, 083101 (2012) Numerical study on the effects of non-dimensional parameters on drop-on-demand droplet formation dynamicsand printability range in the up-scaled model Phys. Fluids 24, 082103 (2012) Photothermal formation and targeted positioning of bubbles by a fiber taper Appl. Phys. Lett. 101, 054103 (2012) Monodisperse water microdroplets generated by electrohydrodynamic atomization in the simple-jet mode Appl. Phys. Lett. 100, 244105 (2012) Additional information on Phys. FluidsJournal Homepage: http://pof.aip.org/ Journal Information: http://pof.aip.org/about/about_the_journal Top downloads: http://pof.aip.org/features/most_downloaded Information for Authors: http://pof.aip.org/authors

PHYSICS OF FLUIDS 24, 082108 (2012)

Bubble pinch-off and scaling during liquid drop impacton liquid pool

Bahni Ray,1 Gautam Biswas,1,2 and Ashutosh Sharma3

1Department of Mechanical Engineering, Indian Institute of Technology,Kanpur, Uttar Pradesh 208016, India2Central Mechanical Engineering Research Institute (CSIR), Durgapur,West Bengal 713209, India3Department of Chemical Engineering, Indian Institute of Technology,Kanpur, Uttar Pradesh 208016, India

(Received 31 October 2011; accepted 19 July 2012; published online 23 August 2012)

Simulations are performed to show entrapment of air bubble accompanied by highspeed upward and downward water jets when a water drop impacts a pool of water sur-face. A new bubble entrapment zone characterised by small bubble pinch-off and longthick jet is found. Depending on the bubble and jet behaviour, the bubble entrapmentzone is subdivided into three sub-regimes. The entrapped bubble size and jet heightdepends on the crater shape and its maximum depth. During the bubble formation,bubble neck develops an almost singular shape as it pinches off. The final pinch-offshape and the power law governing the pinching, rneck ∝ A(t0 − t)αvaries with theWeber number. Weber dependence of the function describing the radius of the bubbleduring the pinch-off only affects the coefficient A and not the power exponent α.C© 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4746793]

I. INTRODUCTION

A drop or bubble breaks up by forming a neck that thins to microscopic dimensions andapproaches singularity both in space and time where pressure grows infinitely large. For the pinch-off of droplet, the dynamics close to pinch-off exhibit self-similar behaviour. The formation andpinch-off of a low-viscosity liquid droplet in air is described by a balance between surface tensionand inertia, resulting in a 2/3 scaling exponent.1 The inverse problem of bubble collapse has beenextensively studied in the case of pinch-off of a bubble rising from a needle or nozzle,2–4 the breakupof gas bubble in straining flow,5, 6 bubble entrapment when a disk quickly pulled through watersurface,7, 8 or bubble entrainment due to Faraday excitation.9 If the dynamics near pinching is solelygoverned by inertia,2–4 then the neck radius rneck is expressed in the time remaining until collapse(t0 − t), where t0 is the time instant of pinching, as r ∝ (t0 − t)1/2, or with a logarithmic correctionas shown by Gordillo et al.5, 6 as rneck(− log rneck)1/4 ∝ (t0 − t)1/2. The collapse may be slowed downby viscosity3 as rneck ∝ (t0 − t) or by surface tension10 as rneck ∝ (t0 − t)2/3or it may be acceleratedby the inertia of the gas flowing inside the neck5 as rneck ∝ (t0 − t)1/3.

Another example of bubble pinch-off phenomena is air-bubble entrapment due to liquid dropimpact on a liquid pool where the process of pinching occurs very fast. Bubble entrapment zone iswithin the transition regime between coalescence and splashing phenomena on drop impact onto aliquid pool.11–17 Pumphrey and Elmore15 showed that bubble entrapment had a lower and an upperboundary on the Weber number-Froude number plane which Oguz and Prosperetti12 determined asWe = 41.3 Fr0.179 and We = 48.3 Fr0.247, respectively.

We solved the Navier-Stokes equations in axisymmetric coordinates along with the surfacetension force as the body force in an incompressible flow. The coupled level-set and volume-of-fluid(CLSVOF) method has been used to track the time-evolution of the interface deformation. TheCLSVOF method was discussed in detail by Sussman and Puckett.18 In the CLSVOF method the LSfunction19 is used only to compute the geometric properties at the interface, while the void fraction

1070-6631/2012/24(8)/082108/11/$30.00 C©2012 American Institute of Physics24, 082108-1

082108-2 Ray, Biswas, and Sharma Phys. Fluids 24, 082108 (2012)

TABLE I. Properties of the reference fluid: air-water system at 20 ◦C.

Property Value

ρw 998.12 kg/m3

ρa 1.188 kg/m3

μw 1.002× 10−3 Pa sμa 1.824× 10−5 Pa sσ 72.8× 10−3 N/m

is advected using the VOF approach.20 The method has been modified by Tomar et al.21 to simulatebubble growth in film boiling and Chakraborty et al.22, 23 used it to simulate the dynamics of airbubble from submerged orifice. In our earlier paper (Ray et al.24) we used a similar method to capturethe complete and partial coalescence phenomena when a liquid drop impacts on a liquid-liquidinterface.

In the present work we extended our earlier study24 for water drop impact on air-water interfaceto investigate the bubble entrapment phenomena. The purpose here is to study the time evolution ofthe bubble pinch-off followed by jet ejection for different Weber numbers at high Froude numberand Reynolds number. An attempt is made to describe the necking stage of the bubble entrapped bynumerical simulation. The results are compared with various experimental and theoretical results inliterature within the limits of our numerical mesh, i.e., times to pinch-off of about 1 μs and spatialresolutions of 100 μm/grid.

II. FORMULATION

A. Computational domain

Complete numerical simulation of the processes of bubble entrapment is performed for a two-dimensional incompressible flow which is described in axisymmetric coordinates (r, z) on a domainwith dimensions 7D × 14D and pool depth 7.2D. The drop is placed at 0.02D from the flat interface,where D is the initial drop diameter. The pool surface is assumed to be flat and motionless and thedrop is assumed to be spherical and travelling at the impact velocity, U. We verified carefully thatour results are independent of mesh size and time stepping. A grid mesh of 350 × 700 is taken.With low spatial resolution of 200 μm/grid (175 × 350) the bubble pinch-off is not observed andfor higher resolution of 50 μm/grid (700 × 1400), the crater depth, bubble size, and jet height areinvariant with respect to the spatial resolution of 100 μm/grid (350 × 700). We chose the timestep as �t = 10−6s for all the simulations satisfying �t ≤ min (�tconv,�tca), where convective

time step, �tconv ≤(

|u|max�r + |v|max

�z

)−1and capillary time step, �tca ≤ 1

2

√(ρw+ρa )(�r )3

πσ. The geomet-

rical dimensions are nondimensionalized by drop diameter D and time is nondimensionalized bytc = D/U, where U is the impact velocity of the drop. The dimensionless numbers deployed todescribe the phenomena are: Weber number We = ρwU 2 D/σ , Froude number Fr = U2/(gD), andReynolds number Re = ρwU D/μw, where ρw is the density of water, μw is the viscosity of water,σ is surface tension, and g is the acceleration due to gravity. The properties of air and water are keptconstant as given in Table I.

B. Numerical approach

Using the level-set formulation due to Chang et al.,25 the mass and momentum equation forincompressible two-phase flow can be written as

∇ · �V = 0, (1)

ρ

[∂ �V∂t

+ ∇ ·(

�V �V)]

= −∇ P + ρ1

Fr+ 1

Re∇ ·

[μ

(∇ ⇀

V +(∇ �V

)T)]

+ 1

Weκ (φ) ∇H (φ) .

(2)

082108-3 Ray, Biswas, and Sharma Phys. Fluids 24, 082108 (2012)

The level-set function φ is maintained as the signed distance from the interface close to the interface.Hence, near the interface,

φ (�r , t)

⎧⎪⎨⎪⎩

< −d in the air cells

= 0 at the inter f ace

> +d in the water cells, (3)

where d = d(�r ) is the shortest distance of the interface from point �r . The unit normal vector �n andthe mean curvature κ of the interface are simply

�n = ∇φ

|∇φ| and κ = −∇ · ∇φ

|∇φ| . (4)

Due to the motion of interface, the interface is captured by solving the advection for the level-setfunction φ and for the volume fraction F in its conservative form,

∂φ

∂t+ ∇ · ( �V φ) = 0, (5)

∂ F

∂t+ ∇ · ( �V F) = 0. (6)

Void fraction F is introduced as the fraction of the liquid inside a control volume (cell), where thevoid fraction takes the values 0 for air cell, 1 for water cell, and between 0 and 1 for a two-phasecell. The density and viscosity are derived from the level-set function as

ρ(φ) = ρw H (φ) + ρa(1 − H (φ)), μ(φ) = μw H (φ) + μa(1 − H (φ)), (7)

where H(φ) is the Heaviside function,

H (φ) =

⎧⎪⎨⎪⎩

1 φ > ε

12 + φ

2ε+ 1

2π

[sin

(πφ

ε

)]|φ| ≤ ε

0 φ < −ε

, (8)

where ε is the interface numerical thickness over which the phase properties are interpolated. Theboundary conditions are symmetry or free slip condition at the left and right boundaries, outflowboundary conditions on the top surface, and no slip and impermeability conditions on the bottomsurface of the domain. The solution algorithm has been described in earlier papers.21–24

III. RESULTS AND DISCUSSIONS

A. Bubble entrapment phenomenon

Figure 1(a) shows the high speed experimental images and the computed shape of the freesurface profiles during bubble entrapment presented by Morton et al.14 and the predictions of thecurrent study. Morton et al.14 used the volume-of-fluid (VOF) method as the interface trackingmethod, which showed good qualitative agreement with the experimental image. The only limitationwas that after the pinch-off, the trapped bubble disappeared since the dynamics of gas-phase wasneglected in their numerical model. Previous attempts to simulate drop impact were carried out byHarlow and Shannon26 using SOLution Algorithm (SOLA-VOF) method and Oguz and Prosperetti12

and Elmore et al.27 using Boundary Integral method. The numerical methods predicted the bubbleentrapment process but could not predict the after effect of bubble pinching. Wang et al.28 usedCLSVOF method to capture entrapped bubble along with the thin jet phenomena and their resultsmatched quite well with Morton et al.14 In the present CLSVOF method the change in profiles alongwith the time sequences shows good agreement with the experiments of Morton et al.14

The cavity depth and jet height also reveal good match with their experimental data at earlyand later stages. The oscillation in cavity depth shown by Wang et al.28 during later stage is avoidedhere. The inward and outward jets which follow after the bubble entrapment are nicely captured inour simulation. The bubble diameter (∼0.18D) is also quite comparable with the experimental data(∼0.2D).

082108-4 Ray, Biswas, and Sharma Phys. Fluids 24, 082108 (2012)

(a () b)

(c)

FIG. 1. Experimental (left column in (a) and solid circles in (c)) and computational (middle column in (a) and triangles in(c)) representing the bubble entrapment phenomena, cavity depth, and jet height by Morton et al.12 and present numericalmethod (right column in (a) and solid line in (c)) for a 2.9 mm drop impacting from a height of 170 mm (Fr = 85, We = 96,Re = 4480). (b) The inward jet in entrapped bubble.

082108-5 Ray, Biswas, and Sharma Phys. Fluids 24, 082108 (2012)

The process of bubble entrapment can be best understood from the vertical and horizontal flowsduring the free surface deformation. The process consists of three stages: expansion, retraction, andnecking. After the drop impacts a wave-swell is created at the liquid surface (radial deformation)and a crater is formed (axial deformation). During the expansion stage, wave-swell height andcrater-depth increase. The crater shape changes from spherical to U-shape (at t = 6.2 in Fig. 1(a)).At retraction stage the wave-swell height and crater-depth decrease and the crater transforms to flatV-shape (at t = 8.6 in Fig. 1(a)). The liquid from wave-swell converges toward the side walls ofthe crater (at t = 10.7 in Fig. 1(a)) leading to neck formation. Finally, the crater side wall collidesand causes pinch-off (at t = 10.9 in Fig. 1(a)). There is a strong pressure drop below the craterbase after pinch-off and a high speed jet is ejected upward (at t = 12.9 in Fig. 1(a)) and downward(Fig. 1(b)). These jets, known as Worthington jets,29 are also observed during solid body impact onwater,7, 8 drop impact on a hydrophobic surface,30 and breakup of bubbles in co-flowing liquid.31

The potential energy which is stored during the crater formation is converted to kinetic energyof the jet during bubble pinch-off. Due to Rayleigh-Plateau instability, the upward jet breaks upinto numerous secondary droplets. Numerically, the downward jet inside the bubble during bubbleentrapment phenomena is captured for the first time (Fig. 1(b)) in this paper. In this case the jetattempts to break the bubble into two but due to surface tension forces the bubbles retain its sphericalshape. Experimental evidences by Elmore et al.27 show that the jet may break into drops inside thebubble or in some cases it can split the bubble into two. Gekle et al.8 and Gekle and Gordillo32

explained the formation of this high speed jet after solid object impact on a liquid surface or afterthe pinch-off of a gas bubble from an underwater nozzle. It is not due to a hyperbolic flow aroundthe singular pinch-off point but due to continuous radial energy focusing along the entire wall of thecavity. The fluid is not accelerated upward continuously from the pinch-off singularity but insteadacquires its large vertical momentum in a small zone located around the jet base.

Bubble entrapment characteristics as a function of Weber number are studied at negligiblegravity (Fr = 100) and viscosity effect (Re = O(103)). Two different types of bubble entrapmentprocesses are observed: large bubble with thin small jet (Fig. 2(a)) and small bubble with long thickjet (Fig. 2(b)) as the Weber number is increased. Figure 2(c) shows the difference in the depth ofcrater base along the symmetry axis. After initial drop impact both the phenomena behave quitesimilarly. During the expansion stage at t = 8, the crater depth gradually increases, after which theretraction begins from crater side walls. The depth stagnates27 during the period t ≈8–10 until thecrater obtains a flat V-shape indicated by A and B. Flow reversal of the crater base at the symmetryaxis begins now. The retraction is stronger at the crater side wall just above the crater base thanat other positions. A sub-crater is formed at the crater base (position C and D). The hydrostaticpressure from the side walls empowers the upward force trying to pull back the interface to initialshape. The crater base again moves downward before pinch-off to form a spherical bubble. Theinset figure indicates that the change in crater base depth during retraction stage is slower for highWe. Another notable difference is the upward jet emanating out during bubble pinch-off. For We= 150, a short, thin, high-speed jet appears which again pinches out numerous small secondarydrops (Fig. 2(a)). Later the velocity of the jet decreases and the jet thickens, rises to a certain height,and finally collapses into the water surface. In the case of We = 160, a similar thin jet is formedwhich produces many secondary droplets and instead of collapsing quickly, it elongates to form along, thick jet (Fig. 2(b)). The jet speed decreases and owing to capillary instability, it breaks up toyield one or two large secondary droplets.

This new small bubble entrapment zone along with the large bubble entrapment zone is shownin Weber-Froude diagram in Fig. 3. At low Froude numbers, the new zone lies above the upperlimit of bubble entrapment and for higher Froude numbers this zone is within the bubble entrapmentregime.

B. Bubble entrapment sub-regimes

The final crater shapes before pinch-off at different Weber numbers are shown in Fig. 4(a). Itis seen that there is a competition between the retraction of crater side-wall and crater base. ForWe > 150, retraction of the base is faster than that of the side wall and the size of sub-crater and

082108-6 Ray, Biswas, and Sharma Phys. Fluids 24, 082108 (2012)

(a) (b) (c)

FIG. 2. (a) and (b) Different crater and jet shapes during large and small bubble entrapment phenomena. The white arrowsindicate the entrapped bubbles. (c) Comparing nondimensional crater depth for two cases, one with solid lines: large bubble:Fr = 100, We = 150, Re = 4941, and other with dashed lines: small bubble: Fr = 100, We = 160, Re = 5294. The craterdepth is measured from the initial water surface position.

the final bubble is much smaller. Based on bubble diameter, maximum jet height and jet speed asa function of Weber number, the bubble entrapment phenomena are divided into three regimes inFig. 4(b): capillary regime (Regime I), inertia-capillary regime (Regime II), and the inertial regime(Regime III). The maximum jet height is the distance of the jet tip from the initial liquid surface justbefore the last secondary drop is ejected and the jet falls back. The jet speed is the speed of the jet

Fr = U2/gD

We

=U

2D

/

100 200 300 400 50050

100

150

200

250

300Upper boundary(Oguz & Prosperetti)

Lower boundary (Oguz & Prosperetti)

Large bubble and small thin jet

Small bubble and long thick jet

ρσ

We = 48.3 Fr 0.247

We = 41.3 Fr 0.179

FIG. 3. Bubble entrapment regime in Weber-Froude parameters.

082108-7 Ray, Biswas, and Sharma Phys. Fluids 24, 082108 (2012)

(a)

(b)

FIG. 4. Effect of Weber number: (a) Crater shape at pinch-off stage, (b) bubble diameter, jet height, and initial jet velocityas a function of Weber number. All the values in axial direction are measured from the equilibrium flat water surface.

tip when it is initially formed during pinch-off. In the capillary regime the bubble size increases asthe surface tension force decreases. More is the surface tension force, lesser is the radial expansionof the sub-crater base and the crater tries to retain its initial shape faster (Fig. 4(a) left row). Furtherincrease in surface tension force does not bring about pinch-off any more. In the inertia-capillaryregime, the interplay of both inertia and surface tension force is seen. These forces accelerate theretraction phase and hence the bubble size reduces. The expansion of the sub-crater base continuesas in Regime I, but the increased inertia force reduces the bubble size.

In Regime III, the inertia force is the dominating force and here the retraction from the craterbase is faster than the side walls. This regime is not always reproducible experimentally as describedby Elmore et al.27 They observed the event “E3” experimentally where a single bubble, multiplebubbles, or no bubble were produced from one impact to next impact unpredictably. According toGekle and Gordillo32 the azimuthal distortions triggered by gas shear or geometrical asymmetriesleads to this unpredictability in experimental results. The numerical result of Elmore et al.27 beforepinch-off is similar to our observations. The jet height in these regimes shows that in Regime I, themaximum jet height is not influenced much, and as the Weber number increases in Regime II, thejet height gradually increases. In the inertia-regime, the jet increases to large height. The thin jet isejected at high speed in Regimes I and II and as the bubble size decreases, the jet is thick, long, andthe initial jet velocity is less (Regime III). The bubble diameter, maximum jet height, and initial jetvelocity are also measured for finer spatial resolution shown in Table II.

C. Bubble pinch-off scaling

The equation describing bubble dynamics near the pinching region is a cylindrical version ofthe Rayleigh-Plesset equation, which describes the collapse of an infinite cylindrical cavity under

082108-8 Ray, Biswas, and Sharma Phys. Fluids 24, 082108 (2012)

TABLE II. Bubble diameter, jet height, and jet speed at two different spatial resolutions.

Parameters Grid Bubble diameter Jet height Jet speed

Fr = 100, We = 130 350 × 700 0.2099 0.379 3.317Fr = 100, We = 130 700 × 1400 0.210 0.377 3.319Fr = 100, We = 190 350 × 700 0.0428 3.968 2.732Fr = 100, We = 190 700 × 1400 0.0428 3.98 2.730

uniform pressure,

(rr + r2

)ln

r

r∞+ 1

2r2 − σ

ρr= gz, (9)

where r is the radius of the bubble, r∞ is the radius where the flow is quiescent (crater opening onthe equilibrium liquid pool surface), z is the depth of the neck below the equilibrium surface, σ isthe surface tension, and ρ is the density of the surrounding liquid. The equation indicates that thephenomenon of bubble pinch-off is driven by inertia of the collapsing flow (first two terms in Eq.(9)) and the surface tension force contributes to accelerate the liquid inward (third term in Eq. (9)).The right hand term is the constant hydrostatic force. For low-viscosity fluids, the inertia term is thedominating force; the asymptotic solution is obtained by neglecting other terms and Eq. (9) becomes

rr + r2 = 0. (10)

Solving Eq. (10), the radius of the bubble is given by rneck ∝ A(t0 − t)0.5. The time scale in thecase of low-viscous fluids is the capillary time defined by (ρR3/σ )1/2, where R is the drop radius.Thus, the coefficient A = (σR/ρ)1/4and for the parameters used in this paper (R = D/2 = 0.0025m, σ = 72.8 × 10−3 N/m, ρ = 998.12 kg/m3) the value of A = 0.02 m/s1/2. Figure 5 shows thebehaviour of rneck with (t0 − t) for different Weber numbers varying as rneck = A(t0 − t)α .

For the three regimes, the characteristic exponent is α = 0.5 ± 0.02 and the coefficientA = 0.02 ± 0.006 m/s1/2 (Figs. 6(a) and 6(b)). Thus, the value of the coefficient A varies morethan the power exponent α with the Weber number. The exponent describing the pinch-off processof bubbles is quite sensitive to the spatial resolution.34 Burton et al.3 experiments on bubble pinch-off in liquids of different viscosities showed three different modes of pinch-off. For lower valuesof the external viscosity, the axial curvature is more hyperbolic and the neck radius undergoes asudden rupture at rneck ≈ 25 μm at a time scale of 10 μs. They showed that bubbles in low viscosity

t0-t

r

10-5 10-4 10-3

(m)

(s)

10-4

5x10-4

5x10-5

neck We =120, 150

We =170

We =180, 190

We =130, 140

We =160

α=0.5

α=

0.5x[1+0.5/{-ln(t0-t)}]α=

0.5x[1+0.5/sqrt{-ln(t0-t)}]

FIG. 5. Variation of computed minimum neck radius rneck with (to − t) during bubble pinching process for different valuesof We. The results are compared with the theoretical predictions for rneck ∝ (t0 − t)α . The dashed line indicates the powerexponent by theoretical Rayleigh-Plesset equation, the dashed-dotted-dotted line indicates the exponent by analysis of Eggerset al.33 and the dotted line indicates the expression of power exponent by Gordillo et al.5, 6

082108-9 Ray, Biswas, and Sharma Phys. Fluids 24, 082108 (2012)

(a) (b)

FIG. 6. (a) Plotting fitting exponent α versus We, (b) plotting coefficient A versus We. Solid line in (a) and (b) is thetheoretical value. Different symbols represent different regimes.

fluids follow (t0 − t)0.5 power law until the shear between the interior and exterior flows causes aninstability that leads to the rupture of the bubble. According to Keim et al.35 the pinch-off appearsto be cylindrically symmetric and proceeds, without rupture, to scales below their camera resolu-tion (4 μm) and derived the power law exponent α = 0.56 ± 0.03. Thoroddsen et al.36 observeda continuous necking down to the pixel-resolution of 10 μm at largest frame rates with slope ofα = 0.57 ± 0.03. The experiment7, 8 of violent collapse of the void created at a fluid surface by theimpact of an object is purely governed by liquid inertia. In the limiting case of large Froude numbers

the inertial scaling (t0 − t) ∝ r2neck

√− ln r2

neck is obtained also shown for bubble pinch-off at high Reby Gordillo et al.5, 6 The value of power law exponent thus can be expressed as α = 0.5 × [1 + 0.5/[ − ln (t0 − t)]]. Analytical work of Eggers et al.33 where the surface tension, gas density, and vis-cosity are neglected, α is time dependent expressed as α = 0.5 × [

1 + 0.5/√− ln(t0 − t)

]. Gekle

et al.34 used boundary integral simulations to show the universal behaviour as Eggers et al.33 withdifferent onset time for this universal pinch-off stage.

Taking into account surface tension, gas density, and viscosity, Gordillo37 derived a systemof two-dimensional Rayleigh-like equation which showed good agreement with the numerical andexperimental36, 38, 39 results. Neglecting surface tension, gas density, and viscosity, Gordillo37 derivedthe equations

ln (R0r1)d ln

(R0 R0

)d (− ln R0)

− 1 = 0, (11)

ln (R0r1)d ln (R0r1)

d (− ln R0)− 1 = 0. (12)

The theoretical prediction of Gordillo37 reduces to Eggers33 theoretical model, α = 0.5× [

1 + 0.5/√− ln(t0 − t)

]. We have compared our numerical results with Gordillo’s37 model and

the model showed good agreement. Figure 5 shows how the numerical evolution matches the asymp-

totic behaviour (t0 − t) ∝ r2neck

√− ln r2

neck (by Gordillo et al.5, 6) for (t0 − t) < 3 × 10−5s, which

slightly differs from the (t0 − t) ∝ r2neck power law and the analyses of Gordillo37 and Eggers et al.33

Here, the comparison with analytical results is shown down to scales of the order of rneck ≈ 60 μm.Our numerical data are in good agreement with the theoretical and experimental data although oursimulations are unable to capture the exact instants of bubble pinch-off due to the limitation ofcomputational resources.

082108-10 Ray, Biswas, and Sharma Phys. Fluids 24, 082108 (2012)

(z-zmin)/rminr/

r min

-10 -5 0 5 100

5

10

15

t0-t = 1.4x10-4 st0-t = 4x10-5 s

t0-t = 2.4x10-4 s

t0-t = 5x10-6 st0-t = 2x10-5 s

FIG. 7. Variation of computed scaled drop profiles r/rneck with scaled axial coordinate (z − zneck)/rneck as the bubbleapproaches pinch-off at Fr = 100, We = 150, and Re = 4941.

It is notable that in Regime II, the pinching for We = 130 and 140 is similar. As the bubblesize reduces at We = 150, the pinching is similar to Regime I (We = 120). At Regime III (We =160 − 190), the pinching time is small and with increase of We from 160 to 180, the value of thecoefficient increases and shows similar pinching characteristics for We = 180 and 190.

In inviscid conditions, the pinch-off dynamics of droplets is governed by a local balance betweensurface tension forces and inertia so the near field structure of drop pinch-off is self-similar anduniversal.3 Bergmann et al.7 showed that the self-similar behaviour appears to hold only in theasymptotic regime of very high impact velocities at limit of infinite Froude number, where theinfluence of gravity becomes negligible and the collapse is truly inertially driven. During bubbleentrapment, the power behaviour is again reflected by plotting the scaled axial and radial coordinatesduring necking.40 Figure 7 shows the variation of computed scaled crater profiles r/rneck as a functionof the scaled axial coordinate (z − zneck)/rneck, where zneck is the axial coordinate at r = rneck. As (t0− t) → 0, the scaled profiles show self-similarity and the extent of axial coordinate over which thesimilarity exists increases as the neck thins.

IV. CONCLUSIONS

In conclusion, the paper shows that the predications due to the numerical method match well withthe experimental results as well as confirm theoretical analysis of pinching dynamics. Simulationsshowed the bubble entrapment, growth of Worthington jets (thin and thick), and subsequent formationof secondary drops. Important phenomena such as downward jet during bubble pinch-off and smallbubble entrapment above the large bubble entrapment zone of Oguz and Prosperetti11 have beenpresented. Three regimes based on bubble diameter versus Weber number have been defined and thescaling theory in each regime has been established.

ACKNOWLEDGMENTS

We heartily thank D. Morton, J. L. Liow, and D. E. Cole for valuable discussions regarding ournumerical results. The authors are grateful to the reviewers for their useful suggestions.

1 R. F. Day, E. J. Hinch, and J. R. Lister, “Self-similar capillary pinchoff of an inviscid fluid,” Phys. Rev. Lett. 80(4), 704(1998).

2 H. N. Oguz and A. Prosperetti, “Dynamics of bubble growth and detachment from a needle,” J. Fluid Mech. 257, 111(1993).

3 J. C. Burton, R. Waldrep, and P. Taborek, “Scaling and instabilities in bubble pinch-off,” Phys. Rev. Lett. 94(18), 184502(2005).

4 M. S. Longuet-Higgins, B. R. Kerman, and K. Lunde, “The release of air bubbles from an underwater nozzle,” J. FluidMech. 230, 365 (1991).

082108-11 Ray, Biswas, and Sharma Phys. Fluids 24, 082108 (2012)

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