elsevier oscillation of bubbles

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Oscillation and breakup of a bubble under forced vibration

Mohammad Movassat, Nasser Ashgriz , Markus Bussmann

Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, ON M5S 3G8, Canada

a r t i c l e i n f o

Article history:

Received 14 August 2014

Received in revised form 5 March 2015Accepted 24 May 2015

Keywords:

Drops and bubbles

Bubble oscillations

Chaotic oscillation

Microgravity flows

a b s t r a c t

Coupled shape oscillations and translational motion of an incompressible gas bubble in a vibrating liquid

container is studied numerically. The bubble oscillation characteristics are mapped based on the bubble

Bond number (Bo) and the ratio of the vibration amplitude of the container to the bubble diameter (A/D).At smallBo andA/D, the bubble oscillation is found to be linear with small amplitudes, and at largeBoand

A/D, it is nonlinear and chaotic. This chaotic bubble oscillation is similar to those observed in two coupled

nonlinear systems, here being the gas inside the bubble and its surrounding liquid. Further increases in

the forcing, results in the bubble breakup due to large liquid inertia.

2015 Published by Elsevier Inc.

1. Introduction

Oscillations and translational motion of gas bubbles in a host

fluid under forced vibrations are encountered in many two phase

flow applications. For instance, in mixers and reactors (Ni andGao, 1996; Krishna and Ellenberger, 2002; Knopf et al., 2006),

and in thermal management systems in microgravity ( Gaul et al.,

2010; Zhang et al., 2009; Weislogel et al., 2009). In most such

devices, bubbles enhance the heat and mass transfer coefficients

and reaction rates in the mixture.

Depending on the imposed amplitude and frequency of the

vibrations, bubble oscillations range from small amplitude and lin-

ear to large amplitude and nonlinear oscillations. Further increase

of the amplitude and frequency of the vibration results in bubble

breakup.

Experimental study of the bubble oscillation under forced vibra-

tion requires a reduced gravity condition to trap the bubble and

study its motion. As a result, there are very limited studies using

forced vibration. The available studies have shown that the bubble

undergoes an oscillatory translational motion at the same fre-

quency as the forcing (Ishikawa et al., 1994; Friesen et al., 2002;

Farris et al., 2004). Vibration amplitude and frequency were kept

small in these studies resulting in a small deviation of the bubble

from the spherical shape. Since the oscillation frequencies were

orders of magnitude lower than the Minnaert frequency (Devaud

et al., 2008), no volume oscillations were considered.

Several studies have investigated bubble oscillation by acoustic

levitation, where an acoustic force was used to trap a bubble and

modulate its oscillation.Eller and Crum (1970)found a threshold

for the beginning of large amplitude shape oscillations of a gas

bubble as a function of the bubble radius. Their results showed that

the pressure threshold for the beginning of shape oscillation

decreases with increasing bubble radius, since surface tensionforce decreases as the bubble size increases. In the case of large

amplitude oscillations, it was shown that the interaction among

volume oscillations, shape oscillations, and translational motion

results in a chaotic bubble response (Akhatov and Konovalova,

2005; Watanabe and Kukita, 1993). Chaotic response is because

of the coupling of large amplitude and nonlinear shape and volume

oscillations, and the translational motion. Due to the coupling,

excitation of any of these motions, e.g. the translational motion,

can excite the rest (Doinikov, 2004; Mei and Zhou, 1991;

Benjamin and Ellis, 1990).

If the acoustic forcing is strong enough, and if the shape of the

bubble is distorted from the spherical shape due to the asymme-

tries in the flow field, a liquid jet can form which penetrates into

the bubble and pierces it (Crum, 1979). The impact of this high

velocity liquid jet is the dominant mechanism leading to cavitation

damage. Bubble breakup has also been studied in the context of

sonoluminescence, where bubbles violently collapse. Calvisi et al.

(2007) and Blake et al. (1999) showed that in the case of surface

instabilities which can be induced by rigid boundaries, neighboring

bubbles, flow disturbances, and asymmetric flows around a bubble,

a breakup liquid jet can form within the bubble core as the bubble

collapses. In a recent study byYoshikawa et al. (2010)large ampli-

tude shape oscillations of a bubble in response to forced vibration

was studied experimentally in a parabolic flight. Results suggested

that as the forcing increases, bubble undergoes large amplitude

shape oscillations and breakup.

http://dx.doi.org/10.1016/j.ijheatfluidflow.2015.05.014

0142-727X/ 2015 Published by Elsevier Inc.

Corresponding author.

E-mail address:[email protected](N. Ashgriz).

International Journal of Heat and Fluid Flow 54 (2015) 211219

Contents lists available at ScienceDirect

International Journal of Heat and Fluid Flow

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In a previous work, the authors studied regular and chaotic

response of a bubble to the forced vibrations in 2D (Movassat

et al., 2012). In the present study, the dynamics of a single bubble

in response to forced vibration is studied using a 3D numerical

scheme. As mentioned, available studies on the bubble oscillation

under forced vibration are limited to small amplitude oscillations

where the bubble shape remains spherical. The focus of the present

study is to understand the coupling between shape oscillations andtranslational motion and characterize the nonlinear large ampli-

tude and chaotic bubble motion, and bubble breakup in frequen-

cies which are orders of magnitude lower than those in acoustic

levitation. Since the imposed frequency in this work is lower than

Minnaert frequency and also because isothermal and adiabatic

conditions are assumed, no volume oscillation is considered and

the bubble response is limited to the shape oscillations and the

translational motion.

2. Numerical model

The equations governing the motion of an incompressible bub-

ble in a liquid domain are the mass and momentum conservation

equations,

r ~V 0 1

@~V

@t ~V r~V

1

qrp

1

qr s

1

q~FSF A2pf

2cos2pft 2

where~Vis the velocity vector, p is the fluid pressure,q is the fluid

density, s is the shear stress tensor,~FSFrepresents the surface ten-sion force per unit volume, which is applied on the interface

between two fluids. The last term on the right hand side of Eq. (2)

is the imposed oscillation force as a result of the forced vibrations,

in whichA and fare the amplitude and frequency of the vibration,

respectively. Since the fluids are Newtonian,

s lr~V r~VT 3

wherel is the fluid dynamic viscosity.The TransAT software was used for the simulations. The simula-

tions were run on 32 processors on SciNet clusters at University of

Toronto. The Level Set (LS) method was used to capture the inter-

face between the two fluids. If the interface is defined by C, a func-

tion,u, is defined asu > 0 in the liquid, u < 0 in the gas, and u = 0on the interface, C. Since the interface moves with the fluids, umust be advected by the following equation,

@u@t

~V ru 0 4

In the LS method, the interface is assumed to have a finite thick-

ness. Thus a smoothed density and viscosity, denoted as qeuandleu, are defined in each computational cell as,

qeu qg ql qgHeu 5

leu lg ll lgHeu 6

where subscriptsgandl represent gas and liquid, respectively. The

modified Heaviside function, He, is defined as, He(u) = 0 for u e, and He(u) = 0.5[1 + u/e+ 1/psin(pu/e)] for|u| 6 e. At each time step the LS function is reinitialized withoutchanging its zero level set. This is achieved by solving the following

partial differential equation,

@d@s signu1 jrdj 7

with initial condition of,d(x, 0) = u(x), wheresign(u) =1 for u < 0,signu 1 foru > 0, andsignu 0 foru 0. In Eq.(7), d is thedistance function representing the interface between the two fluids

ande is the integration variable representing time step. The majordrawback of Level Set method has been its ability to conserve mass.

In TransAT, this error is minimized using a correction factor in the

solver settings. This correction factor is applied locally, for each cell,

and globally, for the whole domain, to assure that the mass is con-

served from one time step to the next. Comparing to other interface

capturing methods such as Volume of Fluid, Level Set provides a

sharper property change across the interface.

The surface tension force is modeled as ~FSTrjd~n, wherer isthe coefficient of surface tension, j is the interface curvature, and dis the Dirac delta function, which is defined as, du dHe=du.Unit normal vectors and Curvature are calculated based on the

level set function as,

~n rujruj

u0

8

j r rujruj

u0

9

The surface tension force is then discretized as a volume force

using a modified CSF method. CSF was first introduced by

Brackbill et al. (1992). This model has been modified in TransAT

to overcome the shortcomings of the original model and minimize

smoothing of this force across the interface (Lakehal et al., 2002;

Liovic et al., 2006; Liovic and Lakehal, 2007). In TransAT the

governing equations are discretized on a collocated mesh in which

velocities are defined at cell centers. Adaptive time stepping is

controlled by specifying the following limits: CFL maxjuijkjDt=Dx;

jvijkjDt=Dy; jwijkjDt=Dz< 0:3, DIFF maxlDt=qDx2i < 0:3, and

STN maxDtffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirjjjdni=qDxi

p < 0:3, which are time restric-

tions for the convective, viscous, and surface tension terms,respectively.

As mentioned in the introduction, experimental studies for bub-

ble oscillation under forced vibration is limited to small amplitude

oscillations in which shape of the bubble remains spherical. In a

previous paper (Movassat et al., 2012), authors showed that simu-

lation results for translational motion of the bubble match well

with experimental data. As frequencies applied in forced vibration

are orders of magnitude smaller than acoustic frequencies, no com-

parison can be made to acoustic studies where bubble undergoes

large amplitude shape oscillations. Also, experimental studies

involve multi bubbles and the interaction force among the bubbles

is an effective force. The focus of this work was to understand the

large amplitude shape oscillations and translational motion, and

chaotic interaction of these two motions, for a bubble under forced

vibration.

To identify the non-dimensional numbers, bubble diameter, D ,

vibration frequency, f, and surface tension coefficient, r , are usedto non-dimensionalized the governing equations. Resulting

non-dimensional numbers are Bond number, BoqAf2D2=r,Reynolds number, Re qAfD=l, and the ratio of the vibrationamplitude to the bubble diameter, A/D. It will be shown that at

large amplitude oscillations, the viscous forces play a smaller role,

andBo and A/D are sufficient to predict the bubble behavior. One

can use Weber number, We, as a non-dimensional number to char-

acterize this flow. However, the Weber number in this case is writ-

ten as WeqAf2D=r, which is a multiplication ofBo and A/D:We BoA=D, and it is not an independent non-dimensional

number in this problem. Therefore, it is sufficient to describe theproblem in terms ofBo and A/D.

212 M. Movassat et al. / International Journal of Heat and Fluid Flow 54 (2015) 211219


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3. Results and discussions

A single bubble withD = 4 mm is positioned in the middle of a

liquid container of dimensions 102010 mm (in x, y, and

z-directions, respectively). The container is vibrated in the vertical

y-direction. A uniform mesh with resolution of 60 cells/D is used.

Fig. 1 illustrates the xy view of the problem configuration and

the mesh. The fluid properties used are ql 1000 kg=m3

, qg1 kg=m3, ll 10

3 Pa s, lg 2 105 Pa s, and r = 0.073 N/m.

These are physical properties of water and air at 20 C. Due to sym-

metry in thex- andz-directions, half of the domain in these direc-

tions is solved using symmetry boundary conditions for symmetry

planes and no-slip boundary conditions for the rest of the

boundaries.

3.1. Regular and chaotic responses

Two vibration frequencies are considered: (i) A frequency of

f= 106.8 Hz with an amplitude ofA= 0.08 mm, which result in a

case with Bo= 0.2, A/D= 0.02, and Re= 34.2, and (ii) a frequency

of f = 75.5 Hz with an amplitude of A= 0.4 mm, which result in

Bo= 0.5, A/D= 0.1, and Re= 120.8.Figs. 2 and 3illustrate the bub-

ble shape, velocity vectors, and contours of velocity magnitude

during the first period at timest=T/4,T/2, 3T/4, andTfor the above

two cases: Bo = 0.2 and Bo= 0.5, respectively. The plane in which

the bubble shape and contours are plotted is the middlexyplane,

which goes through the bubble center-of-mass. Due to the symme-

try, zyplane yields the same results.

Figs. 2 and 3 illustrate the flow patterns during one period.

During the first quarter, the top part of the bubble is pushed down-

ward by the liquid on the top of the bubble. During the third quar-ter, the bottom part of the bubble is pushed upward by the liquid

below the bubble. These top and bottom forces result in a

non-uniform and asymmetric pressure distribution along the

y-direction, which results in an asymmetric velocity distribution

along they-direction. ForBo = 0.5, it is observed that the imposed

oscillation force is strong enough to deform the bubble after one

period, while for Bo = 0.2, the surface tension force is such that

the deviation from the spherical shape is small.

20 mm

10 mm

Fig. 1. Problem configuration (xy view), on the left, and the mesh resolution, on

the right.

Fig. 2. Bubble shape, velocity vectors, and velocity magnitude contours for Bo = 0.2, A/D= 0.02, and Re = 34.2 at (a)t=T/4, (b) t=T/2, (c)t= 3T/4, and (d) t=T.

M. Movassat et al. / International Journal of Heat and Fluid Flow 54 (2015) 211219 213
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Fig. 4shows the location of the center-of-mass of the bubble inthe vertical direction with time for the two cases. For Bo= 0.2

(Fig. 4a), the bubble slowly moves toy=0.05 mm and it then sta-

bly oscillates about this point. ForBo = 0.5 (Fig. 4b), the bubble has

a significant drift along the ydirection. The bubble moves down-

ward and after 44 cycles it impacts the bottom wall. The transla-

tional oscillation of the bubble center-of-mass, as shown in

Fig. 4b, does not appear periodic, even though the forcing fre-quency is periodic. This is due to the coupling of the large ampli-

tude shape oscillations with the translational motion, resulting in

a nonlinear and non-repeating pattern for Bo = 0.5.

In order to characterize the nonlinear behavior of the bubble

oscillation for the large Bo number case of Fig. 4b, the resulting

bubble shape variation is analyzed. The bubble shape in each

Fig. 3. Bubble shape, velocity vectors, and velocity magnitude contours for Bo = 0.5,A/D= 0.1, and Re = 120.8 at (a)t=T/4, (b)t=T/2, (c) t= 3T/4, and (d) t=T.

Fig. 4. Variation of bubble center of mass location in the vertical direction with time for (a)Bo= 0.2, A/D= 0.02, and Re= 34.2, (b) Bo= 0.5, A/D= 0.1, andRe= 120.8. For the

small amplitude case, the bubble oscillates locally with a small drift, while for the large amplitude case the bubble drift is large and the bubble eventually impacts the bottomwall of the container.



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period is decomposed into its Legendre polynomials, which are

mathematical functions used to decompose any arbitrary shape

to a linear sum of different modes:

r D=2 X1n0

cntPncos h 10

where ris the distance from any point to the bubble center-of-mass,Dis bubble diameter, Pncoshis thenth mode harmonic, and cn is

the amplitude of thenth harmonic. The first 10 harmonics were cal-

culated. The first two coefficients, c0 and c1, are zero because c0reflects volume oscillation, and c1 is associated with bubble center

of mass motion. c2 to c9 are calculated from the numerical model

results using a least squares algorithm (Press, 1992).

The variation of the coefficients of various harmonics are

obtained with respect to time. The variation of the second and

third mode coefficients, non-dimensionalized by bubble diameter,

c2/D and c3/D are shown in Fig. 5. Coefficients are calculated at

t=nT. Fig. 5a and b shows c2/D and c3/D for Bo= 0.2, A/D= 0.02,

andRe= 34.2. In this case, c2/Dconverges to0.013 after 50 cycles

indicating that bubble is only slightly deviated from a spherical

shape. Since the bubble shape deformation is small, the bubble

translational motion remains periodic. For this case,c3/Dconverges

to zero indicating that the third mode does not contribute to the

shape of the bubble after several oscillation.

Fig. 5c and d shows the bubble shape coefficients for Bo = 0.5,

A/D= 0.1, and Re = 120.8. Neither c2 nor c3 converge to any value.

In this case, the magnitude of the coefficients is much larger than

the corresponding values for the previous small amplitude case.

The average values ofc2 andc3 coefficients are about 10% and 8%

of the bubble diameter, respectively, resulting in a nonlinear andchaotic shape oscillation. The asymmetry, associated with the third

mode, results in an imbalance between the top and the bottom

parts of the bubble in the y-direction. As a result, the imposed

oscillation force enhances the imbalance as the bubble oscillates

in time.

The variation of the surface area of the bubble for two cases

with time at t=nT is shown in Fig. 6. The area is

non-dimensionalized with the initial spherical area. For Bo= 0.2

case, the maximum deviation from the spherical shape is less than

0.06%, which occurs during the first period. As the oscillations con-

tinue, the area converges to only 0.02% larger area than that of the

initial spherical shape. For the chaotic case ofBo = 0.5, the area of

the bubble in successive cycles oscillates, and as the oscillations

continue, the pattern of the oscillations changes. The maximum

surface area for this case, which occurs during the 36th period,

Fig. 5. Variation of Legendre polynomial coefficients with time for (i)Bo= 0.2,A/D= 0.02, andRe= 34.2, (a)c2/D, (b)c3/Dand for (ii)Bo= 0.5,A/D= 0.1, andRe= 120.8, (c)c2/D,

(d)c3/D. Results suggest that the oscillations are regular for (i) since c2/Dconverges to0.013 while there is almost no contribution from the third mode, but chaotic with noconvergence for (ii), where the amplitude of oscillations is comparable to the bubble diameter.

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reaches a 4% larger area compared to that of the initial spherical

shape.

Shape oscillations and translational motion show that for the

small amplitude case, both translational motion and shape oscilla-

tions are regular, i.e. the bubble behavior becomes repeating in

successive periods. For the larger amplitude case, shape of the bub-

ble is not repeating in successive cycles as there is no convergence

in the Legendre coefficients. Also, there is not a stable position for

the bubble center-of-mass. The results suggest that the bubble

behavior at larger Bo and A/D becomes chaotic.

3.2. Bubble breakup

If the forcing is strong enough, the bubble deformation may

become so large that the bubble breaks up. Fig. 7shows deforma-

tion of a bubble subject to a large amplitude forcing withBo = 0.7,

A/D= 0.125, and Re= 160. At t= 0.25T, the liquid inertia forms a

small dimple at the top of the bubble (Fig. 7a). During the second

quarter, the bubble decelerates and the small dimple on the top

of the bubble grows in time. As a result, a downward moving liquid

flow forms at the core of the bubble at t= 0.35T(Fig. 7b). Since the

Fig. 6. Variation of the surface area of the bubble with time att=nT, (a) Bo= 0.2,A/D= 0.02, and Re = 34.2, (b) Bo= 0.5,A/D= 0.1, and Re = 120.8.

Fig. 7. Bubble shape, velocity vectors, and velocity magnitude contours for Bo= 0.7,A/D= 0.125, and Re= 160 at (a) t/T= 0.25, (b) t/T= 0.35, (c)t/T= 0.45, (d)t/T= 0.5, (e)t/T= 0.55, (f) t/T= 0.66. The bubble is pierced due to the formation of a liquid jet within the core of the bubble.



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bubble is decelerating, the velocity and the inertia of the liquid

cause the core liquid flow to penetrate into the bubble. Breakup

of the bubble by the core liquid flow continues during the remain-

der of the second quarter. After half a period, t= 0.5T, bubble flat-

tens and the core liquid flow continues to penetrate through the

bubble during the third quarter until the front of the liquid flow

reaches the bottom of the bubble. When the jet impacts the bottom

surface of the bubble, the bubble may break up forming tiny bub-bles within the liquid, and tiny liquid drops may form inside the

bubble. After break up, a toroidal bubble is formed. As the forcing

continues, the toroidal bubble undergoes large amplitude shape

deformations and smaller bubbles are formed.

3.3. Viscous effect

Bo and A/D do not depend on the fluid viscosities. To evaluate

the importance of viscous forces in the cases where bubble break,

the inviscid solution for the case with Bo = 0.7 and A/D= 0.125 is

compared to the viscous results presented in the previous sections,

having Re= 160. For the inviscid case, the viscosities of both liquid

and gas are set to zero. Results show that similar to the viscous

case, a liquid jet penetrates into the bubble core, causing the bub-

ble breakup.Fig. 8plots the distance from the front of the liquid jet

to the bottom of the container for viscous and inviscid cases. The

difference between the two cases is very small: the mesh size is

0.067 mm, and the maximum difference between the two cases

is about two mesh sizes. Although the difference is small, the figure

does show that in the inviscid case the front of the jet is closer to

the bottom of the container, i.e. that the jet has penetrated further

at the same time compared to the viscous case. Based on the above

comparison it is observed that the viscous forces do not alter the

bubble behavior. Consequently, for the range of parameters studies

here, the bubble response to the vibrations, Bo and A/D are suffi-

cient to characterize the bubble behavior.

3.4. Bubble oscillation map

A parametric study is performed to predict the effect ofBo and

A/D on the bubble response to the forced vibrations. Fig. 9illus-

trates the summary of the parametric study. Three types of

responses are characterized: regular oscillations, chaotic oscilla-

tions, and bubble breakup. Bubble breakup corresponds to the

cases in which shape oscillation is so large that breakup occurs.

Three lines are also shown in this figure. These are constant

frequency lines corresponding to the second, third, and fourth res-

onant frequencies of the shape oscillations. For a 4 mm air bubble

in water these frequencies are 52.6, 96, and 144 Hz for the second,

third, and fourth modes, respectively (Lamb, 1930).

Based on the map inFig. 9, regular oscillations occur at small Bo

and smallA/D. Cases withA/D6 0.08 andBo 6 0.4 (except for two

cases) respond regularly to the forced vibrations. For regular cases,

the second mode contributes to the bubble shape, while the asym-

metric third mode does not contribute. Two chaotic (without

breakup) cases are observed within the regular region, Bo= 0.4,

A/D= 0.02, and Bo = 0.4,A/D= 0.05. Frequencies of these two cases

coincide with the third and fourth mode resonant frequencies.Since the forcing frequency and the resonant frequency are very

close for these two cases, large amplitude oscillations are excited

and the bubble response deviates from the regular oscillations.

The triangular points represent cases with large amplitude

oscillations in which no breakup occurs within the first 10 periods

of oscillations. These points mainly lie between the regular oscilla-

tion and the breakup cases. In these cases, higher harmonics than

the second mode are excited, and the amplitude of the oscillations

are much larger compared to the regular oscillation cases. As a

result of a coupling of two nonlinear systems (liquid and the gas)

both a chaotic shape oscillation and a chaotic translational motion

is observed.

As A/D and Bo increase, the inertia of the surrounding liquid

interacts with the bubble shape oscillations to result in bubblebreakup. Since the shape oscillation is chaotic, and because bubble

breakup depends on the shape of the bubble, the transition

between the chaotic with no breakup and chaotic with breakup

is not precise. For instance, at Bo= 0.6, breakup occurs for

A/D= 0.05, while increasing the amplitude to A/D= 0.06 results in

a no-breakup case. Detailed analysis of the results show that the

bubble shape for A/D= 0.05 provides the condition for breakup to

occur, while forA/D= 0.06, the bubble shape and the flow field pre-

vents the formation of a core flow to break up the bubble.

To summarize, there are two factors that combine to lead to

bubble breakup under forced vibrations:

i. Bubble shape: Nonlinear large amplitude oscillations of the

bubble shape allow for the onset of bubble breakup. At smallamplitude shape oscillations the bubble does not deviate

Fig. 8. The variation of the distance between the liquid jet front and the container

bottom wall with time for viscous and inviscid solutions for Bo = 0.7 andA/D= 0.125.

Fig. 9. Bubble response to forced vibration as a function ofBo and A/D. Three types

of responses are characterized: regular oscillations, chaotic oscillations without

piercing, and chaotic oscillations with piercing. The three straight lines are constant

frequency lines which correspond to the first three resonant frequency modes.

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greatly from the equilibrium spherical shape. Surface ten-

sion is strong enough to prevent the formation of a core liq-

uid flow that pierces the bubble. As the oscillation amplitude

increases, regions of larger and smaller curvature are

formed. As well, the bubble shape changes the flow field of

the surrounding liquid. A large amplitude shape oscillation

is a necessary condition for the occurrence of bubble

breakup.ii. Liquid inertia: Due to the density difference between the gas

in the bubble and the surrounding liquid, any small change

in the velocity of the liquid results in a great change in the

nearby gas velocity. As a result, when the surrounding liquid

pushes the bubble, the velocity of the gas which is close to

the liquid, increases. As well, the liquid is able to deform

and penetrate into the bubble. Because of the symmetry of

the problem considered here, the maximum liquid velocity

and consequently, the maximum inertia transfer from the

liquid to the gas occurs at the bubble core. As a result,

breakup initiates and occurs within this region.

3.5. Effect of mesh size

Finally, to demonstrate that the mesh resolution which was

used in this work is good enough to capture the physics of the

problem, the case of Bo= 0.7 and A/D= 0.125 was modeled with

a finer mesh. The mesh resolution was refined to 90 cells per diam-

eter, for a total of 5,695,312 cells in the domain. Instead of using 32

processors as for the coarse case, 64 processors were used to model

the fine case.Fig. 10compares the shape of the bubble and the ver-

tical velocity contours for the coarse and fine meshes. The compar-

ison shows that while there are tiny differences in the velocity

contours, especially at t=T/2 when the bubble center of mass is

at rest and the velocities are small, the coarse mesh can still predictthe breakup and the penetration of the liquid jet within the bubble

core. The results are very similar in the two cases. It should be

noted that the formation of the smaller bubbles and the film drai-

nage is a micro-scale problem, and it is beyond the scope and

numerical capacity of this work. The goal of this mesh refinement

study is to show that the bubble breakup is a physical phe-

nomenon rather than a numerical error.

4. Summary and conclusions

The response of a bubble in a liquid container in response to

forced vibrations was studied. Forced vibration induces an oscilla-

tory force on the bubble. As a result, the bubble undergoes an oscil-

latory translational motion. Both the bubble shape and the bubblelocation inside the container oscillate. The volume oscillation is

small and it is neglected as the applied frequency is much lower

than the Minnaert frequency. Bond number (Bo), A/D and, and

Reynolds number (Re) can describe the dynamics of this type of

bubble oscillation. It is shown that the viscous forces do not play

an important role in the range of parameters studies here, and

the effect of the Re is not considered. It was shown that:

IncreasingBo and A/Dchange the bubble behavior from a regu-

lar oscillation to a chaotic one. The second mode of the oscilla-

tion is the dominant mode contributing to the shape of the

bubble in regular oscillations. In the chaotic oscillations, the

higher modes are also excited resulting in large amplitude non-

linear shape oscillations. The translational motion also becomeschaotic and non-repeating for the large amplitude cases.

Further increase of the amplitude and frequency of oscillations

results in bubble breakup. As a result of the inertia of the oscil-

lating liquid, a core liquid flow is formed within the bubble core.

This flow penetrates into the bubble and results in a pierced

bubble with a toroidal shape. The toroidal bubble undergoes

through large amplitude oscillations and the main bubble

breaks into smaller bubbles.

A map indicating the outcome of the bubble oscillation, i.e. reg-

ular, chaotic, and breakup, as a function ofBo and A/D is pro-

vided by performing a parametric study. The bubble shape

oscillation is chaotic in large forcing, and there is not always a

clear transition from regular to chaotic and then to breakup as

the shape of the bubble plays an important role in the oscilla-tion outcome. Vicinity of the applied frequency to the natural

frequency of different modes also changes the oscillation

outcome.

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t/T=0.25

t/T=0.5

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Fig. 10. Bubble shape and vertical velocity contours for Bo = 0.7 and A/D= 0.125,coarser mesh (60/D) on the left and finer mesh (90/D) on the right.

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