Growth and collapse of a vapor bubble in a narrow tube E. Ory, H. Yuan, A. Prosperetti, S. Popinet, S. Zaleski

To cite this version:

E. Ory, H. Yuan, A. Prosperetti, S. Popinet, S. Zaleski. Growth and collapse of a vapor bubble in a narrow tube. Physics of Fluids, American Institute of Physics, 2000, 12 (6), pp.1268-1277. ￿10.1063/1.870381￿. ￿hal-01445440￿

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Growth and Collapse of a Vapor Bubble in a Narrow Tube



E Ory H Yuan and A Prosp eretti

Department of Mechanical Engineering

The Johns Hopkins University

Baltimore MD USA

S Popinet and S Zaleski

Laboratoire de Modelisation en Mecanique UMRCNRS

Universite Pierre et Marie Curie

place Jussieu Paris Cedex France

January

Abstract

The uid mechanical asp ects of the axisymmetric growth and collapse of a

bubble in a narrow tub e lled with a viscous liquid are studied numerically

The tub e is op en at b oth ends and connects two liquid reservoirs at constant

pressure The bubble is initially a small sphere and growth is triggered by a

large internal pressure applied for a short time After this initial phase the

motion pro ceeds by inertia This mo del simulates the eect of an intense lo

calized brief heating of the liquid which leads to the nucleation and growth of

a bubble The dimensionless parameters governing the problem are discussed

and their eects illustrated with several examples It is also shown that when



Also Department of Applied Physics Twente Institute of Mechanics and Burgerscentrum

University of Twente AE Enschede The Netherlands

the bubble is not lo cated at the midp oint of the tub e a net ow develops ca

pable of pumping uid from one reservoir to the other The motivation for

this work is oered by the p ossibility to use vapor bubbles as actuators in

uidhandling micro devices

PACS Nos Dz i j

I INTRODUCTION

In an inkjet printer drops are pro duced and ejected due to the rapid growth of bubbles

in a narrow ow passage just upstream of the nozzle exit see eg Ref This is but

one example of the p ossible application of vapor bubbles in small devices where they oer

the interesting p otential of mechanical actuation pumping and other ow eects without

mechanical moving parts In these situations bubbles are pro duced by small heaters in the

tub e wall to which a brief current pulse is applied A very small mass of liquid reaches a

temp erature comparable with the critical temp erature and a vapor bubble nucleates and

rapidly grows In a recent pap er we have studied the thermal and attendant phasechange

pro cesses that take place in such a situation We found that due to the work of expan

sion and vapor condensation on the cold liquid surface the initial high vapor pressure in the

bubble is very quickly dissipated The rest of the bubble evolution essentially pro ceeds by

inertia In that study the uid mechanical asp ects of the problem were treated by means of

a simplied mo del based on p otential ow and an approximate treatment of viscous eects

The purp ose of this article is to simulate the actual ow b ehavior by solving numerically the

full NavierStokes equations We shall however simplify the thermal problem by assuming

that the bubble internal pressure at time is briey raised to a level p ab ove the satura

tion level p corresp onding to the undisturb ed liquid temp erature This high internal

0

overpressure is maintained for a time after which it falls to zero again In another recent

pap er we have compared the results of this simple mo del with the one of Ref nding

a very go o d agreement

The advancing of a gas bubble in a tub e is a classic problem treated by many others

since the early work of Bretherton Ref gives a go o d review and some more recent

work can b e found in Refs The situation considered in this pap er is however dierent

as rst the bubble undergo es a signicant volume change and second the ow is highly

transient as opp osed to the steady or quasisteady situations considered b efore In addition

inertial eects are signicant and therefore we deal with the full NavierStokes equations

rather than the Stokes equations as many of the previous authors The situation considered

in the study of b oiling in p orous media see eg Ref is also quite dierent

from the present one as in that case heat is added to the system continuously from the

entire p ore wall rather than lo cally and impulsively as here Among the earlier work on

the sp ecic problem studied here the closest one is that of Asai et al who however

developed a quasionedimensional mo del and were chiey concerned with drop formation

in the inkjet system rather than with sp ecic uid mechanical phenomena

An interesting and unexp ected result that we nd is that a single bubble cyclically

growing and collapsing away from the midp oint of the channel is capable of inducing a mean

unidirectional ow capable of pumping liquid from one reservoir to the other

I I PRELIMINARY CONSIDERATIONS

We consider a tub e of length L and inner diameter D connecting two liquid reservoirs at

the same pressure P Fig We simplify the problem by assuming axial symmetry and

1

at the initial instant place a small spherical bubble of diameter D on the axis at the center

i

of the tub e The pressure in the bubble is maintained at the level p p for t after

0

which it falls to p for the remainder of the growth and collapse cycle p is the saturated

0 0

vapor pressure at the undisturb ed liquid temp erature

On the basis of a simple conceptual mo del according to which in its growth the bubble

1

L with an acceleration pushes in opp osite directions two equal liquid columns of length

2

1

L we nd that a characteristic velocity scale for the problem is p p P

0 1

2

p p P

0 1

V

1

L

2

where is the duration of the highpressure phase In terms of this characteristic velocity

we dene Reynolds Strouhal and Weber numbers in the usual way by

2

D D V V D

S t W e R e

V

Here and are the density viscosity and surface tension co ecients of the liquid

It should b e noted that the velocity V dened by is of the order of the p eak velocity

during the pro cess and therefore the value of the Reynolds number dened here tends

to underestimate the magnitude of viscous eects In the literature the capillary number

C a W eR e is also frequently encountered but its signicance here is somewhat obscured

by the highly transient nature of the phenomenon investigated

Other parameters characterizing the problem are the ratio of the applied overpressure to

the pressure dierence P p that tends to collapse the bubble

1 0

p

P

P p

1 0

the asp ect ratio of the channel

L

A

D

and the ratio of the initial bubble diameter D to the channel diameter

i

D

i

D

D

Other quantities of interest can b e expressed in terms of the previous parameters For

example the ratio of the viscous diusion time to the capillary wave p erio d is

s

2

D R e

p

T

d

3

D

W e

This quantity which equals the square ro ot of the Laplace number or the inverse of the

Ohnesorge number may also b e interpreted as the order of the number of p erio ds necessary

for viscosity to damp out the longest capillary waves that the system can supp ort

We shall compare the results with those of a simplied mo del developed in earlier work

in which the bubble is approximated as a cylinder o ccupying the entire cross section of

the tub e with some provisions made for surface tension and viscous eects A full

description and a complete numerical solution are presented in If the mo del is further

simplied neglecting surface tension eects interfacial curvature and viscosity it lends itself

to an approximate analysis and a consideration of the results found in this way is useful to

interpret the numerical ones presented b elow For simplicity we only consider the simplied

solution valid provided the bubble surfaces remain suciently far from the tub e ends With

this assumption it can b e shown that the maximum horizontal extension X of the bubble

m

is

2

p p P p

0 1

X

m

1

P p

L

1 0

2

In the context of this simple mo del the bubble is a cylinder and therefore its maximum

1

2

D X or in dimensionless form volume is V

m max

4

P V

max

1

2

A S t D L

4

On the basis of the same simple mo del one nds that the pro cesses of growth and collapse

are nearly the mirror image of each other and that the bubble lifetime t is given by

B

t

B

P

or

P V t

B

L A S t

With the aid of these relations we have the following expression for the ratio of the

viscous diusion length to the tub e diameter

s

p

t P

B

T

v

D S t R e

This parameter can also b e interpreted as the ratio of the viscous diusion time to the

bubble lifetime Another signicant parameter is the ratio of the p erio d of capillary waves

with wavelength of the order of D to the bubble lifetime

s

3

p

S t D

T W e

2

t P

B

This parameter gives an idea of the number of capillary wave p erio ds during the bubble life

I I I THE MODEL

We solve numerically by a metho d briey describ ed later the standard full incompress

ible viscous NavierStokes equations in the interior of the tub e Since only a minute amount

of liquid is heated and only for a brief fraction of the duration of the pro cess we treat the

uid prop erties constant The usual noslip condition is applied at the walls At the bubble

surface the normal stress is taken to equal the internal bubble pressure corrected for surface

tension eects while the tangential stress vanishes The b oundary conditions at the tub e

ends are less straightforward and require some discussion

The simplest option is to imp ose that the pressure at the tub e ends is the undisturb ed

reservoir pressure P This choice can b e justied during the growth phase of the bubble as

1

due to ow separation the uid issues from the tub e ends similarly to a jet Approximating

the ow as parallel and neglecting viscous eects in the ow induced by the jet in its

surroundings we then nd that the pressure at the tub e outlets is indeed approximately

equal to P The accuracy of this prescription during the collapse phase however is harder

1

to justify as the liquid now enters the tub e ends essentially omnidirectionally in a fashion

reminiscent of a sink ow If we approximate the ow as p otential an application of the

Bernoulli integral shows that the pressure at the tub es ends index e is given by

e

2

u P P

e 1

e

t

where is the velocity p otential and u the lo cal velocity assumed uniform over the cross

e e

section In the context of p otential ow it can b e proven that the velocity p otential

e

satises

D u

e e

where the upp er sign is for the left end and the lower sign for the right end of the tub e

With this relation we then nd

u

e

2

u D P P

e 1

e

t

The last term can b e shown to b e negligible on the basis of the following argument Integrate

the momentum equation along the axis of the tub e from the bubble surface to the tub e end

assuming a uniform velocity This pro cedure gives

du P P

e e b

t

dt

where is the length of the liquid column b etween the free surface and the tub e end the

pressure in the liquid at the bubble surface P diers from the bubble internal pressure

b

p b ecause of surface tension Up on using this result to eliminate the liquid acceleration in

0

we nd

D

2

P P u P P

e 1 e b

e

t

In this pap er we only consider situations such that D and we therefore feel justied

in dropping the last term In principle at the end of the collapse if opp osite sides of the

bubble were to collide P would suddenly b ecome very large and this approximation could

b

b e inaccurate We do not include this nal stage of the collapse in the present simulations

In summary then we set P P during bubble growth while during collapse we

e 1

imp ose

2

u P P

e 1

e

The switching b etween the two conditions is done at the moment of maximum bubble expan

sion when the velocity is essentially zero which maintains a desirable degree of continuity

To estimate the relative imp ortance of the second term in it is useful to rewrite this

equation in dimensionless form

2

P P u

e 1

e

P p P p P p

1 0 1 0 1 0

Since p P the rst term in the righthand side is very close to The second term can

0 1

b e written as

2 2

u u P

e e

2

P p AS tV

1 0

Since V is of the order of the maximum velocity and for the cases considered here the com

bination of dimensionless parameters app earing in this equation is not large one would only

exp ect a small contribution from this end correction which is conrmed by the numerical

results presented later in connection with Fig

All the results presented here are computed using a freesurface version of the co de

describ ed in The full incompressible axisymmetric NavierStokes equations are

solved by a pro jection metho d The interface is discretized using a set of marker p oints

linked by cubic splines This description supplies the accurate knowledge of the p osition

and curvature of the interface necessary to include the surface tension contribution The

marker p oints are advected by a simple bilinear interpolation scheme and are redistributed

at every time step to insure a uniform distribution as the bubble deforms Axial symmetry is

enforced in the calculation In the radial direction the integration domain extends b etween

the axis and the wall of the tub e while in the axial direction it includes the entire tub e

length

The interface velocity is calculated by extrap olation from the liquid region which requires

that a sucient number of grid p oints exist b etween the bubble surface and the tub e wall

To avoid an excessive grid renement as this liquid layer gets thinner we arbitrarily set the

outward radial velocity of the interface to zero when it has advanced as far as a few grid

p oints from the tub e wall By varying the number of grid p oints b etween and and by

using dierent grids we have veried that this pro cedure leads only to minute dierences

that are undetectable within the thickness of the lines used in the graphs When the bubble

is as close as this to the tub e wall the radial velocity of the interface is very small anyway

We have also conducted the standard grid renement tests to verify the grid indep endence

of the solutions that will b e presented in the next section Typically we used square cells

with grid p oints in the radial direction

IV RESULTS

Figure shows the bubble shap e at dierent instants of time during growth upp er panel

and collapse In this as well as in all the examples that follow the bubble originates at the

midp oint of the tub e and therefore we only show the right half of the bubble Note that the

tub e extends well b eyond the right b oundary of the gures all the way to z D The

dimensionless parameters have the following values R e S t W e

P A D see Table Here and in the following gures except the last

3

two the duration of the internal overpressure is V L

It may also b e useful to have in mind a concrete physical realization of this system in



terms of dimensional quantities With the physical prop erties of water at C the previous

parameters would corresp ond eg to a mmlong tub e with a diameter of m The

reservoir pressure is P atm the overpressure p is bars and the overpressure

1

duration is s the characteristic velocity V is ms and the initial bubble diameter

m

As mentioned b efore in the practical applications we envisage the initial temp erature of

the bubble surface would b e of the order of the critical temp erature see eg The

corresp onding saturation pressure would b e much higher than the value of bars used here

However in that case the bubble would start from a small nucleus and a large part of the

initial overpressure would b e exp ended in balancing surface tension and in expanding the

nucleus Since here we start already from a macroscopic bubble and we keep the pressure

overpressure constant until it falls to zero rather than allowing it to gradually decay the

value that we use is realistic This p oint was examined in Ref where it was shown that

assuming the pro duct p equal to the time integral of the bubble internal overpressure

resulted in a go o d comparison with the results of a thermally more sophisticated mo del in

which the initial bubble temp erature was higher but the co oling of the vapor was accounted

for

3

In Fig successive proles are separated by V tL dimensionless units

corresp onding for the dimensional values given b efore to s hence the internal pres

sure has already fallen to p b efore the second prole shown

0

The bubble remains approximately spherical until it grows to o ccupy ab out of the

cross section At that p oint preferential growth in the axial direction b egins with a relatively

high curvature that gradually b ecomes more blunt as the ow slows down The curvature

of the end surfaces of the bubble during collapse remains relatively small and the motion

is very nearly onedimensional until the very end The rapid reversal from an oblate to a

prolate shap e b etween the next to the last and the last contours shown is reminiscent of a

similar phenomenon encountered in the collapse of a free initially oblate bubble The

phenomenon is caused by the fact that when inertia dominates the liquid velocity tends to

b e inversely correlated with the lo cal radius of curvature of the interface

The maximum dimensionless volume of the bubble predicted by is in this case

to b e compared with a calculated dimensionless volume of In addition to the ap

proximations inherent in this dierence is due to the fact that the precise bubble shap e

is rounded at the ends which further decreases the volume with resp ect to the cylindrical

shap e assumed in The bubble dimensionless lifetime estimated from is while

the computed one is The dierence b etween the two is enhanced by the fact that the

bigger bubble predicted by the simple mo del takes longer to collapse and has therefore a

longer lifetime The parameter T dened in has the value implying that viscous

v

eects are only signicant over a small fraction of the channel diameter The thickness of

the residual lm near the wall is limited by the numerical considerations describ ed in the

previous section and therefore is not quantitatively signicant here

The capillary number evaluated from the formal relation C a W eR e has the value

However by using directly the denition C a V with the computed timedep endent

velocity one nds values ranging approximately b etween and with most of the time

C a Thus surface tension is considerably less imp ortant than viscosity although

due to the transient nature of the ow the results applicable to the steady case see eg

Ref are not directly relevant here The parameter T dened in equals so

that the longest capillary waves have a p erio d comparable with the bubble characteristic

time which explains the relative bluntness of the bubble ends The parameter T of is

d

ab out implying that capillary waves are only very lightly damp ed by viscosity

A sixfold decrease in the Reynolds number from to Fig pro duces a

bubble ab out shorter with a dimensionless lifetime of Neither eect is contained

in and which are evidently inapplicable to this relatively highviscosity case In

qualitative agreement with the increased value of T the liquid lm now is not limited by

v

the numerics and is thicker than b efore but the other general features of the pro cess are

not signicantly aected The increased damping of capillary waves T is now ab out

d

p ermits a greater curvature of the bubble end surfaces With a larger Reynolds number

one nds that the bubble maximum volume increases but little else changes with resp ect to

the case of Fig The bubble volume vs time is shown in Fig for dierent values of the

Reynolds number Here as well as in Figs and time is made dimentionless

according to

t V t



T

L A S t

The dashed line shows the eect of mo difying the b oundary condition at the tub e ends

discussed in section III by simply imp osing P P during b oth growth and collapse The

1

eect on the bubble dynamics is evidently rather small The maximum volume and the

bubble lifetime are shown vs the Reynolds number in Fig

The bubble shap e is markedly aected by an increase of the Weber number Figure

is for an increase to W e In the rst place the bubble grows longer as the pressure

drop across the interface necessary to balance the eect of surface tension is less Secondly

the liquid lm clearly has a nonuniform thickness as the reduced surface tension decreases

the pressure gradient in the liquid due to curvature Finally the larger value of T allows

for the presence of capillary waves that are particularly evident during the collapse phase

If the Weber number is reduced to Fig the picture is similar to that of Fig for

W e except that due to the capillary pressure drop the bubble do es not grow quite

as long

All other things b eing equal an increase in the Strouhal number means a shorter duration

of the overpressure phase The bubble grows therefore less as implied by and as shown in

Fig for S t The simple expression predicts for this case a bubble dimensionless

volume of against the exact value of For this case T and therefore

capillary waves are prominent on the bubble surface If S t is reduced to Fig the

bubble grows so large that the hypothesis under which the result is derived is violated so

that the predicted dimensionless volume is quite dierent from the actual computed

value of Figure shows the bubble volume vs time for several values of S t According

to the relation the dimensionless bubble lifetime should b e inversely correlated with

S t which is qualitatively veried Quantitatively is only approximately correct for the

larger Strouhal numbers due to the neglect of end eects

Figures and illustrate the eect of increasing and decreasing the pressure parameter

P whose inuence on the volume vs time is shown in Fig The prop ortionality of the

bubble lifetime to P predicted by is well veried provided the bubble do es not grow to o

long

Finally the inuence of the asp ect ratio is illustrated in Figs A and A

When plotted versus z D there is remarkably little dierence b etween the two results

in spite of the large variation in the asp ect ratio This nding is in approximate agreement

with the result that would predict X D P S t indep endent of A The dierence is

max

however in the exp ected direction when the maximum volume is normalized as in the

bubble grows to o ccupy ab out of the channel volume for A but only ab out for

A The bubble lifetime increases by ab out going from A to A

The eect of the bubble initial diameter is small It is found that increasing D D from

i

to only leads to a increase in the maximum volume and a increase in the

bubble lifetime

V SINGLEBUBBLE PUMPING

A very interesting phenomenon is encountered when the bubble grows and collapses away

from the tub e midp oint An example is shown in Fig where the initial bubble nucleus is

p ositioned of the tub e length away from the left end of the tub e Here all the dimensional

parameters have the same values as in Fig except that s As in the previous cases

the area shown in the gure is a small fraction of the computational domain that extends

for z m

It is evident from the gure that the axial p osition where the collapse is completed

m from the tub e left end is dierent from that of the original bubble nucleus m

away from the tub e left end Thus the bubble moves m ie tub e diameters

away from the closest end of the tub e Since the bubble essentially o ccludes the tub e during

most of its life this result suggests that the growth and collapse cycle is capable of imparting

a net displacement to the liquid This fact is conrmed by Fig which shows the liquid

velocities on the tub e axis at the two ends of the tub e vs time for this case It is seen here

that at the end of the collapse there is a residual p ositive net velocity ie directed toward

the end of the tub e farther from the bubble

Thus by generating successive bubbles in a p erio dic fashion one would b e able to pump

the liquid from one reservoir to the other one along the tub e Further considerations on this

interesting and p otentially useful eect can b e found in Ref

Figure presents also other elements of interest In the rst place it is clear that the

direction of motion do es not reverse at the same time at the two ends of the bubble Due

to the greater inertia of the liquid column on the right outward motion in this direction

continues somewhat longer than that on the left Secondly there is a marked dierence in

shap e b etween the two ends of the bubbles during growth and esp ecially during collapse

Due to the dierent inertia of the liquid on the two sides of the bubble the velocity with

which the two interfaces displace is also dierent The velocity of the left interface is rel

atively large and the shap e here has several features reminiscent of those encountered in

Fig where the large value of the Weber number W e also caused a prevailing

of inertia over surface tension The motion of the right bubble interface is instead relatively

slow due to the large inertia of the longer right liquid column Here particularly at the

b eginning of the collapse one encounters a tendency for the bubble surface to detach from

the tub e wall which is reminiscent of other slow collapse cases found b efore eg in Fig

for S t or Fig for P

VI CONCLUSIONS

We have studied the uid dynamics of the growth and collapse of a bubble in a small

tub e under the action of a shortlived internal overpressure In a practical application one

would b e interested in using the bubble growth to achieve mechanical actuation p ossibly

with a high rep etition rate The growth would b e triggered by p owering an electric heater

for a short time to bring the temp erature of a small liquid volume in the vicinity of the

critical temp erature

In this note we have avoided the complexity of phase change and heat diusion by

applying a constant internal overpressure for a short time of the order of of the total

bubble lifetime In spite of this brief duration the expansion is mechanically very energetic

without requiring a large amount of thermal energy It should therefore b e relatively easy to

avoid a progressive increase of the systems temp erature by dissipating heat by conduction

through the tub e walls which is an essential requirement for achieving a fast rep etition rate

As shown in Ref in an actual application the co oling of the bubble o ccurs on a

scale of microseconds b ecause of the work of expansion and the conduction of heat into the

surrounding cold liquid After this time the temp erature of the bubble remains essentially

constant at the ambient value the vapor pressure b ecomes negligible and the entire dynam

ics are governed by inertia This b ehavior is a consequence of the fact that for a liquid such

as water the vapor density at normal temp erature is so low that the latent heat required to

maintain the bubble lled with saturated vapor can b e supplied without altering appreciably

the temp erature of the liquid at the liquidvapor interface The phenomenon is analogous

in normal ow cavitation and the reader may consult eg Ref for more details and a

quantitative argument

Finally a very interesting and somewhat surprising nding of this pap er has b een the

pumping action describ ed at the end of the previous section This eect which we have in

the meantime demonstrated exp erimentally could have useful practical applications and

will b e pursued in future work

ACKNOWLEDGMENT

The p ortion of this work carried out at the Johns Hopkins University was supp orted by

AFOSR under grant F

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Asai A Hara T Endo I Onedimensional mo del of bubble growth and

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spherical cavity in a liquid J Basic Eng

Chikanawa K Yoshida S Hong H Horio H Generation growth and

collapse of a bubble in conductive liquid heated directly by electric current JSME Int

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Giavedoni MD Saita FA The axisymmetric and plane cases of a gas phase

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Giavedoni MD Saita FA The rear meniscus of a long bubble steadily

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Oguz HN Prosp eretti A The natural frequency of oscillation of gas bubbles

in tub es J Acoust Soc Am

Oguz HN Zeng J Axisymmetric and threedimensional b oundary integral

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Olbricht WL Porescale prototypes of multiphase ow in p orous media Ann

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Peng XF Hu HY Wang BX Boiling nucleation during liquid ow in

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ference Hewitt G ed Ind Eng Chem

Plesset MS Prosp eretti A Bubble dynamics and cavitation Ann Rev Fluid

Mech

Popinet S Zaleski S A fronttracking algorithm for accurate representation

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Satik C Yortsos A p orenetwork study of bubble growth in p orous media

driven by heat transfer J Heat Transfer

Scardovelli R Zaleski S Direct numerical simulation of freesurface and in

terfacial ow Ann Rev Fluid Mech

Wilson S K Davis S H Banko S G The unsteady expansion and

contraction of a long twodimensional vapour bubble b etween sup erheated or sub co oled

parallel plates J Fluid Mech

Yuan H Oguz HN Prosp eretti A Growth and collapse of a vapor bubble

in a small tub e Int J Heat Mass Transfer

Yuan H Prosp eretti A The pumping eect of growing and collapsing bubbles

in a tub e J Micromech Microeng

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PT PT

TABLES

Figure P Re We St A T T T

v

d

TABLE I Values of various dimensionless parameters used in the gures of this article

TABLE CAPTION

I Values of various dimensionless parameters used in the gures of this article

FIGURE CAPTIONS

Sketch of the situation mo delled in this pap er The computational domain is limited to

the interior of the tub e and is b ounded by the dashed lines at the tub e ends

Bubble shap e at dierent instants during growth upp er panel and collapse lower

panel here and in the following the last computed shap e in the upp er panel is rep eated in

the lower one The dimensionless parameters have values A R e S t

W e see Table Because of symmetry only the right half of the bubble is shown

Note that the actual computational domain extends b etween z D and z D

and therefore considerably b eyond the left and right frame b oundaries Successive bubble

3

shap es are separated by dimensionless time units

As in Fig except that R e Successive bubble shap es are separated approxi

3

mately by dimensionless time units

Bubble volume vs time for dierent values of the Reynolds number all other parameter

values as in Fig The dashed line diers from the adjacent solid line only in that the

condition P P is applied during b oth the growth and collapse phases

e 1

Bubble lifetime circles left scale and maximum bubble volume squares right scale

vs Reynolds number all other parameter values as in Fig

As in Fig except that W e Successive bubble shap es are separated approxi

3

mately by dimensionless time units

As in Fig except that W e Successive bubble shap es are separated approxi

3

mately by dimensionless time units

As in Fig except that S t Successive bubble shap es are separated approxi

2

mately by dimensionless time units

As in Fig except that S t Successive bubble shap es are separated approximately

3

by dimensionless time units

Bubble volume vs time for dierent values of the Strouhal number all other parameter

values as in Fig

As in Fig except that P Successive bubble shap es are separated approximately

3

by dimensionless time units

As in Fig except that P Successive bubble shap es are separated approxi

3

mately by dimensionless time units

Bubble volume vs time for dierent values of the pressure ratio all other parameter

values as in Fig

As in Fig except that A The computational domain extends b etween z D

3

and z D Successive bubble shap es are separated approximately by dimension

less time units

As in Fig except that A The computational domain extends b etween z D

3

and z D Successive bubble shap es are separated approximately by

dimensionless time units

Successive shap es during growth upp er panel and collapse lower panel of a bubble

initiated at a distance equal to of the tub e length from the leftend of the tub e The

dimensional parameters have the same value as in Fig except that s Note that the

computational domain extends b etween z and z m and therefore considerably

b eyond the left and right frame b oundaries The separation b etween consecutive bubble

shap es is approximately s

Liquid velocity induced by the bubble growth and collapse on the axis of symmetry

at the left solid line and right dashed line ends of the tub e for the case of the previous

gure Note that at the end of the collapse there is a residual velocity away from the left

end of the tub e FIGURES

Tube wall

P8 P8

r Liquid D z D i

L

FIG 0.5

r / D 0

0.5 0 0.5 1 1.5 2 z / D

0.5

r / D 0

0.5 0 0.5 1 1.5 2

z / D

FIG 0.5

r / D 0

0.5 0 0.5 1 1.5 2 z / D

0.5

r / D 0

0.5 0 0.5 1 1.5 2

z / D

FIG Re = ρVD / µ

0.08

0.06 L/4) 2 D π

0.04

bubble volume / ( Re = 408.33 0.02 Re = 306.26 Re = 102.08 Re = 51.04

0 0 0.05 0.1 0.15 0.2

T* = t / (A St τ)

FIG 0.2 0.09

0.19 0.08 L/4) 2 ) D τ π 0.18 0.07

0.17

0.06 0.16 maximum bubble volume bubble lifetime / (A St bubble lifetime 0.05 0.15 maximum bubble volume / (

0.14 0.04 0 100 200 300 400 500

Reynolds number

FIG 0.5

r / D 0

0.5 0 0.5 1 1.5 2 z / D

0.5

r / D 0

0.5 0 0.5 1 1.5 2

z / D

FIG 0.5

r / D 0

0.5 0 0.5 1 1.5 2 z / D

0.5

r / D 0

0.5 0 0.5 1 1.5 2

z / D

FIG 0.5

r / D 0

0.5 0 1 2 3 4 z / D

0.5

r / D 0

0.5 0 1 2 3 4

z / D

FIG 0.5

r / D 0

0.5 0 1 2 3 4 z / D

0.5

r / D 0

0.5 0 1 2 3 4

z / D

FIG St = D / Vτ

0.15

St = 4.78 0.125 St = 9.57 St = 19.13 St = 38.27 0.1 L/4) 2 D π

0.075

0.05 bubble volume / (

0.025

0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

T* = t / (A St τ)

FIG 0.5

r / D 0

0.5 0 1 2 3 4 z / D

0.5

r / D 0

0.5 0 1 2 3 4

z / D

FIG 0.5

r / D 0

0.5 0 1 2 3 4 z / D

0.5

r / D 0

0.5 0 1 2 3 4

z / D

FIG ∆ p / (P∞ − p0)

0.125

∆ p / (P∞ − p0) = 12.5 ∆ 0.1 p / (P∞ − p0) = 25 ∆ p / (P∞ − p0) = 50 ∆ p / (P∞ − p0) = 100 L/4) 2

D 0.075 π

0.05 bubble volume / (

0.025

0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

T* = t / (A St τ)

FIG 0.5

r / D 0

0.5 0 0.5 1 1.5 2 z / D

0.5

r / D 0

0.5 0 0.5 1 1.5 2

z / D

FIG 0.5

r / D 0

0.5 0 0.5 1 1.5 2 z / D

0.5

r / D 0

0.5 0 0.5 1 1.5 2

z / D

FIG 31.25 µ r ( m) 0 31.25 375 500 625 750 875 1000 z (µm)

31.25 µ r ( m) 0 31.25 375 500 625 750 875 1000

z (µm)

FIG

12

10 left end 8 right end

6

4

2

0

−2 velocity (m/s)

−4

−6

−8

−10

−12 0 25 50 75 100 125

time (µs)

FIG