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Chemical Engineering Science 54 (1999) 17591774

Dynamics of drop formation in viscous flowsXiaoguang Zhang

Air Products and Chemicals, Inc., 7201 Hamilton Boulevard, Allentown, PA 18195-1501, USA

Received 22 April 1998; accepted 21 December 1998

Abstract

This paper presents numerical results of the dynamics of a viscous liquid drop that is being formed directly at the tip of a vertical,

circular tube and breaks into an ambient, viscous fluid. A model is developed to predict the evolution of the drop shape and its

breakup based on the volume-of-fluid/continuum-surface-force method, which is a solution algorithm for computing transient,

two-dimensional, incompressible fluid flow with surface tension on free surfaces of general topology (Richards et al. (1995 Physics of

Fluids, 257, 111145)). The full NavierStokes system is solved by using finite-difference formulation on a Eulerian mesh. The mesh is

fixed in space, with the flow and interface moving through it to ensure accurate calculations of complex free surface flows and

topology, including surface breakup and coalescence. The nonlinear dynamics of drop growth and breakup are simulated for

describing and predicting the universal features of drop formation. The focus here is on dynamic effects of a quiescent or flowing

ambient fluid on drop breakup and the subsequent generation of satellite droplets. The effects of finite inertial, capillary, viscous, and

gravitational forces are accounted for in order to classify drastically different formation dynamics and to elucidate the fate of satellite

droplets. The numerical predictions are compared with experimental measurements for a typical system of 2-ethyl-1-hexanol drops

forming and breaking into quiescent water, and the results show excellent agreement. 1999 Elsevier Science Ltd. All rights

reserved.

Keywords: Drops; Satellites; Evolution; Deformation; Breakup; Dynamics; Free-surface flows; Volume-of-fluid; Continuum-surface

force

1. Introduction

A common way of dispersing a liquid in an immiscible

fluid is to flow it continuously through a nozzle or an

orifice plate from which it emerges into the ambient fluid

and breaks into drops. At low flow rates, the liquid being

ejected emanates from the nozzle as discrete drops under

its own weight. At high flow rates, the liquid is ejected

from the nozzle as a jet that subsequently breaks up into

small drops because of well-known Rayleigh instability

(Clift et al., 1978). The formation of liquid drops from

nozzles has long been a topic of interest because of its

occurrence in a wide variety of engineering applications,

such as distillation and extraction processes and spraying

and emulsifying technologies, among others. The pre-

vious applications have demonstrated the importance of

a fundamental understanding of the dynamics of drop

formation in designing and controlling these processes to

obtain certain desired drop characteristics. The dynamicsof a viscous liquid drop forming from a capillary tube

and breaking into a quiescent, ambient fluid that is

inviscid and dynamically inactive has been extensively

investigated in an earlier study by Zhang (1999). The

evolution with time of surface profile and internal flow

of the drop is simulated by using a numerical method

based on algorithms of volume of fluid (VOF) (Hirt and

Nichols, 1981) and continuum surface force (CSF)

(Brackbill et al., 1992). In the study by Zhang (1999), the

effects of finite inertial, capillary, viscous, and gravi-

tational forces are accounted for in order to classify

drastically different formation dynamics and, in particu-

lar, to elucidate the feature of satellite generation. The

present paper follows this earlier work very closely and

extends the results to the situation where a viscous liquid

drips from a circular capillary tube at low flow rates and

breaks up as drops into another immiscible, viscous

liquid. The focus here is on the dynamic effects of the

ambient fluid on drop formation. Despite the fact that

the overall process of drop formation may not be quali-

tatively altered, the dynamic effects of an ambient fluidbring about an additional hydrodynamic force on the

forming drop, which may change the surface profile of

0009-2509/99/$see front matter 1999 Elsevier Science Ltd. All rights reserved.

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the drop and features of drop breakup and satellite

creation. Moreover, an external flow of the ambient fluid

has been found to lead to a smaller-volume breakoff drop

and a longer drop length prior to its breakup (Oguz and

Prosperetti, 1993; Zhang and Stone, 1997).

During liquid dripping from a capillary tube, the vol-

ume of a drop that emerges from the tube increases by the

continuous addition of the drop liquid. When the volumeof the forming drop exceeds a critical value, the drop

necks and a large portion of it starts to fall rapidly and

eventually breaks off from the tube. The most interesting,

yet complicated, phenomena in the drop formation are

the evolution of a liquid thread that connects the detach-

ing drop and the remainder of the liquid on the tube

during the drop breakup and generation of satellite drop-

lets subsequent to the thread breakup. A thorough know-

ledge of the drop evolution and breakup is indispensable

for complete predictions and designs of practical pro-

cesses that involve interfacial contact and flows of two

liquids.

Drop formation from a nozzle or an orifice has been

studied extensively from theoretical and experimental

perspectives. Much of the previous theoretical, as well as

experimental, work has been aimed at predicting the size

of drops breaking from a nozzle as a function of fluid

properties, nozzle geometry, and flow rate of the liquid

through the nozzle (Clift et al., 1978; Kumar and Kuloor,

1970). The theoretical analyses for this purpose have been

based primarily on macroscopic force balances and have

assumed that drop formation occurs in two stages. The

first stage takes place with a pure static growth of thedrop, which ends with a loss of equilibrium of forces. The

second stage corresponds to the necking and breaking of

the drop from the nozzle (Scheele and Meister, 1968;

Heertjes et al., 1971). These studies have concluded that

the volume of the drops that are so formed depends on

not only the nozzle size and liquid properties, as is the

case with static pendant drop (Michael, 1981), but also

the liquid flow rate through the nozzle. Although the

previous studies differ slightly in their approaches to

analyzing the second stage of drop formation, these sim-

plified models can, at best, approximate reality when the

liquid flow rate is vanishingly small and the predictions

of the size of breakoff drops exhibit deviations from

experimental measurements, with errors around 20%

(Clift et al., 1978).

Although the previously cited studies of drop forma-

tion in the dripping mode have captured some of the

gross features of the phenomenon, they have done little

to elucidate the fundamental fluid mechanics of the pro-

cess, viz., the evolution of the shape of the forming drop

in particular, the development, extension, and breakup of

the liquid thread and satellite generation. Complete

simulations of drop formation involve the solution of theNavierStokes system with specified boundary condi-

tions. Cram (1984) and Eggers and Dupont (1994) have

derived and solved one-dimensional equations of mass

and axial momentum conservations to simulate drop

dripping. These approximate equations have been

derived from the NavierStokes system by either

(1) neglecting the radial component of the velocity

and variation of the axial component of the velocity

and pressure in the radial direction (Cram, 1984) or (2)

extending the velocity and pressure variables in a Taylorseries in the radial direction and retaining only

the lowest-order terms in these expansions (Eggers

and Dupont, 1994). Notwithstanding the failure to

profile actual velocity fields observed inside growing

drops and to describe the dynamics of drop breakup

(Schulkes, 1993), these models are surprisingly successful

in predicting the evolution with time of the drop shapes,

as made evident by the qualitative comparison of cal-

culated shapes with experimental observations in a few

specific situations when the liquid flow rate is vanishingly

small.

Qualitative features of the dynamics of the thread

breakup have been explored experimentally by Peregrine

et al. (1990) and, in more detail, by Shi et al. (1994). These

studies show the details of the evolution with time of

a liquid thread at the time preceding, at the instant of,

and at the time following drop detachment. Peregrine et

al. (1990) have documented photographic sequences of

events occurring during breakup of the thread in their

paper. Their results exhibit the process of double break-

age of the liquid thread. Under the weight of a detaching

drop underneath, the thread necks and breaks at its

lower end, where the thread joins with the falling drop, toform a free primary drop. Because of unbalanced capil-

lary forces on the thread after its first breakup, the thread

recoils; secondary breakup then occurs at its upper end,

leading to the generation of satellite droplets. These ex-

perimental observations have been well modeled by

Schulkes (1994), who numerically integrated a potential

flow formulation of Eulers equations for the formation

of inviscid drops. The primary contribution of Shi et al.

(1994) is the demonstration of the breakup of drops with

a large range of viscosities by experiment and computa-

tion, using the one-dimensional model developed by Egg-

ers and Dupont (1994). Shi and his coworkers show that

liquid threads can spawn a series of smaller necks with

ever-thinner diameters prior to breakup. In spite of their

demonstration of these phenomena during drop detach-

ment, the authors do not provide any quantitative evalu-

ation of the process or systematic description of the roles

of operating conditions on the drop formation and

thread breakup. Recently, Zhang and Stone (1997) have

developed a numerical model to simulate the formation

of viscous drops from a capillary tube using a boundary

integral method. These researchers consider the evolu-

tion and breakup of a drop to assess quantitativelythe effects of viscous, buoyant, and capillary forces.

In particular, they reveal the important role of the

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viscous effects and imposed flows of an ambient fluid on

the dynamics of drop deformation and breakup. In the

low-Reynolds-number flow limit, this model cannot

identify the importance of the inertial force, which has

been found to be effective as the flow rate is increased

(Zhang and Basaran, 1995; Zhang, 1999).

It is noteworthy that Richards et al. (1995) have de-

veloped a dynamic simulation method based on theVOF/CSF numerical technique to investigate the full

transient of liquid drop and jet formation from startup to

breakup. In their study, a transition from dripping to

jetting of an emerging liquid is obviously identified when

the liquid flow rate exceeds a critical value. Results of

their numerical simulations show significantly more ac-

curacy than previously simplified analyses in predicting

the jet dynamics, including the jet evolution, velocity

distribution, and volume of breakoff drops. The novel

feature of this method is the use of a Eulerian volume-

tracking approach designed to simulate flows with free

interfaces of arbitrarily complex topology (e.g., merging

and breakup), which are of particular interest to us in our

research. However, Richards and coworkers have fo-

cused, in their paper (Richards et al., 1995), on predicting

the volume of breakoff drops under the conditions cor-

responding to liquid flow rates near and above the

formation of a jet with the Reynolds number exceeding

400. A detailed understanding of liquid dripping into

another immiscible, viscous fluid at the level of Zhang

(1999) is still lacking.

The major goal of this study, along with the sub-

sequent study of numerical solutions, is to remedy theseand related inadequacies so as to complete our under-

standing of drop formation. The transient NavierStokes

equation has been solved for the axisymmetric free-

boundary problem of a Newtonian liquid that is dripping

vertically and breaking as drops into another immiscible

Newtonian liquid. The VOF/CSF-based numerical ap-

proximations used by Richards et al. (1995) have been

extended to simulate the complete process of drop forma-

tion from the time a drop emerges from the tube to its

breakoff from the tube with continuous feeding of the

drop liquid at a certain flow rate. In contrast to most of

the previous studies, the only assumptions involved in

the present numerical model are (1) Newtonian fluids of

drop and ambient phases with constant physical proper-

ties and (2) laminar flows. The special feature of

VOF/CSF, which allows free surfaces to cross the com-

putational mesh smoothly, ensures that the calculations

pass the point of necking followed by natural breakup of

drops without interruption, which is a major incentive

for using the VOF/CSF method in this study. We pay

particular attention to the dynamic effects of an ambient

fluid on breakup of a liquid thread and generation of

satellite droplets. The numerical results are comparedwith the available experimental data, which are obtained

using an ultra-high-speed motion analysis and video

system. Section 2 presents the numerical model, which

includes the problem definition and formulation as well

as a brief discussion of the VOF/CSF algorithm. A more-

detailed description of the VOF/CSF and its validation

on several test problems can be found elsewhere

(Richards, 1994; Hirt and Nichols, 1981; Brackbill et al.,

1992). The approach used to solve the governing equa-

tions and associated boundary conditions is alsodescribed in Section 2. Section 3 briefly describes the

experimental apparatus and methods of data acquisition

and analysis. Typical experiments with 2-ethyl-1-hexanol

(2EH) drops forming in distilled water are performed,

and the resulting data have been compared with the

numerical simulation for different conditions. The com-

putational results and analyses of the findings are

the subject of Section 4. Most of the results are presented

in dimensionless forms and show the importance of

dynamic effects of an ambient fluid and inertial,

viscous, capillary, and gravitational forces on drop

formation. For illustration and verification purposes,

comparisons of numerical simulations and experimental

measurements for typical processes are also presented

in Section 4. The concluding remarks are provided in

Section 5.

2. Mathematic formulation and numerical method

The system of interest is an axisymmetric drop of an

incompressible Newtonian liquid, density and viscosity

, forming into an immiscible, incompressible New-tonian fluid, density#and viscosity, at the tip ofa circular cylindrical capillary tube. The drop liquid is

injected at a constant flow rate, Q, as shown in Fig. 1. The

tube has inner and outer radii, R and R, respectively,

and its axis lies along the direction of the gravity vector,

g; therefore, an axisymmetric free-boundary problem is

imposed. The ambient fluid is contained in a cylindrical

tank that has an inner radius, R, and is coaxial with the

capillary tube. The surface tension,, of the liquidliquidinterface is spatially uniform and constant in time. It is

convenient to define a cylindrical coordinate systemr,z, , whose origin lies at the center of the outlet planeof the tube and where rdenotes the radial coordinate; zis

the axial coordinate measured in the opposite direction

of gravity, g; and is the azimuthal angle. For the

axisymmetric configuration of interest in this study, the

problem is independent of the azimuthal coordinate.

Isothermal, transient flows of liquids of the drop and

ambient phases are governed by the NavierStokes

system

) v"0, (1)

v

t# ) (vv)"!

1

p#

1

) #g#

1

F

, (2)

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Fig. 1. Schematic of a drop forming from a capillary tube and breaking

into an ambient fluid.

where v is the velocity vector, t is time, p is the scalarpressure, andFdenotes body forces that may be present

in the system. The viscous stress tensor, , is defined as

follows for the Newtonian liquid:

"[(v)#(v)]. (3)

The transient NavierStokes system (Eqs. (1) and (2)) is

solved by using a finite-difference formulation on a

Eulerian mesh, which is fixed in space with the flow and

interface moving through it. The basic algorithm is to

break up a time descretization of the momentum equa-

tion into two steps. In the first step, a velocity field is

computed from incremental changes resulting from vis-

cosity, advection, gravitational acceleration, and body

forces. In the second step, the velocity field is projected

onto a zero-divergence vector field, resulting in a single

Poisson equation for the pressure field. The details of the

overall solution scheme can be found in a study by Kothe

and Mjolsness (1992).

A nonconventional approach, referred to as the CSF

method (Brackbill et al. 1992), is used to represent the

effect of the surface tension at free surfaces. It interprets

the surface tension as a continuous, three-dimensional

effect across free surfaces and incorporates it as a localiz-ed volume force in the NavierStokes equation rather

than as a boundary-value condition. The volume force,

which is nonzero only within free surfaces, is given in the

CSF model by

F"F

"F, (4)

where is the surface tension, is the local free surfacecurvature, andF denotes a VOF volume function that is

used to track the profile of free surfaces.Free surfaces are represented using the VOF technique

pioneered by Hirt and Nichols (1981). The VOF method

provides a means of following fluid regions through a

Eulerian mesh of stationary cells and enables a finite-

difference representation of free surfaces that are arbit-

rarily oriented with respect to the computational mesh.

The scalar function, F, is defined as the fractional volume

of the drop fluid in the respective cells of the computa-

tional mesh and is given by

F(x,z,t)"

1, in the drop,

'0,(1, at the free surface,

0, in the ambient fluid.

(5)

The free surface position in the respective cells is divided

from its neighboring cells and is governed by

F

t#(v ))F"0. (6)

This equation states that the volume function, F, moves

with the fluid and provides the information necessary to

reconstruct the free surfaces.The NavierStokes system (Eqs. (1) and (2)) is solved

for drop formation beginning at the instant at which the

free surface of the drop is flat, situated at the tip of the

tube with the entire system being at rest att)0, subject

to the following boundary conditions. The three-phase

contact line, where the drop liquid, the ambient fluid, and

the solid surface meet, remains pinned to the sharp inner

edge of the tube surface for all times t*0:

r

R"1, at

z

R"0. (7)

In contrast to liquidgas systems, when a liquid drop

forms from a tube into another immiscible liquid (the

focus of the present paper), whether the contact line pins

on the inner edge or the outer edge depends on the

relative wettabilities of the two liquids on the flat surface

of the tube (Berg, 1993). For simplicity, the contact line is

assumed to pin to the inner edge of the tube and a tube

having a wall sufficiently thin is used in this study. Never-

theless, the simulations can also be applied for systems in

which the contact line pins to the outer edge of the tube

as long as the tube wall is sufficiently thin that its effectscan be neglected (Zhang, 1999). The fixed contact line

eliminates the troublesome determination of the contact


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angle in this problem. Far upstream of the tube outlet,

the flow inside the tube is fully developed,

v"0, v

"2

Q

R 1!r

R

, (8)as

z

RP!R, for 0)

r

R)1,

where v

and v

are the radial and axial components of

the velocity of the drop liquid, respectively. In computa-

tions reported in this paper, the tube length has been set

to be twice of the inner radius of the tube. Greater lengths

have been tested, and the calculated results have shown

no noticeable difference. The ambient fluid may be qui-

escent or may be set in motion by continuously supplying

the ambient fluid at a constant flow rate,Q. In the latter

case, a fully developed flow in the annulus far upstream of

the outer container is imposed (Bird et al., 1960),

v"0,

v"2

Q

(R!R

)

1!r

R#

1!(R

/R)

ln(R/R

)

ln(r/R), (9)

as z

RP!R, for

R

R)

r

R)

R

R,

wherev

and v

are the radial and axial components of

the velocity of the ambient fluid, respectively. Along

surfaces where the liquid and solid come in contact, theliquid obeys the conditions of no slip and no penetration:

v"0, at z

R"0, for 1)

r

R)

R

R, (10)

at r

R"1 and,

r

R"

R

R, for!R(

z

R)0,

at r

R"

R

R, for !R(

z

R(R.

A continuative outflow boundary condition is used at

the top of the computational domain. The length in#zdirection is set to be sufficiently longer than the estimated

drop breakoff length (i.e., the length of a drop at instant

of its breakup). Greater lengths have been tested, and the

calculated results have shown no noticeable difference in

features of drop breakup and satellite generation.

Nondimensionalizing the governing NavierStokes

system and boundary conditions, using R as the length

scale and the average velocity of liquid inside the tube

u"Q/Ras the velocity scale, yields three dimension-

less parameters that describe the fluid mechanics of drop

formation:

Re"u

R

, Ca"

u

and G"

Rg

.

The Reynolds number, Re, measures the importance of

inertial forces relative to viscous forces; the capillary

number, Ca, measures the importance of viscous forces

relative to surface tension forces; and the gravitational

Bond number, G, measures the importance of gravi-

tational forces relative to surface tension forces. The

viscosity ratio of the drop liquid and ambient fluid, , is

introduced to account for the dynamic effects of theambient fluid. The wall effects of the ambient fluid con-

tainer are described by the ratio of the inner radii of the

container and the capillary tube,R/R. If an axial flow is

imposed in the ambient fluid surrounding the capillary

tube with a average velocity u"Q

/(R

!R

), a di

mensionless characteristic velocity,u/u

, enters the prob-

lem description.

The numerical solutions for the axisymmetric free-

boundary problem of drop formation are closely fol-

lowed the previous studies (Richards, 1994; Zhang, 1999).

For a comprehensive description and analysis of the

numerical method, the reader is referred to Richards

(1994). The VOF/CSF algorithms are used to solve the

governing NavierStokes system and the boundary con-

ditions on a Eulerian rectangular, staggered mesh in

cylindrical geometry. In order to achieve large cell-wise

resolution of the free surface change and high computa-

tional efficiency, a finer local mesh is used around the

drop region, whereas a coarser mesh is used in the ambi-

ent region. The complete process of drop formation is

simulated from the time a drop emerges from the tube

until its detachment from the tube, with continuous feed-

ing of the drop liquid at a specified flow rate.

3. Experimental approach

The experiments have been designed to obtain quanti-

tative information on the dynamics of drop formation of

a typical liquidliquid system of 2-ethyl-1-hexanol (2EH)

drops forming and breaking into distilled water in order

to compare the measurements with and verify the predic-

tions of the numerical model presented earlier. In the

experiments, attention is particularly paid to the evolu-

tion with time of the shape of a growing drop as it necks

and then breaks up and also to the creation of satellite

droplets subsequent to drop breakup.

The apparatus used to form drops has been described

elsewhere (Zhang and Basaran, 1995). It consists essen-

tially of a fine-capillary tube through which the liquid

used to form drops is delivered at a constant volumetric

flow rate by means of a liquid syringe pump (ATI Orion

M361). A liquid drop is formed at or emerges directly

from the tip of the tube. The capillary tubes used in the

experiments are chosen to have sufficiently large ratios of

inner and outer radii, with R/R'0.8 that the walleffects of the tubes can be neglected (cf. Zhang, 1999). The

capillary tubes are submerged in a water container


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Fig. 2. Time sequence of shapes of a 2EH drop forming in quiescent water from a tube ofR"0.16 cm at the liquid flow rateQ"5 ml/min, wheret is

the time of the drop formation measured from the instant that the drop emerges from the tube.

having a sufficiently large inner radius as compared with

the tube radius with their ratio R/R"62.5 so that the

wall effects of the container can be neglected.

An ultra-high-speed video camera by Kodak (Ektapro

Electronic Memory Motion Analyzer Model EM1012),

which is used for continuously capturing images of the

drop formation process, and the associated hardware for

recording, storing, and analyzing drop-shape data areessential to the experimental study. The camera system is

composed of an intensified imager, which can record

1000 full images or 12,000 partial images per second and

allows rapid and accurate determination of the loci of

instantaneous interface profiles from which the various

measures of drops are evaluated. The entire apparatus is

placed on a vibration isolation table from Newport.

The drop liquid is 2-ethyl-1-hexanol with the viscosity

and density of 0.089 g cm sand 0.83 g cm, respec-

tively; and the ambient liquid is distilled water which

has a viscosity and density of 0.01 g cm s and

1.0 g cm, respectively. The interfacial tension is meas-

ured to be 13.2 g s(Harris and Byers, 1989). All experi-

ments are performed at room temperature of 22$0.5C.

In a typical run, a steady flow with a desired rate is

established through the capillary and the ambient water

is quiescent. The water and 2EH are mutually saturated

prior to each experiment. The system is allowed to run

for about 5 mins before measurements are taken. A peri-

odic flow situation is then reached in which the drops

form, grow, and detach from the outlet of the capillary.

Since, in this dripping regime, primary and satellite drops

of uniform size are continuously created, this techniqueprovides a reliable and repeatable illustration of the

dynamics of drop formation. Reproducibility of results

for the drop shape and volume has been found to be

within 5% by making measurements under the same

conditions, but at different times.

4. Results and discussion

This section presents the results of an investigation of

all major effects governing the dynamics of drop forma-

tion, which are represented by Re, Ca, G, , R/R, and

u/u

. Calculations are performed by systematically vary-

ing one parameter while keeping the others fixed. The

quantitative results to be reported have thus been made

over wide ranges of the governing parameters to provide

insight into the dynamics and to classify drastically

different formation processes. In the nature of its im-

portance and complexity of drop breakup and satellite

generation, studies directed towards parameter correla-tions and/or maps to characterize satellite generation are

underway in our laboratory. Whereas it is straightfor-

ward to vary any one of the dimensionless governing

parameters while holding all others fixed in numerical

modeling, it is not possible to do so in laboratory experi-

ments. Therefore, a typical case of 2EH drop formation

in quiescent water is considered for comparative and

verification purposes. The experimental investigation for

drop formation has been performed by varying the vol-

umetric flow rate, Q, to cover certain ranges of the

Reynolds number and the capillary number: that is, the

Reynolds number varies as 10(Re(70, the capillary

number varies as 710(Ca(410, and the

gravitational Bond number and viscosity ratio are fixed

atG"0.32 and"8.9, respectively. Besides the surfaceprofile of forming drops, two measurements are used to

characterize the features of drop evolution and breakup:

the maximum length which a drop can reach prior to its

breakup, , measured along the tube axis from the tip of

the tube to the apex of the drop (see Fig. 1); and the

volume of a primary (or breakoff ) drop, . As shown in

Section 4.1, the excellent results obtained by comparisons

of numerical simulations and experimental measure-ments give confidence as to the accuracy of the numerical

model and its predictions.

4.1. Typical case

Fig. 2 shows a time sequence of interface profiles ob-

tained by the numerical model (presented by the curved

images on the right side of each interface profile) and

experiments (the black images on the left side) for a 2EH

drop forming in quiescent, ambient water from a tube of

radius R"0.16 cm, at a constant liquid flow rate of

Q"5 ml/min. As the volume of the drop increases


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successively with the continuous addition of the drop

liquid, the drop rises on the tip of the tube. The geometry

of the drop gradually transforms from nearly spherical to

pear shaped during its growth period. As the drop vol-

ume becomes sufficiently large that the buoyancy force

on the drop exceeds that of the surface tension (at

t'1.21 s), the drop necks rapidly and a large portion

starts to detach irreversibly from the tube. During thenecking sequence, the drop elongates rapidly and con-

tracts in the middle to form a liquid thread. The liquid

thread, which connects the detaching drop and the re-

mainder of the liquid on the tube, is stretched along with

rapid rise of the detaching drop and eventually breaks at

its upper end at t"1.2712 s, resulting in a free primary

drop. Immediately after the thread breaks at the upper

end, the newly freed end of the thread recoils due to the

unbalanced force of the surface tension. Meanwhile, the

cone-shaped liquid remaining on the tube tends to re-

trieve its apex and become spherical because of the capil-

lary force. A large curvature, therefore, develops at the

joining point of the liquid cone and the thread, leading to

a large capillary pressure at and high velocities of the

liquid out of that point (Zhang, 1999). Then, in a similar

fashion to the initial breakup, the thread breaks again,

this time at its lower end at t"1.275 s. As a result of the

thread rupturing at both ends sequentially, a satellite

droplet is created, lying between the primary drop and

the cone-shaped liquid remaining on the tube. In fact, the

satellite droplet is found to be very small, typically hav-

ing a volume less than 1% that of the primary drop.

Comparison of the numerical prediction with the experi-mental measurements shows excellent agreement. The

numerical model accurately predicts the evolution of the

interface profile of the drop, particularly, the double

breakage of a liquid thread and generation of a satellite

droplet subsequent to the thread breakup.

The dynamic response of drop formation is especially

notable during the period of drop breakup. Fig. 3 shows

a summary of breakup features in terms of the maximum

lengths that a drop can attain at the instant when the

primary drop is about to detach (or the limiting lengths),

, and the volume of the primary drop,, as functions of

the flow rate,Q. The drop lengths and volumes presented

in Fig. 3 are obtained for 2EH drops forming in water

from a tube of the inner radius, R"0.16 cm. Three

interface profiles of drops at the instant they are about to

break are included in Fig. 3 for flow rates ofQ"5, 10,

and 15 ml/min. The corresponding experimental results

of /R, /R, and photographic profiles of drops are

also shown in Fig. 3 by points and black images on the

left, respectively. Again, excellent agreement between the

numerical simulation and experimental measurements

has been demonstrated in Fig. 3. As the flow rate in-

creases, the pressure at the tube exit and the mass (orvolume) of liquid accumulating in the cone directly above

the tube rise rapidly, due in part to the large axial

Fig. 3. Dimensionless limiting length and breakoff volume of 2EH

drops forming into quiescent water as a function of the flow rate from

a tube ofR"0.16 cm.

momentum of the entering liquid and also to the resist-

ance that the thread exerts on the large amount of liquid

exiting the tube during the necking and drop detachment

steps. As a consequence of the pressure and mass buil-

dup, a larger elongation of the liquid cone occurs as

Q increases. As shown in the inserted profiles in Fig. 3,the increased deformation of the liquid cone retards, or

even abolishes, development of the liquid thread at the

larger Q, a determining factor that changes the features of

thread breakup and satellite generation. The behavior of

thread development and breakup is so interesting and

important in drop formation, particularly in the creation

of satellite droplets, these features will be discussed fur-

ther in a separated part. Fig. 3 also shows that the

breakoff volume of a drop increases with the flow rate,Q.

The rate of increase in the breakoff volume of a drop

decreases and tends to level off as Q becomes sufficiently

large. Indeed, as the flow rate continues to increase, over

a critical value, transition to jetting occurs and then the

volume of a primary drop decreases (Richards et al.,

1995). Moreover, in spite of the fact that the elongated

liquid cone and enlarged detaching drop may contribute

to the increase in the limiting length of a drop, the results

in Fig. 3 indicate only a slight increase in the limiting

length as Q increases. The primary reason for the weak

dependence of the limiting length on the flow rate is the

oblate deformation of the detaching drop caused by the

viscous effects of the ambient fluid. Figure 3 depicts

clearly that the additional deformation or flattening ofthe top of a detaching drop, brought about by the ambi-

ent fluid, increases as Q increases because of the increased


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Fig. 4. Variation of interaction time with the flow rate for 2EH drops

forming in quiescent water from a tube ofR"0.16 cm. The times for

the three images in inserts ofQ"5 ml/min are t"1.271, 1.2748, and

1.277 s, respectively. The times for the three images in inserts ofQ"10

ml/min are t"0.643, 0.646, and 0.65 s, respectively.

Fig. 5. Dimensionless limiting length and breakoff volume of drops as

a function of the Reynolds number at Ca"0.01, G"1, "1, andu/u

"0.

forming rate of the drop; therefore, it tends to counteract

the elongation of the liquid cone.

As a result of the double breakage of a liquid thread,

a satellite droplet is predicted by the numerical model

and is observed in experiments to be created under cer-

tain operating conditions (see Fig. 2). Detailed examina-

tion of the thread breakage and satellite creation reveals

the importance of deformation of the liquid cone that issessile on the tube and joins the lower end of the thread.

A quantitative measurement is defined to describe the

breaking feature of the liquid thread as the time interval

that elapses between the breakup of the two ends of the

thread, commonly referred to as interaction time. Fig.

4 shows the variation in the time interval, t, with theflow rate obtained by the numerical model (curve) and

experiments (points) for 2EH drops forming into water

from a tube of R"0.16 cm. Two series of interface

profiles of breaking drops, one obtained from the model

(the right side) and the other from experiments (the left

side), are inserted in Fig. 4 for flow rates ofQ"5 and 10

ml/min. These inserted profiles demonstrate typical situ-

ations of drop formation with and without satellite

generation. Evidently, as the flow rate increases, the in-

creased axial momentum of the entering liquid causes the

liquid cone to sustain a large axial elongation. It is more

difficult for the surface tension force to restore the cone-

shaped liquid mass to near-spherical form when compet-

ing with increasing inertial force and then to develop

a localized, large curvature at the point where the cone

joins with the thread. Therefore, more time is required for

the secondary rupture of the thread at the joining point.As shown in Fig. 4, the interaction time, t, increaseswith elevated flow rate. When the flow rate is sufficiently

high, the secondary breakup of the thread can no longer

occur before the thread recoils and coalesces with the

liquid cone. In that case, no satellite droplet is created

and the interaction time becomes infinity, as indicated in

Fig. 4. Apparently, whether or not double breakage of

a thread occurs and a satellite droplet is created are

determined substantially by the relative importance of

the inertial and capillary forces, which can be measured

by a Weber number,e"ReCa. For the present study

of the 2EH drops forming into water at different flow

rates, the results from numerical model and experi-ments show that the threshold of the flow rate is

8.5$0.5 ml/min, which corresponds to a critical Weber

number of 0.031.

4.2. Effect of the Reynolds number

Fig. 5 demonstrates computational predictions of the

variation of the dimensionless limiting length, /R, and

the volume of a breakoff drop, /R, as a function of

the Reynolds number, while holding other governing

parameters fixed as Ca"0.01, G"1, and "1. Fornumerical results shown in Sections 4.24.5, a quiescent

ambient fluid is considered with a outer container of the

fixed size ofR/R"3. An obviously smaller value ofR

/R

has been used in numerical calculations in comparison

with that in experiments in order to attain a high com-

putational efficiency. In fact, as shown in Section 4.6, the

wall effects of an ambient fluid container can be neglected

as R/R*2.5. Moreover, the boundary condition of

rigid free slip is used in calculations shown in Sections

4.24.5 for the container wall to further minimize the wall

effects. The effects of axial flow of the ambient fluid are

shown in Section 4.7. Four profiles of interface shapes of


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Fig. 6. Evolution with time of interface profiles of detaching drops at

Reynolds numbers of (a) Re"1 and (b) Re"10, with Ca"0.01,

G"1,"1, and u/u

"0.

drops at the instant they are about to detach are inserted

in Fig. 5 for Re"0.1, 1, 10, and 100. As expected, the

limiting length, , of a detaching drop increases mono-

tonically as the Reynolds number increases, due prim-

arily to the increase in the inertial force to elongate the

liquid cone remaining on the tube (see Fig. 5). Moreover,

the larger axial momentum of the liquid leaving the tube

forces more liquid to squeeze into the drop and causes

the volume of the detaching primary drop to increase as

Reincreases. At a certain criticalRe, transition to jetting

occurs, at which time the volume of the primary drop

attains its maximum, as shown in Fig. 5.

As the Reynolds number increases, the dynamic effects

of the ambient fluid becomes pronounced on a detaching

drop because of the increased speed of the drop rising in

the medium. The increased tangential viscous stress on

a detaching drop from the ambient fluid causes the drop

to change its shape from prolate (at Re)1) to oblate(at Re&10) and, then, to dimpled ellipsoidal

(at Re*100), as shown in inserted drop profiles in Fig. 5.

Fig. 6 demonstrates breakup features of drops at

Reynolds numbers ofRe"1 and 10, while holding other

governing parameters fixed as Ca"0.01, G"1, "1,and u

/u

"0. At low Reynolds numbers (Fig. 6a), the

liquid thread ruptures at both ends sequentially because

of the stretching under the weight of the detaching pri-

mary drop and development of large local mean curva-

tures at the joining points of the thread with the twobodies of liquid. Peaks of capillary pressure are then

induced inside the thread at its two ends and force the

liquid out of these breaking points, accelerating thread

rupture (see also Zhang, 1999). Consequent to the double

breakage of the thread, a satellite drop is generated. In

contrast, at the large Reynolds number (Fig. 6b), the

liquid thread is no longer obvious as a result of the

increasing elongation of the liquid cone on the tube

during the drop necking sequence. Following breakoff of

the primary drop, the tip of the liquid cone rebounds and

forms a bulbous head under unbalanced capillary forces.

Eventually, as shown in Fig. 6b, the bulbous head merges

into the liquid cone and a new drop emerges. No satellite

droplet is generated at this large Reynolds number.

4.3. Effect of the capillary number

The effects of the capillary number on drop breakup, in

terms of the dimensionless limiting length, /R, and

detaching volume of the drop, /R, are shown in Fig.

7 with other parameters fixed at Re"1, G"1, "1,and u

/u

"0. Four drop profiles at the instant of detach-

ment are inserted for Ca"0.001, 0.01, 0.05, and 0.1.Similar to the cases of drop forming into air (Zhang,

1999), the limiting length, , and the volume of the

primary drop,, are shown to increase monotonically as

Ca increases.

Noteworthy, as the capillary number becomes suffi-

ciently large (such asCa"0.1 in Fig. 7), the capillary (or

surface tension) force cannot restore the detaching pri-

mary drop to spherical shape and, therefore, a large

curvature can no longer be developed at the joining point

of the detaching drop and the liquid thread. Moreover,

the increased viscous force relative to the capillary force

holds a relatively long liquid thread prior to its rupture.

Therefore, in contrast to drop formation at small capil-

lary numbers, interface disturbances may cause the

thread to rupture at its middle portion prior to the

capillary-driven breakup of the thread at its upper end

joining the detaching primary drop, as shown in Fig. 8

for a series of interface profiles of a breaking drop under

conditions ofCa"0.1 and all other conditions the same

as those in Fig. 7. As a result of the sequential rupture of

the thread at its top portion, a small satellite droplet is

created. The rest of the thread then recoils rapidly and

merges into the liquid cone. A similar breakage feature ofa drop of a highly viscous liquid has been observed in

experiments with glycerol drops (Zhang and Basaran,


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a function of the capillary number at Re"1, G"1, "1, andu/u

"0.


Re"1, Ca"0.1,G"1,"1, and u/u

"0.

1995). At intermediate range of the capillary number, the

effects of the viscous force is no longer sufficiently great

to cause a long liquid thread, and the surface tension

force is still small relative to the inertial force, leading to

drop formation at a small Weber number. Similar to the

situation of large Reynolds numbers, the development of

a thread is suppressed by the large elongation of the

liquid cone on the tube, and obviously, the double break-

age of the thread and satellite generation will no longeroccur (cf. Fig. 6b). At a very small capillary number, the

capillary force becomes dominant and causes the double


a function of the gravitational Bond number at Re"1, Ca"0.01,

"1, and u/u

"0.

breakage of a thread at its two ends and subsequent

satellite generation (cf. Fig 6a).

4.4. Effect of the gravitational Bond number

The gravitational Bond number, G, has been varied in

calculations by changing the magnitude of the density

difference between the two phases,. Fig. 9 shows thevariation of the dimensionless volume of a primary drop,

/R, and the limiting length of the drop at the instant of

its detachment, /R, as functions of the gravitational

Bond number with the other parameters fixed at Re"1,

Ca"0.01,"1, and u/u

"0. In Fig. 9, instantaneous

interface profiles of detaching drops are inserted for

G"0.1, 1, 3, and 10. Fig. 9 shows monotonical decrease

in the volume of the detaching drop and the limiting

length with increasing values of G. In particular, the

volume of the primary drop is shown to be a strong

function of the magnitude of the gravitational force, orG,

as G(2. The rate of decrease in the breakoff volume

with increasing G falls and tends to level off asG becomes

sufficiently large and the volume of the primary drop is

sufficiently small, indicating the difficulty of reducing the

volume of breakoff drops. Because of the large driving

force presented to pull the liquid out of the tube at a large

value of G, the drop is stretched immediately after its

emergence from the tube and breaks rapidly. The large

pulling effect of gravity significantly reduces the volumeof liquid remaining on the tube during drop breakoff, and

the development of a liquid thread is no longer as evident


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Fig. 10. Dimensionless limiting length and breakoff volume of drops asa function of the viscosity ratio at Re"1, Ca"0.01, G"1, and

u/u

"0.

as with smallGvalues. Obviously, at such large values of

G, the double breakage of the thread and satellite genera-

tion cannot be expected, as predicted by the numerical

model. As a result of increasing velocity of drop de-

tachment at large G, the increasing viscous force of the

ambient fluid causes the detaching drop to deform to

a dimpled ellipsoidal shape; a similar phenomenon

occurred as Re is increased.

4.5. Effect of the viscosity ratio of two phases

Despite the fact that liquid viscosities of drop and

continuous phases have little influence on the volume of

a breakoff drop (Kumar and Kuloor, 1970), the viscous

force has been found to affect drop necking and breakup

greatly, particularly the development, extension, and

breakup of a liquid thread and generation of satellite

droplets, as briefly discussed in Section 4.3. To address

the viscous effects on drop formation, the viscosity ratio

of the drop liquid to the ambient fluid, , is used and hasbeen changed by varying the viscosity of the drop liquid

alone in present numerical simulations. Fig. 10 shows the

variation of dimensionless breakoff volume, /R, and

limiting length, /R, as a function of with other gov-

erning parameters fixed atRe"1,Ca"0.01,G"1, and

u/u

"0. Instantaneous interface shapes when drops are

about to break are also displayed in Fig. 10 for"0.01,1, 5, and 10. As the viscosity ratio (or the viscosity of the

drop liquid) increases, the dimensionless limiting length,

/R, increases substantially due to the significant in-

crease in lengths of the liquid thread and cone, as shown

in the inserted drop profiles in Fig. 10. Viscosity of the

drop liquid plays a great role in stabilizing the drop

interface, which makes possible greater thread elongation

and extension by damping, and even eliminating, oscilla-

tions of the interface. As a result of the increased exten-

sion of the thread, a longer elapsed time is required for

drop breakup and, thus, more liquid is retained in the

outlet of the tube, resulting in a longer liquid cone on thetube. In spite of its weak dependence on the viscosity

ratio, , the volume of breakoff drops varies with ina peculiar wayit exhibits a maximum in the neighbor-

hood of"1, as shown in Fig. 10, for the ranges ofRe,Ca, andG studied. This feature of nonmonotonical vari-

ation of the volume of breakoff drops with the liquid

viscosity is similar to that observed in early experiments

(Kumar and Kuloor, 1970).

Along with the increased elongation and extension of

a liquid thread at large viscosity ratio, the creation of

satellite droplets subsequent to the thread break is


viscosity ratios of (a)"0.01 and (b) "5, with Re"1, Ca"0.01,G"1, and u

/u

"0.


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expected to be interesting and complicated. Fig. 11 shows

two series of interface profiles of drops. Each takes a close

look at thread breakup and creation of satellite droplets,

with viscosity ratios of "0.01 and 5 and all otherconditions the same as those in Fig. 10. Apparently, the

stabilizing effects of the larger viscosity of the drop liquid

(or the viscosity ratio) leads to the production of a longer,

thicker liquid thread, as shown in Fig. 11b. Immediatelyafter its first breakup at the upper end, the thread recoils

under the unbalanced capillary force to form a round

knob at its top; this leads to the formation of a main,

large satellite droplet following the secondary breakup of

the thread. As a result of the large viscous force at the

large viscosity ratio, a noticeably large interaction time

(or the time interval between the two breakups), t, isobserved, namely that t"14 milliseconds for "5(Fig. 11b) as compared witht"1.6 milliseconds at thesmall viscosity ratio of"0.01 (Fig. 11a). Therefore, asshown in Fig. 11b, waves have sufficient time to develop

and propagate on the thread surface. Along with the

thread recoil and the secondary breakage, these waves

grow and eventually break the thread when, at certain

points, the amplitude of the waves is compatible with the

radius of the thread. This leads to the creation of a sub-

satellite droplet that has a much smaller size and lies

between the main satellite and the cone-shaped liquid

remaining on the tube. In contrast, the development and

extension of a liquid thread are no longer so evident at

the small viscosity ratio (Fig. 11a). As a result, a consider-

ably smaller satellite droplet is created subsequent to

double breakage of the thread. Thus, we expect the cre-ation of satellite droplets to cease or to be unmeasurable

as the viscosity ratio becomes extremely small, for

example, in the case of bubble generation in liquids (cf.

Oguz and Prosperetti, 1993).

4.6. Wall effects of the outer container

When drops are formed in a bounded medium, the

dynamics of their deformation and breakup is affected by

the wall of the ambient fluid container. Wall effects have

always been present, to a greater or lesser extent, as

a result of disturbances of the ambient fluid by forming

drops. Fig. 12 depicts computational results of the wall

effects on drop formation, which are described by the

variation of dimensionless breakoff volume, /R, and

limiting length, /R, of a drop as a function of the ratio

of the inner radius of the container to that of the capillary

tube,R/R. The results in Fig. 12 are obtained by holding

other governing parameters fixed, with Re"1,

Ca"0.01, G"1, "1, u/u

"0, and the boundary

condition of rigid no slip for the container wall. Four

interface profiles of detaching drops are inserted in

Fig. 12 for R/R"1.35, 1.4, 2, and 4. Quantitativemeasurements of /R and

/R in Fig. 12 indicate

clearly that the wall effects become significant and the


a function of the dimensionless inner radius of the ambient fluid

container at Re"1,Ca"0.01,G"1,"1, and u/u

"0.

container radius, R, becomes a determining factor gov-

erning the deformation and breakup of a forming drop at

small values ofR/R. The presence of the wall causes an

increased deformation of a detaching drop, with elonga-tion occurring in the axial direction to yield an approx-

imately prolate ellipsoid shape (see Fig. 12). At a value of

R/R sufficiently small that the drop size is compatible

with the outer boundary, the drop fills most of the con-

tainer cross section and a slug flow regime results.

Apparently, the increased viscous effects of the ambient

fluid delay drop breakup considerably, resulting in a sig-

nificant increase in the breakoff volume of a drop asR/R

decreases. As a result of increased volume of the detach-

ing drop and its axial elongation, the limiting length of

the drop increases dramatically when the slug flow oc-

curs atR/R(1.5, as shown in Fig. 12. It is expected that

as the distance between the wall and a drop, or R/R,

increases, the wall effects will decrease. Beyond a certain

value of R/R so that the drop no longer sees the

container wall, the wall effects become negligible. As

illustrated in Fig. 12, when the radius ratio R/Rexceeds

a value of about 2.5, the variation of breakoff volume and

limiting length of the drop with R/R becomes virtually

unmeasurable, indicating negligible effects of the con-

tainer wall.

Besides the change in shape of a detaching drop with

R/R, the inserted interfacial profiles of breaking drops inFig. 12 indicate clearly that the shape of the liquid cone

remaining on the tube varies as well. As R/Rdecreases,


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Re"1, Ca"0.1,G"1,"1, R/R"1.35, and u

/u

"0.

the increased viscous effects delay the drop breakup, in

a manner similar to that of increasing (discussed in

Section 4.5). Therefore, more liquid is accumulated in theliquid cone, causing the cone shape to change from con-

cave to convex prior to drop breakup. Fig. 13 depicts this

breakup feature of a drop at the slug flow regime with

R/R"1.35 and all other conditions the same as those in

Fig. 12. Unlike drops forming into an unbounded me-

dium (cf. Fig. 6a), no measurable thread is developed

between the breakoff drop and the liquid cone and hence

the drop breaks sharply at its middle portion, as the

result of the accumulation of a large volume of liquid in

and convexity of the liquid cone at such a small value of

R/R. Obviously, in such a situation, a satellite droplet

can no longer be created subsequent to the drop breakup.Moreover, a comparison of the breakup features of drops

within a tightly bounded medium (Fig. 13) with those in

an infinite medium (Fig. 6a) indicates clearly that the

breakup time of the drop, along with its breakoff volume

and limiting length, is significantly increased by the wall

effects as R/R decreases.

4.7. Effect of the externalflow

Flowing an ambient fluid concurrently with the dis-

persed phase has been used as an effective and practicalmethod for reducing the volume (or size) of breakoff

drops by accelerating drop detachment with an

Fig. 14. Evolution with time of the dimensionless length of drops

forming in an ambient fluid which flows at different characteristic

velocities ofu/u

"0, 50, and 200 at Re"1, Ca"0.1, G"1, "1,

and R/R"2.

additional viscous shear force (Clift et al., 1978). Besides,

an external uniform flow is found to vary the interface

profile and breaking length of a drop during drop forma-tion considerably at infinitesimal inertial effects (Zhang

and Stone, 1997). Fig. 14 presents the entire history of the

formation of three drops that are each subjected to an

external flow of different velocities. The drop evolution is

presented quantitatively in terms of the drop length, ,

nondimensionalized by the inner radius of the tube, R.

The results shown in Fig. 14 are calculated for the three

situations of drop forming in an ambient fluid for which

fully developed flows are applied concurrently at differ-

ent average velocities of u/u

"0, 50, and 200, respec-

tively. Other controlling parameters are held constant:

Re"1, Ca"0.01, G"1, "1, R/R"2, and R

/R

"1.1 (see Fig. 1). Evidently, the external flow of the

ambient fluid plays an important role in drop evolution

and breakup, as clearly depicted by the two series of

instantaneous profiles of drops at u/u

"0 (the lower

series) and 200 (the upper series), respectively, in Fig. 14.

When injected into a container with a specified flow rate

(or average velocity), the ambient fluid interacts with

a forming drop and brings about an additional hy-

drodynamic force on it. Initially, at the growing stage, the

drop of small volume grows and rises slowly at the tip of

the tube. The entering ambient fluid, if possessing a suffi-ciently high velocity as shown in Fig. 14 with u

/u

"200,

passes over the drop rapidly and expands, resulting in


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a reduction in its velocity and, consequently, an increase

in the ambient pressure (Batchelor, 1967). As a result of

the large pressure on its top, the drop deforms and

flattens at its tip; then the drop length, , is measured to

be shorter than that in quiescent or slowly flowing me-

dium, as shown in Fig. 14 at t(0.12 s. Along with the

increase in the interface area of the drop at longer time,

the viscous force on the drop interface induced by theexternal flow becomes important and stretches the drop

into a spheroid in the axial direction, whereas the drop is

seen to be pear shaped under the gravitational force

when the external flow is vanishing, as shown in the

inserted profiles in Fig. 14. Differing from body forces,

which pull the drop liquid away from the tube, such as

the gravitational force (cf. Fig. 8), the external flow acts

on the drop interface as a viscous shear stress, which

stretches and elongates the drop and, therefore, leads to

a great increase in the drop length. At the extremely high

velocity of the external flow, such as that ofu/u

"200 in

Fig. 14, a thread about three time longer than the tube

radius is developed during necking and breakup of the

drop. It is expected that a large satellite droplet, or

multiple satellites, will be created subsequent to the

thread breakup. This phenomenon is similar to that

observed in the formation of conducting or insulating

drops driven by an external electric field, which acts with

electric charges and induces an electrostatic force on

drop surfaces (Zhang and Basaran, 1996).

Fig. 15 shows variations of breakoff volume and limit-

ing length of drops as a function of the average velocity of

an external flow, u, normalized by the average velocity ofthe drop liquid ejecting from the tube, u

under condi-

tions that are otherwise the same as those used in Fig. 14.

Three typical interface profiles of drops at the instant

when they are about to detach are inserted for external

flows ofu/u

"1, 100, and 200. As expected, the external

flow, acting as an additional force assisting to break up

drops, reduces the breakoff volume of the drops. As the

velocity of the external flow (or its dimensionless term

u/u

) increases, the volume of breakoff drops decreases

and, correspondingly, the dripping rate increases mono-

tonically. It is noteworthy that since the volume of

breakoff drops decreases as the velocity of an external

flow increases, the increase in the limiting length, /R,

withu/u

(shown in Fig. 15) is obviously attributed to an

increase in the thread length. The concurrent increase in

drop elongation and decrease in the volume of breakoff

drops greatly affect the dynamics of thread rupture and

the subsequent generation of satellite droplets. Fig. 16

depicts the feature of a drop breaking into a flowing

medium with u/u

"200 and all other conditions the

same as those in Fig. 14. Because of the large shear stress

on the drop interface generated by the external flow, the

drop produces a long, thick thread prior to its breakup.Immediately following the detachment of a primary

drop, the newly freed tip of the thread tends to retract


a function of the dimensionless velocity of the flowing media atRe"1,

Ca"0.01,G"1,"1, and R/R"2.


Re"1, Ca"0.1, G"1,"1, R/R"2, and u

/u

"200.

downward under the unbalanced capillary force. How-

ever, owing to the great opposition from the external

flow, the tip of the distorted thread reaches a minimum

height and then starts to ascend att"0.154 s. As it does

so, a bulbous head develops at the tip of the thread

and continues to expand. A secondary neck then be-gins to develop immediately below the head at

t"0.17 s. Eventually, a second breakup occurs to create


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a secondary drop with the volume almost 10% of that of

the primary drop. This is in contrast to the situation with

quiescent or slowly flowing ambient fluid where the sec-

ondary (or satellite) drop has a volume less than 1% of

that of the primary drop (cf. Fig. 6(a)). The large differ-

ence in the thread breakup is also indicated by difference

in the interaction time. Namely, in a quiescent ambient

fluid, the time interval that elapses between the twobreakups is about 3.6 ms as shown in Fig. 6a, while with

an external flow ofu/u

"200, it takes more than 17 ms

(see Fig. 16). Hence, for the latter case, the secondary

breakup can be regarded as a breakup entirely separate

from the first one; evidently the liquid jetting occurs to

generate drops of nonuniform sizes.

5. Concluding remarks

Through both theoretical and experimental means, we

have investigated the effects of inertial, viscous, gravi-

tational, and surface tension forces on drop formation.

The emphasis has been on determining the importance of

the dynamic effects of an ambient viscous fluid on drop

evolution and breakup. The VOF/CSF numerical algo-

rithm, on which our numerical model is based, allows

calculations to pass the breaking point during drop

formation continuously without numerical modifications

to overcome the singular nature of the interface rupture.

This feature makes it possible to characterize numerically

the double breakage of a liquid thread and the sub-

sequent generation of satellite droplets under differentconditions. Reassuringly, the numerical predictions show

excellent agreement in comparison with experimental

measurements for a typical liquidliquid system of 2EH

drops forming in water.

According to the results obtained in the present study,

the maximum (or limiting) length that a drop attains

prior to its breakup and the volume of a drop that breaks

off from the tube increase significantly with increased

Reynolds and capillary numbers and with decreased

gravitational Bond number of the drop liquid. During

necking and breakup of a drop, a liquid thread, which

connects to a detaching drop and a cone-shaped liquid

mass sessile on the tube, develops and stretches. Under

certain conditions, the thread breaks at both ends se-

quentially, resulting in satellite droplets. When drops

form into a quiescent ambient fluid, whether the satellite

droplet is created subsequent to the thread breakup is

substantially dependent on the relative importance of the

inertial and surface tension forces. Dynamic effects of an

ambient fluid have been shown to play an important role

in drop formation. In particular, the increasing wall ef-

fects of an ambient fluid container cause a significant

increase in the volume of breakoff drops and a largereduction in the length of the liquid threada factor that

may alter the possibility of creation of satellite droplets.

In contrast, an external flow of the ambient liquid leads

to a great increase in the thread length and radius but to

a decrease in the volume of breakoff drops. Moreover,

as a result of viscous damping of the interface distur-

bances, a long, thin thread is developed when the ratio of

the viscosity of the drop liquid to that of the ambient

fluid becomes sufficiently large, resulting in the genera-

tion of multiple satellite droplets subsequent to dropbreakup.

Acknowledgments

This work was supported by the Division of Chemical

Sciences, Office of Basic Energy Sciences, U.S. Depart-

ment of Energy under contract DE-AC05-96OR22464

with Lockheed Martin Energy Research Corp. The

author performed this study when he was working at

Oak Ridge National Laboratory, Tennessee. The author

thanks Dr. J. R. Richards of Dupont for providing the

details of his calculations.

References

Batchelor, G.K. (1967). An introduction to fluid dynamics, New York:

Cambridge at the University Press.

Berg, J.C. (1993). ettability. New York: Marcel Dekker.

Bird, R.B., Stewart, W.E., & Lightfoot, E.N. (1960). ransport phe-

nomena. New York: Wiley.Bogy, D.B. (1979). Drop formation in a circular liquid jet. Annual

Review of Fluid Mechanics, 11, 207228.

Brackbill, J.U., Kothe, D.B., & Zemach, C. (1992). A continuum method

for modeling surface tension.Journal of Computational Physics 100,

335354.

Clift, R., Grace, J.R., & Weber, M.E. (1978). Bubbles, drops, and par-

ticles. New York: Academic Press.

Cram, L.E. (1984). A numerical model of drop formation. In J. Noye,

& C. Fletcher (Eds.), Computational techniques and applications:

CAC-83. Amsterdam: Elsevier.

Eggers, J., & Dupont, T.F. (1994). Drop formation in a one-dimensional

approximation of the NavierStokes equation. Journal of Fluid

Mechanics262, 205221.

Harris, M.T., & Byers, C.H. (1989). An advanced technique for inter-facial tension measurements in liquidliquid systems. ORNL/TM-

10734.

Heertjes, M.P., de Nie, L.H., & de Vries, H.J. (1971). Drop formation

in liquidliquid systems I Prediction of drop volumes at

moderate speed of formation. Chemical Engineering Science 26,

441449.

Hirt, C.W., & Nichols, B.D. (1981). Volume of fluid (VOF) method for

the dynamics of free boundaries.Journal of Computional Physics39,

201225.

Kothe, D.B., & Mjolsness, R.C. (1992). Ripple: A new model for

incompressible flows with free surfaces. AIAA Journal 30,

26942700.

Kumar, R., & Kuloor, N.R. (1970). The formation of bubbles and drops.

Advances in Chemical Engineering 8, 255368.

Michael, D.H. (1981). Meniscus stability.Annual Review of Fluid Mech-

anics13, 189215.


8/13/2019 1-s2.0-S0009250999000275-main

16/16

Oguz, H.N., & Prosperetti, A. (1993). Dynamics of bubble growth and

detachment from a needle. Journal of Fluid Mechanics 257,111145.

Peregrine, D.H., Shoker, G., & Symon, A. (1990). The bifurcation of

liquid bridge.Journal of Fluid Mechanics 212, 2539.

Richards, J.R. (1994). Fluid mechanics of liquid-liquid system. Ph.D.

dissertation, University of Delaware.

Richards, J.R., Beris, A.N., & Lenhoff, A.M. (1995). Drop formation in

liquidliquid systems before and after jetting. Physics of Fluids, 7,

26172630.Scheele, G.F., & Meister, B.J. (1968). Drop formation at low velocities

in liquidliquid systems: Part I. Prediction of drop volume.

A.I.Ch.E. Journal 14, 919.

Schulkes, R.M.S.M. (1993). Nonlinear dynamics of liquid columns:

A comparative study. Physics of Fluids A, 5, 21212130.

Schulkes, R.M.S.M. (1994). The evolution and bifurcation of a pendant

drop. Journal of Fluid Mechanics 278, 83100.

Shi, X.D., Brenner, M.P., & Nagel, S.R. (1994). A cascade of structure in

a drop falling from a faucet. Science, 265, 219222.

Zhang, D., & Stone, H.A. (1997). Drop formation in viscous flows at

a vertical capillary tube. Physics of Fluids 9, 22342242.

Zhang, X. (1999) Dynamics of growth and breakup of viscous

pendant drops into air. Journal of Colloid and Interface Science,

In press.Zhang, X., & Basaran, O.A. (1995). An experimental study of dynamics

of drop formation. Physics of Fluids 7, 11841203.

Zhang, X., & Basaran, O.A. (1996). Dynamics of drop formation from

a capillary in the presence of an electric field. Journal of Fluid

Mechanics326, 239263.


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