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Effects of insoluble surfactants on the nonlinear deformation and breakupof stretching liquid bridges

Bala Ambravaneswaran and Osman A. Basarana)

School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907-1283

Received 13 July 1998; accepted 27 January 1999

During the emission of single drops and the atomization of a liquid from a nozzle, threads of liquid

are stretched and broken. A convenient setup for studying in a controlled manner the dynamics ofliquid threads is the so-called liquid bridge, which is created by holding captive a volume of liquid

between two solid disks and pulling apart the two disks at a constant velocity. Although the stability

of static bridges and the dynamics of stretching bridges of pure liquids have been extensively

studied, even a rudimentary understanding of the dynamics of the stretching and breakup of bridges

of surfactant-laden liquids is lacking. In this work, the dynamics of a bridge of a Newtonian liquid

containing an insoluble surfactant are analyzed by solving numerically a one-dimensional set of

equations that results from a slender-jet approximation of the NavierStokes system that governs

fluid flow and the convection-diffusion equation that governs surfactant transport. The

computational technique is based on the method-of-lines, and uses finite elements for discretization

in space and finite differences for discretization in time. The computational results reveal that the

presence of an insoluble surfactant can drastically alter the physics of bridge deformation and

breakup compared to the situation in which the bridge is surfactant free. They also make clear how

the distribution of surfactant along the free surface varies with stretching velocity, bridge geometry,

and bulk and surface properties of the liquid bridge. Gradients in surfactant concentration along the

interface give rise to Marangoni stresses which can either retard or accelerate the breakup of the

liquid bridge. For example, a high-viscosity bridge being stretched at a low velocity is stabilized by

the presence of a surfactant of low surface diffusivity high Peclet number because of the favorable

influence of Marangoni stresses on delaying the rupture of the bridge. This effect, however, can be

lessened or even negated by increasing the stretching velocity. Large increases in the stretching

velocity result in interesting changes in their own right regardless of whether surfactants are present

or not. Namely, it is shown that whereas bridges being stretched at low velocities rupture near the

bottom disk, those being stretched at high velocities rupture near the top disk. 1999 American

Institute of Physics.S1070-66319902705-1

I. INTRODUCTION

Studies of long, cylindrical fluid columns and their sta-

bility have been carried out since the 19th century. Through

the works of Plateau,1 Rayleigh,2 and Mason,3 among many

others, it has been determined theoretically as well as experi-

mentally that in the absence of gravity, the critical value of

the ratio of the length of a cylindrical column of liquid to its

diameter above which the column cannot be held in stable

equilibrium is .

A liquid bridge is a column of liquid held between two

coaxial solid disks. When such a static bridge is impulsively

set into motion and stretched uniaxially, it deforms gradually

and contracts at its middle portion. Of great interest in the

dynamics of the stretching liquid bridge is the fate of a slen-

der liquid thread that develops as time advances and eventu-

ally thins and breaks. This process subsequently creates two

large drops whose volumes may nevertheless be unequal de-

pending on various parameters. Moreover, one or more sat-

ellite drops may be formed following the rupture of the fluid

interface. In this paper, a theoretical study is presented of the

effects of surfactants which are insoluble in the bulk liquid

on the dynamics of uniaxially stretched liquid bridges. As

the bridge stretches, the surfactant redistributes on the inter-

face, thereby causing gradients in surface tension. This in

turn causes flows due to the Marangoni effect which can

change considerably the dynamics of bridge deformation and

breakup compared to the situation in which the bridge is

surfactant free. The goal here is to understand the role of the

physical properties of the bridge liquid and the surfactant,

stretching speed, and bridge size on the dynamics, which has

heretofore been lacking.

The statics and dynamics of liquid bridges have attracted

much attention for more than a century. Interest in them has

grown in the last few decades because of applications in

diverse fields. For example, a liquid bridge serves as an ide-

alized but useful model in studying the floating zone tech-

nique for crystal growth.4 Anilkumar et al.5 have investi-

gated controlling thermocapillary convection in such a liquid

aAuthor to whom correspondence should be addressed. Electronic mail:

[email protected]

PHYSICS OF FLUIDS VOLUME 11, NUMBER 5 MAY 1999

9971070-6631/99/11(5)/997/19/$15.00 1999 American Institute of Physics

Downloaded 09 Oct 2012 to 150.140.190.160. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissions


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bridge by vibrating one of the supporting rods. Other ex-

amples include industrially important processes like spraying

and atomization of liquids where a fundamental understand-

ing of the breakup of liquid columns is essential Ref. 6; see

also Refs. 79. Another reason for studying liquid bridges

comes from the fiber spinning process10 which is industrially

practiced on a large scale. Tsamopoulos et al.11 and Tirtaat-

madja and Sridhar12 have exploited the dynamics of liquid

bridges for developing techniques for the measurement ofsurface tension, shear viscosity, and extensional viscosity of

molten Newtonian and non-Newtonian liquids. Studies by

Ennis et al.13 and Chen et al.14 have been motivated by the

application of liquid bridges to agglomeration of particles.

Of special interest to the present authors and yet another

motivation for studying dynamics of liquid bridges is the

close analogy between interface rupture during drop forma-

tion from a capillary tube and liquid bridge breakup cf.

Refs. 1517. The stretching liquid bridge provides a con-

trolled method of studying the dynamics and breakup of a

fluid neck connecting an about-to-form drop from the rest of

the liquid in the capillary tube.

There have been many theoretical and experimental

studies of static liquid bridges since the pioneering works of

Plateau and Rayleigh. These have addressed various physical

situations such as drops, or liquid bridges, held captive be-

tween parallel surfaces, crossed cylinders, and nonparallel

surfaces and those undergoing gyrostatic rotation, as re-

viewed by Zhang et al.16 It was not until after the work of

Fowle et al.,18 however, that the dynamics and breakup of

liquid bridges began to be studied. The majority of the sub-

sequent theoretical work on liquid bridge dynamics has ei-

ther relied on one-dimensional models or taken the bridge

liquid and the surrounding liquid to be inviscid. Furthermore,

virtually all of the theoretical and experimental works untilthe 1990s have either considered the dynamical response of

the bridge due to oscillations of one of the rods or breakup

that results when a bridge near its static limit of stability is

subjected to a disturbance. The one-dimensional models

have been developed under the assumption that the bridge is

sufficiently slender so that the axial velocity is independent

of the radial coordinate and depends solely on the axial co-

ordinate and time. Until very recently, two fundamentally

different one-dimensional models have been used to analyze

the dynamics of liquid bridges. The first one is the inviscid

slice model due to Lee19 and the second one is the model

based on the so-called Cosserat equations.20 Meseguer21 and

Meseguer and Sanz,22

among others, have studied thebreakup of liquid bridges with these models. Schulkes6,7 has

carried out careful studies evaluating the validity and limita-

tions of the one-dimensional approximations. Sanz and

Diez23 have studied the nonaxisymmetric but linearized os-

cillations of inviscid liquid bridges. Borkar and

Tsamopoulos24 and Tsamopoulos et al.11 have studied using

linear stability analysis the linearized oscillations of liquid

bridges of small and arbitrary viscosities. Chen and

Tsamopoulos25 have used the finite element method to study

finite amplitude oscillations of liquid bridges of arbitrary vis-

cosity. Sanz26 and Mollot et al.27 have studied experimen-

tally the oscillations of liquid bridges. More recently, Nico-

las and Vega28 and Mancebo et al.29 have studied the

nonlinear dynamics of nearly inviscid liquid bridges under-

going weakly nonlinear oscillations.

Virtually all of the previous theoretical and experimental

work on stretching liquid bridges has been motivated by and

aimed at characterizing the rheological response of poly-

meric liquids to uniaxial extension. Sridhar et al.30 studied

stretching liquid bridges to measure the extensional viscosity

of polymer solutions. Similar studies for Boger fluids were

conducted by Solomon and Muller.31 More recently, McKin-

ley and co-workers32,33 have studied both computationally

and experimentally the response of viscoelastic liquid

bridges to uniaxial extension. Shipmanet al.34 used the finite

element method to solve the free boundary problem that de-

scribes the nonlinear deformation of a stretching bridge of a

viscoelastic fluid and thereby attempted to simulate some of

the experiments of Sridhar et al.30 Especially relevant to the

present paper is the work of Kroger et al.35 who studied the

effects of inertial and viscous forces on the dynamics of

stretching liquid bridges. These authors correctly observed

that while the interfacial tension force causes contraction andeventual breakup of a stretching liquid bridge, inertial and

viscous forces tend to stabilize the bridge surface and

thereby significantly slow down its breakup.

The dynamics of stretching liquid bridges have been

studied in detail both theoretically and experimentally by

Zhanget al.16 and theoretically in the creeping flow limit by

Gaudet et al.36 In contrast to most earlier works on liquid

bridges, Zhang et al.16 used a recently derived set of one-

dimensional equations whose predictions they showed were

in excellent agreement with their experiments. The set of

one-dimensional equations used by these authors were ar-

rived at from the NavierStokes system and interfacial

boundary conditions by either i retaining the leading-order

terms in a Taylor series expansion in the radial coordinate of

the velocity and the pressure fields and the free surface lo-

cationRef. 37 or ii carrying out an asymptotic expansion

under a slender-jet or long-wave approximation to capture

the leading order dynamics Ref. 8. Although they address

physical situations that are somewhat different than those

considered by Zhang et al.16 and Gaudet et al.36 and that

considered in this paper, Padday et al.38 have studied experi-

mentally the stability and breakup of pendant liquid bridges

and Chen and Steen39 have studied theoretically the

capillary-driven breakup of inviscid bridges. Unfortunately,

how the presence of surfactants would affect the dynamics ofstretching liquid bridges is unknown and forms the subject of

this paper.

Surfactant effects have, of course, been studied in other

free boundary problems. Prior to 1990, however, virtually all

studies devoted to the effects of surfactants on the fluid me-

chanics of drops were restricted to situations in which either

the drops remained spherical see, e.g., Refs. 40 and 41 or

the drop deformations were small see, e.g., Ref. 42. Stone

and Leal43 studied the effects of insoluble surfactant on the

finite-amplitude deformation and breakup of a drop in a

steady flow under Stokes flow conditions. Here, as in later

papers, Leal and co-workers used the boundary integral tech-

998 Phys. Fluids, Vol. 11, No. 5, May 1999 B. Ambravaneswaran and O. A. Basaran



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nique to solve for the flow. Stone and Leal43 used a linear

equation of state to relate interfacial tension and the local

concentration of surfactant on the drop surface. Milliken

et al.,44 while restricting the surfactant to be insoluble and

the flow to the creeping flow regime, extended the work of

Stone and Leal43 by adopting a nonlinear relationship be-

tween the interfacial tension and the surfactant concentra-

tion. Milliken et al.44 performed simulations over a wider

range of drop viscosities and subjected the drops to a widervariety of flow conditions than those in the earlier work of

Stone and Leal.43 The effect of surfactant solubility on drop

deformation and breakup was taken up by Milliken and

Leal,45 who assumed that surfactant transport in the bulk was

diffusion dominated. Effects of finite fluid inertia in such

problems have recently been considered by Leppinen et al.,46

who studied theoretically the steady and transient behavior

of surfactant-laden drops falling through air. Leppinen et al.,

too, assumed that the surfactant is insoluble in both the drop

liquid and surrounding air and adopted a linear equation of

state relating surface tension and surfactant concentration.

Pawar and Stebe47 have extended the work of Milliken

et al.44 on drop deformation in extensional flows by account-

ing for surface saturation and nonideal interaction among

surfactant molecules for the case of insoluble surfactants.

Much of the experimental work on free boundary prob-

lems in the presence of surfactants has been motivated by the

desire to measure dynamic surface tension DST. Franses

et al.48 have provided a comprehensive review of virtually

all nonoptical techniques that have been developed until

1996 to measure DST for its own sake, for inferring surfac-

tant concentrations along fluid interfaces, and for under-

standing dynamic interfacial phenomena due to DST effects.

More recently Hirsa et al.49 have used the optical method of

second harmonic generationSHGto measure instantaneousprofiles of surfactant concentration along fluid interfaces.

Building on earlier works of Eggers and Dupont,37

Papageorgiou,8,9 and Zhang et al.,16 a set of one-dimensional

evolution equations is presented in Sec. II that governs the

shape of, axial velocity in, and surfactant distribution along

the surface of a stretching liquid bridge. The numerical

method used to solve the set of evolution equations is de-

scribed in Sec. III. Section IV presents detailed results of

computations, including ones that highlight the relative im-

portance of surfactant convection to surfactant diffusion

along the liquidgas interface. Concluding remarks form the

subject of Sec. V.

II. PROBLEM FORMULATION

The system is an axisymmetric bridge of fixed volume V

of an incompressible, Newtonian liquid of spatially uniform

and constant viscosity and density . The bridge is sur-

rounded by a dynamically inactive ambient gas phase that

exerts a constant pressure and negligible viscous drag on the

bridge. As shown in Fig. 1, the bridge is captured between

and is coaxial with two solid circular disks, or rods, of equal

radii R which are separated by an initial distance L o from

each other. The common axis of symmetry of the bridge and

the disks is vertical and lies along the direction of the gravity

vectorg. The two contact lines are circles that remain pinned

to the edges of the disks throughout the motion. The free

surface separating the liquid from the ambient gas has a fixed

amount of an insoluble surfactant deposited on it. The sur-

factant is taken to wholly reside on the liquidgas interface

and hence does not penetrate into, or get adsorbed on, the

disk surfaces. Here either the top disk moves upward along

the axis of symmetry at a constant velocity Um while the

bottom disk is stationary or else the two disks are taken to

move with velocities Um/2 and Um/2, respectively, as

shown in Fig. 1. The case of symmetric stretchingmoving

the top and the bottom disks in opposite directionsremoves

any asymmetry that might arise when the bottom disk is held

stationary and the velocity of the top disk is impulsively

changed from 0 to Um; this is a point which is returned to in

the next section. The surface tension of the liquidgas inter-

face is spatially nonuniform and depends on the local con-

centration of the surfactant. In what follows, it is convenient

to define a cylindrical coordinate system r,,z whose ori-gin lies at the center of the lower disk surface, where rde-

notes the radial coordinate, z the axial coordinate measured

in the direction opposite to gravity, and the azimuthal

angle. For axisymmetric configurations of interest in the

present study, the problem is independent of the azimuthal

angle.

Isothermal, transient flow of a viscous liquid inside a

stretching bridge is governed by the NavierStokes system

and appropriate boundary and initial conditions. The dynam-

ics of the insoluble surfactant along the liquidgas interfaceis governed by the convection-diffusion equation Refs. 50

and 51, see also Ref. 52. Following Eggers and Dupont37

and Papageorgiou,8 this spatially two-dimensional system of

partial differential equations is reduced to a spatially one-

dimensional system by expanding the axial velocity v(r,z ,t)

and the pressure p(r,z ,t) in a Taylor series in the radial

coordinate:

vr,z ,tv0z ,tv2z ,tr2, 1

pr,z, tp 0z, tp 2z, tr2. 2

FIG. 1. A stretching bridge of a surfactant-laden liquid held captive between

two rods under gravity.

999Phys. Fluids, Vol. 11, No. 5, May 1999 B. Ambravaneswaran and O. A. Basaran



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In 1 and 2, vn(z,t) and p n(z ,t), where n0,2,4,..., are

unknown functions of the axial coordinate and timetthat are

to be determined. Substitution of 1 into the continuity

equation yields the following expression for the radial veloc-

ity u(r,z ,t):

ur,z ,tr

2

v0

z

r3

4

v2

z . 3

Substitution of 13 into the remaining governing equa-

tions and boundary conditions yields the following equations

at the leading order from the z-component of the Navier

Stokes equation, the normal stress balance, the convection-

diffusion equation, and the kinematic condition:

v0

t v0

v0

z

1

p 0

z

4v2

2v0

z2g , 4

p 0v0

z 2H, 5

tv0

z

2

v0

z D s

2

z 2

1

h

h

z

z , 6

h

tv0

h

z

h

2

v0

z , 7

where h (z,t) is the bridge profile, (z, t) is the surface con-

centration of surfactant, g is the magnitude of the accelera-

tion due to gravity, (z ,t) is the surface tension of the inter-

face, 2H is twice the local mean curvature of the interface,

and D s is the surface diffusivity of the surfactant. At the

leading order, the tangential stress balance yields an expres-

sion for v2 ,

v2

1

2h

z

3

2h

h

z

v0

z

1

4

2v0

z2 , 8

which can be used to eliminate this second-order quantity

from4. The leading order equations that govern the shape,

axial velocity, and surfactant concentration follow once 5 is

substituted into 4. Thenceforward, subscripts attached to

the leading order terms have been dropped for simplicity.

The equations that govern the dimensionless axial veloc-

ity v v(z, t), the bridge profile h h(z, t), and the surface

concentration of surfactant (z, t), where tis the dimen-

sionless time, are

v

tv

v

zOh

p

z3Oh

1

h2

zh2 v

z2

zG , 9

h

tv

h

z

1

2h v

z , 10

t

1

Pe2

z2

1

h

h

z

z v

z

1

2 v

z. 11

In this paper, the surface tension of the liquidgas inter-

face is related to the surfactant concentration by the non-

linearSzyskowsky equation of state see Ref. 53

1ln 1. 12

Equations912are already dimensionless because length

is measured in units of R and time in units of

R 3/o, where o is the surface tension of the pureliquid, or the solvent. With these choices for the length and

time scales, the velocity scale is not independent but is given

byUR/o/R . In912 and below, variables thatappear with a tilde over them are the dimensionless counter-

parts of those without the tilde. In Eq. 9, Oh/Ro isthe Ohnesorge number, which measures the importance of

viscous forces relative to inertial forces and GR 2g/o is

the gravitational Bond number, which measures the impor-

tance of the gravitational forces relative to the surface ten-

sion forces. In Eq. 11, PeR 2/D s is the Peclet number

which determines the importance of convection of surfactantrelative to its diffusion along the free surface. The parameter

mRT/o, where m is the maximum packing concen-

tration of the surfactant, Ris the universal gas constant, and

T, the temperature, provides a measure of the strength of the

surfactant. Moreover, the modified dimensionless pressure p,

which is measured in units ofo/R and whose axial deriva-

tive appears in Eq. 9, is related to twice the dimensionless

local mean curvature of the interface by

Ohp

h1h/z21/2

2h/z2

1h/z23/2. 13

The dimensionless pressure Pinside the liquid bridge to the

leading order is then given by37

PpOhv

z. 14

As shown by Papageorgiou,8 keeping the full curvature term,

as in Eq. 13, in the asymptotically correct slender bridge

equation9 is not rational. However, Eggers and Dupont,37

who studied drop formation, and Ruschak,54 Kheshgi,55 and

Johnsonet al.,56

who studied the dynamics of thin films overflat and cylindrical substrates, and Zhang et al.16 who studied

stretching liquid bridges without surfactants, have also

adopted this approach, because doing so results in a better

description of the nonlinear evolution of interface shapes

than truncating the curvature expression at the order de-

manded by the slender jet asymptotics.

Equations 911 are solved subject to the boundary

conditions that the three phase contact lines, where the

bridge liquid, the ambient fluid, and the solid surfaces meet,

remain pinned for all time, t0,

hz0,t1, hzL/R , t1, 15




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and the axial velocity at the disk surfaces follow the adher-

ence conditions

vz0,t0 or Um/2,

vzL/R , tUm or Um/2. 16

In these equations L is the dimensional instantaneous length

of the liquid bridge and the dimensionless disk velocity Um

Um/UUmR/o measures the importance of inertialforces relative to surface tension forces. Moreover, because

the surfactant cannot penetrate the disks, the surfactant con-

centration must obey

z z0,t0,

z zL/R , t0, 17

in the context of the one-dimensional theory being consid-

ered here.

Initial conditions must be specified to complete the

mathematical statement of the problem. In this paper, situa-

tions are considered in which the bridge is impulsively setinto motion from an initial state of rest that corresponds to a

stable equilibrium shape of a captive bridge of volume V/R 3,

initial slenderness ratio Lo/R , and under the condition that

the gravitational Bond number equals some specific value G.

Moreover, the surfactant is taken to be distributed with a

uniform concentration o along the surface of the static

bridge. The initial conditions are

hz, t0 hoz, 18

vz, t0 0, 19

z, t0o, 20

where ho is the interface shape function of the equilibrium

shape. The equilibrium bridge shape is, of course, governed

by the YoungLaplace equation

2HKGz, 21

where H is the dimensionless local mean curvature and Kis

the reference pressure, and the constraint that the bridge vol-

ume is fixed.

Therefore, the dynamics of stretching and breaking ofsurfactant-laden liquid bridges are governed by eight param-

eters, namely the Ohnesorge number Oh, the gravitational

Bond number G, the dimensionless disk velocity Um , the

dimensionless volume V/R 3, the slenderness ratio L o/R, the

Peclet number Pe, the so-called strength of the surfactant ,

and the initial surfactant concentration o.

III. FINITE ELEMENT ANALYSIS

The set of one-dimensional, nonlinear equations 9

11 that governs the transient response of a stretching liquid

bridge is solved numerically by using the Galerkin/finite el-

ement method57,58 for spatial discretization and finite differ-

ences for time integration. The problem is reformulated by

introducing a new variable , so that the highest-order de-

rivative appearing in the governing equations is of second

order with respect to the spatial coordinate z. This reformu-

lation requires that Eqs. 911be augmented by the equa-

tion

hz0. 22

With this reformulation, it is required that the basis functions

which represent the unknowns h, ,, and vbe continuous

or that they fall into a class of interpolating functions known

as Co basis functions.57 In this work, the domain 0z

L/R is divided into NE elements. The unknowns are then

expanded in terms of a series of linear basis functions i(z):

hz, ti1

N

h i tiz, 23

z, ti1

N

i tiz, 24

z, ti1

N

i tiz, 25

vz, t

i1

N

v i tiz, 26

where h i, i, i , and v i are unknown coefficients to be

determined and NNE1 is the number of nodes.

The Galerkin weighted residuals of Eqs. 911 and

22are constructed by weighting each equation by the basisfunctions and integrating the resulting expressions over the

computational domain. The weighted residuals of Eqs. 9

and 11 are then integrated by parts to reduce the order of

the highest-order derivative appearing in them and the result-

ing expressions are simplified through the use of boundary

conditions1517. The residual equations are next cast to

a fixed isoparametric coordinate system 01 by the iso-

parametric mappingz i1N

z ii(), 57 where thez is denote

the locations of the nodes or the mesh points. Because one or

both disks are moving, the domain length changes as time

advances. This is accounted for in this paper by allowing the

nodes of the finite element mesh z i to move proportionally to

the motion of the disks Refs. 59 and 60; see also Ref. 61.For example, when only the top disk is moving and the bot-

tom one is stationary,

z i tz i t0L

L o, i1,..,N. 27

The evaluation of the residuals then requires that time de-

rivatives at fixed locations in physical space be cast onto

time derivatives at fixed isoparametric locations by

d

d t

tvm

z, 28




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where vm(z, t)Umz/ (L/R). Extension to the situation

when both disks are moving is straightforward. With these

manipulations, the residual equations become

RIi

0

1 dvd t vvm

v

zi 1log 1

h121/2

1log 1/z

123/2 di

dz

3 Oh2h

v

z

h

zi

v

z

di

dz

21

d

dzG izd, 29

RIIi0

1

h

zizd, 30

RII Ii

0

1 dd t vvm

zi

2

v

zi

1

Pe

1

h

z

h

zi

1

Pe

z

di

dzzd, 31

RIVi

01

dh

d tv

vm

h

z

h

2

v

zi

zd, 32

where zdz/d and i1,...,N.

The Galerkin weighted residuals 2932 are a set of

nonlinear ordinary differential equations in time. In this

work, time derivatives are discretized at the pth time step,

tp tp tp1 , by either first-order backward differences

or second-order trapezoid rule. With time discretization in

place, the resulting system of 4N nonlinear algebraic equa-

tions is solved by Newtons method. Four backward differ-

ence time steps with fixed tp provide the necessary

smoothing before the trapezoid rule is used.

62

Moreover, inthis work a first-order forward difference predictor is used

with the backward difference method and a second-order

AdamsBashforth predictor is used with the trapezoid rule.

The norm of the correction provided by Newton iterations,

dp1, is an estimate of the local time truncation error ofthe trapezoid rule. The time step is chosen adaptively by

requiring the norm of the time truncation error at the next

time step to be equal to a prescribed value so that tp1 tp(/dp1)

1/3 Ref. 63. Relative error of 0.1% per

time step, 103, is prescribed in the computations.

The algorithm for computing the transient evolution of

shapes of stretching bridges and the concentration profiles of

the surfactant has been programed in FORTRAN. Once the

initial or equilibrium bridge profile is known, the top rod is,

or both rods are, impulsively set into motion and the compu-

tations are continued until dimensionless minimum radius at

some node falls below a specified value, which is typically

set to 103 unless otherwise stated. The length of the bridge

at breakup is called the limiting length and denoted by L d.

Several tests were done to ensure the accuracy of the

calculations. The volume of the bridge and the total amount

of surfactant on its surface were monitored throughout the

computations. In all of the cases reported in this paper, the

change incurred by these quantities was always less than

0.01%. The correctness of the algorithm was also verified by

accurately predicting static stability limits of liquid bridges

in the absence and presence of gravity.1,2,64 The ability to

carry out comparisons between predictions made with the

present algorithm and well-established results from the lit-

erature when the disk velocityvelocities is are zero is one

reason why the velocity scale based on the rod radii and the

capillary time scale is preferred in this paper over that based

on Um. The sensitivity of the computed solutions to meshrefinement was also studied. All results to be reported in the

next section were shown to be insensitive to further system-

atic increases in the number of elements or mesh points.

Most important, that the predictions made with the present

algorithm of situations in which the bridge is surfactant free

are in excellent agreement with the experimental results of

Zhang et al.,16 ensures that the one-dimensional model is

true to reality.

At first glance, the bridge response ought to be identical

in two situations in the first of which the bottom disk is

stationary and the top disk is moving upward with a velocity

Um and in the second of which both disks are in motion, the

top one in the upward direction with a velocity Um/2 and the

bottom one in the downward direction with a velocity

Um/2, both systems observed from an inertial frame of

reference fixed to the laboratory. The mathematical equiva-

lence of these two problems is readily apparent if an observer

moves in another inertial frame of reference with respect to

the fixed frame of reference in the first situation with a ve-

locity equal to one-half of the top disk velocity in the upward

direction. However, the condition of an inertial frame of ref-

erence for the moving frame is violated because the top disk

in the first situation is not always moving with a constant

velocity but suffers an initial acceleration when the disk ve-

locity is abruptly changed from 0 to Um at t

0. Computa-tions have shown that when the disk velocity is sufficiently

low, virtually identical results are obtained between the two

situations. However, at high disk velocities and in the ab-

sence of gravity, a bridge that is set in motion by moving the

top disk alone deforms asymmetrically about zL/2 whereas

the same bridge that is set in motion by moving both disks in

opposite directions deforms symmetrically about zL/2.

However, the asymmetry in bridge deformation in the first

situation can be removed computationally by artificially im-

posing an initial velocity distribution that varies linearly

from zero at the bottom disk to Um at the top disk. Hence, in

what follows, it is to be understood that, unless otherwise




8/20

stated, the top disk is moving and the bottom disk is station-

ary.

IV. RESULTS AND DISCUSSION

In the experiments of Zhang et al.,16 typical values of

the rod radii were R0.16 cm. For a water-like liquid this

corresponds to Oh2.9103 and for a glycerol-like liquid

this corresponds to Oh

4.2. Because these authors did nothave access to a high-speed translation stage, Um0.6 cm/sec in their experiments. However, disk velocities

of about 5 cm/sec are achievable.65 In this section, the vari-

ous parameters introduced in Sec. II are varied over wide

ranges to develop a quantitative appreciation of the effect

that they have on the dynamics of bridge breakup. Hence the

Ohnesorge number Oh is varied from 103 to 10 and the

gravitational Bond number G is varied from 0 to 2, thereby

covering the extreme cases of small bridges of high surface

tension liquids and large bridges of low surface tension liq-

uids. The Peclet number Pe is varied from 102 to 105 and

is varied from 0ineffective surfactantto 1.0a very

strong surfactant. Zhang et al.16 have shown that initialbridge aspect ratio virtually has no effect on limiting bridge

length so long as the bridges have the same volume. Thus,

L o/R is taken to equal 2 in most of the cases to be consid-

ered.

In what follows, situations in which viscous effects are

large are discussed first and the impact of various dimension-

less groups on the dynamics is analyzed. Attention is then

turned to situations in which viscous effects are small and

where the effects of some of these parameters on the dynam-

ics will be shown to differ drastically from those in theformer situation.

A. High-viscosity liquid bridges

Figure 2 shows the variation with the dimensionless

axial coordinate of the interface shape of and the dimension-

less concentration, dimensionless axial velocity, and the di-

mensionless total pressure inside a bridge of glycerol-like

liquid in its final state just before breakup. The bridge is

being held captive between two rods of radii R0.16 cm and

stretched at a velocity Um0.5 cm/sec. Moreover, gravity is

absent, the initial bridge profile is cylindrical, and the initial

slenderness ratio L o/R2. The liquid bridge also has a sur-

factant of very high surface diffusivity deposited on its sur-face such that Pe0.1. Values of all the dimensionless

groups are given in the caption to Fig. 2. Figures 3 and 4

show the evolution in time of the shape and the concentra-

tion profiles as the bridge approaches the state shown in Fig.

2. In Figs. 24 and certain others to follow only one-half of

the various profiles are shown as the problem is axisymmet-

ric. The evolution in time of the surfactant concentration

profiles depicted in Fig. 4 makes plain that as the bridge

narrows and necks as shown by the corresponding evolution

in time of the bridge shape depicted in Fig. 3, the dominant

physical response is dilution of surfactant on the surface of

the bridge accompanied by a surfactant concentration profile

FIG. 2. Variation with axial position of the dimensionless bridge radius h/R

of and dimensionless concentration /m, dimensionless axial velocity v

and dimensionless pressure pinside a bridge of glycerol-like liquid at the

incipience of breakup ( t125.55). Here Oh4.202, G0, Um0.028,

L0/R2,V/R32, Pe0.1, 0.5, and 00.5.

FIG. 3. Evolution in time of the shape of the bridge whose profile at the

incipience of breakup is shown in Fig. 2.




9/20

that remains virtually uniform along it. This finding accords

with intuition because diffusion of surfactant dominates its

convection so long as the bridge is far from breakup and

fluid velocities are low everywhere within the bridge. How-

ever, as the bridge nears breakup, there is rapid flow of fluidout of the neck in either direction, as made evident by the

velocity field in Fig. 2: this causes surfactant to be convected

away forcefully from the region where the neck is thinnest,

as can be seen from the humps in the concentration profile on

either side of the axial location where the neck is about to

break. As the latter phenomenon occurs over very short

times preceding breakup, there is insufficient time for the

surfactant to redistribute itself along the surface before the

interface ruptures. The calculations predict that the limiting

length, L d/R , of the surfactant-laden bridge at breakup is

higher than that of a pure glycerol bridge. This outcome is

due to the overall reduction in the surface tension of the

liquidgas interface of the surfactant-laden bridge comparedto that of the surfactant-free bridge and the accompanying

reduction in the capillary pressure which drives the liquid out

of the neck region and causes bridge breakup.

Figures 57 depict a situation in which all of the dimen-

sionless groups except Pe are identical to those in Figs. 24.

In contrast to Figs. 24, the liquid bridge of Figs. 57 has a

surfactant of very low surface diffusivity deposited on its

surface such that Pe105. Although the distribution of sur-

factant in this case is convection dominated as opposed to

the previous case, Fig. 6 shows that the shape of the bridge

evolves in a similar fashion compared to that of the low Pe

bridge shown in Fig. 3. However, comparison of Figs. 7 and

4 reveals that the evolution in time of concentration profilesin the two situations is quite different. Figure 7 shows that in

the high Pe limit, there is depletion of surfactant in the neck

from the outset as the surfactant is convected away from it.

By contrast, in the low Pe limit, there is just dilution of

surfactant everywhere and virtually no depletion of it in the

neck until the final stages of breakup. In the high Pe case,

surfactant being convected out of the neck accumulates near

the two disks and results in concentration gradients from the

neck to the two disks. These concentration gradients in turn

give rise to surface tension gradients and cause Marangoni

stress-induced flows from the disks towards the neck. In-

deed, the proper view of the dynamics in this case emerges

if one moves with a frame of reference that is based at

zL/2, whereLis the dimensionless instantaneous length of

the bridge, and translates upward with a velocity Um/2 rela-

tive to the stationary bottom plate: while the capillary pres-

sure gradient-induced flows are symmetrically evacuating

the neck they are opposed by the Marangoni stress-induced

flows. Therefore, at these low stretching velocities the Ma-

rangoni stresses delay bridge breakup and consequently the

limiting length of the high Pe bridge turns out to be larger

than that of the low Pe bridge.

It is noteworthy that in the high Pe limit, the bridge

profiles depicted in Fig. 6 and the concentration profiles de-

FIG. 4. Evolution in time of the concentration profile for the same bridge as

that of Fig. 2. The concentration profiles are at the same instants in time as

the shape profiles shown in Fig. 3.FIG. 5. Variation with axial position of the dimensionless bridge radius h/R

of and dimensionless concentration /m , dimensionless axial velocity v,

and dimensionless pressure p inside a bridge of glycerol-like liquid at the


L0/R2, V/R32, Pe105, 0.5, and00.5.




10/20

picted in Fig. 7 are similarly shaped at each instant in time.

This observation, of course, can be readily appreciated by

noting that in the limit as Pe, Eq. 10 which governs the

bridge profilehbecomes identical to Eq.11which governs

the surfactant concentration

.Figure 8 shows the variation with Pe of the limiting

lengths of bridges of glycerol-like liquids on the surface of

which a surfactant is depositedindicated by the curve la-

beled as mobile surfactantin situations in which the

stretching velocity is low. The bridges are held captive be-

tween two rods of radii R0.16 cm and stretched at a veloc-

ity of Um0.5 cm/sec. Here gravity is absent, the initial

bridge profiles are cylindrical, and the initial slenderness ra-

tiosL o/R2. Values of corresponding dimensionless groups

are given in the caption to Fig. 8. At low Peclet numbers, L dincreases with increasing Pe because of the role played by

Marangoni stresses in delaying bridge breakup as explained

earlier in the context of Figs. 57. However, once the Pecletnumber exceeds a critical value Pec , the higher the Peclet

number the sooner after the stretching begins that most of the

surfactant ends up near the two disks and leaves a large

portion of the bridge near its middle section completely de-

pleted of surfactant cf. Figs. 4 and 7. Therefore, L d does

not continue to increase indefinitely with Pe as Peclet num-

ber exceeds Pec1000, but in fact decreases slightly with

increasing Pe. This is because the capillary pressure is higher

and the stabilizing Marangoni stresses are inoperative in

the middle of the neck on account of the total depletion of

the surfactant there at early times when the Peclet number

PePec compared to situations at intermediate Peclet num-

bers when 1PePec. The correctness of these predictions

have been verified by demonstrating that they remain un-

changed upon doubling the number of mesh points used in

obtaining the results shown in Fig. 8. Figure 8 also showsthat the limiting length of a bridge along the surface of which

the surfactant is free to move, the mobile surfactant case, is

bound above and below by two limiting cases. The limiting

length of a surfactant-free bridge is always lower than that of

a bridge covered with a mobile surfactant. However, the lim-

iting length of a bridge whose surface tension is kept con-

stant at a value equal to that of a bridge having surfactant

uniformly distributed on its surface at the initial concentra-

tion othe so-called uniform surfactant caseis always

higher than that of a bridge covered with a mobile surfactant.

In other words, in the uniform surfactant case 1

ln(1

o) at each point along the bridge surface for alltime, the Marangoni stresses are absent, and thus only the

effect of the overall reduction in surface tension but not

surface tension gradients due to presence of surfactant is

consideredcf. Refs. 43 and 44.

The results which have been shown until this point and

in particular by Fig. 8 highlight the roles played by dilution/

diffusion and Marangoni flows at the two extremes of low

and high Peclet numbers, respectively. The entire range of

Peclet numbers from Pe1 to Pe1 is considered in this

paper to observe these two different effects. In systems that

are easily realizable in the laboratory, however, these oppos-

ing effects that surfactants can exhibit can be observed by

FIG. 7. Evolution in time of the concentration profile for the same bridge as

that of Fig. 5. The concentration profiles are at the same instants in time as

the shape profiles shown in Fig. 6.

FIG. 6. Evolution in time of the shape of the bridge whose profile at the

incipience of breakup is shown in Fig. 5.




11/20

using different surface coverages, as shown by Stebe and

co-workers.47

The trends discussed so far apply to situations in which

high viscosity, surfactant-laden liquid bridges are stretched

slowly in the absence of gravity. These trends undergo subtlechanges as the stretching speed is increased discussed next

and in Sec. IV D, or the gravitational Bond number is made

nonzero or viscosity is lowered.

The limiting length of the bridge increases as the stretch-

ing speed is increased because the relative importance of the

destabilizing capillary force falls compared to the inertial

force. Indeed, when a liquid bridge is stretched axially at a

low velocity, it takes on at each instant in time a profile that

closely resembles the equilibrium shape that it would have

were the moving disk instantaneously brought to rest and

sufficient time were to elapse for any flow transients to die

down due to viscosity. Hence, at low stretching velocities it

is no surprise that the limiting length that the bridge attainsexceeds the maximum stable length of a static bridge by only

a small amount. By contrast, at higher stretching velocities,

the departure of the transient shapes from the equilibrium

shapes is so large and the breakup of the bridge is delayed

significantly that its limiting length is increased substantially

over the maximum stable length of a static bridge. Figure 9

shows the variation with the dimensionless axial coordinate

of the interface shape of and the concentration, dimension-

less axial velocity, and dimensionless total pressure inside a

bridge of glycerol-like liquid as it is nearing breakup. All the

dimensionless groups in Fig. 9 are the same as those in Fig.

2 except that the stretching velocity is now about an order of

magnitude larger, viz. Um0.28. Moreover, in order to off-

set the asymmetry arising from initial transients, the bridge

of Fig. 9 is stretched symmetrically. In other words, the disks

are pulled in opposite directions with velocities Um/2 and

Um/2. Because the Peclet number is low, here again as in

the case of low-velocity stretching there is just dilution of

surfactant until the last stages of breakup when convection

finally becomes important and leads to the humps in the

concentration profile. The shape profile shown in Fig. 9

points to the formation of a satellite drop which has been

observed even for bridges of pure liquids of intermediate

viscosity see Ref. 16.

Figures 10 and 11 correspond to situations in which all

the dimensionless groups except Pe are the same as those in

Fig. 9. The results shown in Fig. 10 highlight the effect of anintermediate Peclet number, Pe10, on the dynamics of the

bridge breakup. Figure 10 shows that in this situation a fa-

vorable concentration gradient arises away from the neck and

the two disks which causes flow toward the neck and results

in the formation of a large satellite drop. This effect is, of

course, absent when Pe1 and surfactant concentration is

nearly uniform across the bridge surface. Figure 11, where

Pe105, shows that when Pe1 all the surfactant is quickly

swept to the vicinity of the two disks, which causes the neck

to be totally depleted of any surfactant. In the high Pe limit,

gradient in the surfactant concentration that exists occurs so

far away from the neck that any back flow that does arise is

FIG. 8. Variation of the dimensionless limiting length L d/R with Pe of a

bridge of glycerol-like liquid on the surface of which a surfactant is depos-

ited solid curve. Here Oh4.202, G0, Um0.028, L0/R2, V/R3

2, 0.5, and00.5. Also shown are the limiting lengths of a bridge

of pure glycerol and a glycerol-like liquid on the surface of which surfactant

is uniformly distributed, with other parameters being the same.

FIG. 9. Variation with axial position of the dimensionless bridge radius h/R

of and dimensionless concentration /m , dimensionless axial velocity v,

and dimensionless pressure p inside a bridge of glycerol-like liquid at the


L0/R2, V/R32, Pe0.1, 0.5, and 00.5 and the rods are

stretched in the opposite directions with speeds Um/2.




12/20

too far removed from the middle of the long bridge to pro-

duce a satellite of appreciable size.

Figure 12 shows the variation with Pe of the limiting

lengths of bridges of glycerol-like liquids along the surface

of which a surfactant is depositedindicated by the curve

labeled as mobile surfactantin situations in which the

stretching velocity is high. All the dimensionless groups in

Fig. 12 are the same as those in Fig. 8 except the stretching

velocity which is an order of magnitude larger. Figure 12

shows that the stretching velocity is so high that the deple-

tion of surfactant in the neck that occurs with increasing Pe

dominates the stabilizing influence exerted by the Marangonieffect. Consequently, the limiting length as Pe is lower

than that as Pe0. The fall in L d/R at Pe10 and the rise in

L d/R for slightly higher values of Pe are due to the appear-

ance and disappearance of satellite drops cf. Figs. 10 and

11.

Figure 13 summarizes the variation of the limiting

bridge length with stretching velocity at low and high Peclet

numbers. Figure 13 shows that whereas L d/R is larger for a

bridge with Pe105 than one with Pe0.1 at low stretching

velocities, the opposite is true at high stretching velocities.

According to the results presented up to this point, there is

depletion of surfactant from the neck due to convection as

Peclet number increases. As shown in Fig. 7, at low stretch-

ing velocities the neck is relatively short that the Marangoni

stresses that arise from the resulting concentration gradient

are sufficient to drive an appreciable backflow toward the

neck to increase the limiting length of the liquid bridge. By

contrast, as shown in Fig. 11, at high stretching velocities the

surfactant-depleted neck becomes so long that the Marangoni

stresses are inoperative in delaying the rupture of the neck.

Indeed, in the absence of appreciable Marangoni stresses and

the presence of high surface tensions along the neck due to

the total absence of surfactant there, it accords with intuition

that at high stretching velocities the limiting length of the

high Pe bridge is lower than that of the low Pe bridge.

Figure 14 shows the effect of gravity on the variation

with the dimensionless axial coordinate of the shape of and

the dimensionless concentration, dimensionless axial veloc-ity, and dimensionless total pressure inside a bridge of a

glycerol-like liquid at the incipience of breakup. All of the

dimensionless groups in Fig. 14 are identical to those in Fig.

2 with the exception of the gravitational Bond number,

which equals 0.503 here but 0 in Fig. 2. As in previous

studies of equilibrium shapes and stability of static bridges64

and those of stretching liquid bridges without surfactants,16

Fig. 14 shows that an increase in G hastens bridge breakup

and hence results in a decrease in limiting length. The pres-

ence of gravity of course breaks the symmetry of the bridge

profile about its midplane zL/2 and causes liquid to accu-

mulate near the bottom disk.

FIG. 10. Variation with axial position of the dimensionless bridge radius

h/R of and dimensionless concentration /m, dimensionless axial velocity

v, and dimensionless pressure pinside a bridge of glycerol-like liquid at the


L0/R2, V/R32, Pe10, 0.5, and 00.5 and the rods are






L0/R2, V/R32, Pe105, 0.5, and 00.5 and the rods are





13/20

B. Low-viscosity liquid bridges


axial coordinate of the interface shape of and the dimension-

less concentration, dimensionless axial velocity, and dimen-sionless total pressure inside a bridge of water-like liquid as

it is nearing breakup. The bridge is being held captive be-

tween two rods of radii R0.16cm, has volume

V0.04 cm3, and is being stretched at a velocity

Um0.6 cm/sec. Moreover, gravity is present in this case,

the initial slenderness ratio Lo/R2, and the initial bridge

profile is that of the equilibrium shape. The bridge surface is

also covered with a surfactant of high diffusivity such that

Pe0.1. Values of all the dimensionless groups are given in

the caption to Fig. 15. As in the case of the bridge of the

glycerol-like liquid discussed earlier in connection with Fig.

2, the surfactant distribution along the bridge in the present

case also remains nearly uniform until times close to

breakup. However, as is known from studies of surfactant-

free drops forming from capillaries66,15,17

and stretchingbridges,16 low-viscosity liquids give rise to fluid interfaces

that exhibit large slopes or even approach overturning close

to interface rupture. Figure 15 shows that large axial veloci-

ties in the vicinity of the two ends of the neck are the con-

sequences of this interface topology. These large velocities

in turn cause convection of surfactant out of the neck and

result in the two sharp concentration peaks seen in Fig. 15.

The resultant surfactant distribution shown in Fig. 15 causes

surface tension to be locally low at the two ends of the neck.

Therefore, Marangoni stresses in this case cause flows that

accelerate the rupture of the interface instead of slowing it as

in the case of high-viscosity bridges discussed earlier. The

FIG. 12. Variation of the dimensionless limiting lengthL d/R with Pe of a

bridge of glycerol-like liquid on the surface of which a surfactant is depos-

ited solid curve. Here Oh4.202, G0, Um0.28, L 0/R2, V/R3

2, 0.5, and00.5 and the rods are stretched in the opposite direc-

tions with speeds Um/2. Also shown are the limiting lengths of a bridge of

pure glycerol and a glycerol-like liquid on the surface of which surfactant is

uniformly distributed, with other parameters being the same.

FIG. 13. Variation of the dimensionless limiting length L d/R with the di-

mensionless stretching velocity Um/U at two extremes of Pe. All other

parameters are the same as those of the bridge of glycerol-like liquid the

governing dimensionless groups for which are given in the caption to Fig. 2.




incipience of breakup ( t92.15). Here Oh4.202, G0.503, Um

0.028,L 0/R2, V/R32, Pe0.1,0.5, and00.5.




14/20

surfactant-laden bridge in this case breaks faster than a

bridge of pure water. The results depicted in Fig. 15 and

discussed in this paragraph demonstrate that surfactants can

have apparently unexpected effects on the dynamics of

stretching liquid bridges as they approach breakup.

Figure 16 depicts a situation in which all of the dimen-

sionless groups except Pe are identical to those in Fig. 15. In

contrast to Fig. 15, the liquid bridge of Fig. 16 has a surfac-

tant of very low surface diffusivity deposited on its surface

such that Pe105

. Several features distinguish the high Pe-clet number case depicted in Fig. 16 from the low Peclet

number case depicted in Fig. 15. First, there is depletion of

surfactant from the neck due to convection even at early

times. Second, less surfactant is left along the neck during

the final stages of breakup. Therefore, as opposed to the low

Pe case, Marangoni stresses come into play at early times

and remain in effect until breakup in the high Pe case. Given

the concentration profile shown in Fig. 16, the Marangoni

effect is stabilizing and allows a bridge that is laden with a

high Pe surfactant to attain a higher limiting length than a

bridge that is free of surfactant. It is again noteworthy that

the bridge shape and the surfactant concentration distribution

shown in Fig. 16 have similar profiles, as demanded by the

governing equations 10 and11.

When all the dimensionless groups are kept at the values

they have in Fig. 15 or 16 but the stretching velocity is

increased, the trends summarized in the previous two para-

graphs continue to be observed with the following exception.

For a low Pe bridge, the spikes in the surfactant concentra-

tion profile that arise at large times are less effective in ac-

celerating bridge breakup at high stretching velocities than at

low ones. This finding accords with intuition because the

necks at high stretching velocities are longer than ones at

low stretching velocities, which tends to reduce gradients insurfactant concentration and concomitant Marangoni

stresses.

Figure 17 shows the variation with Pe of the limiting

lengths of bridges of water-like liquids along the surface of

which a surfactant is deposited in situations in which the

stretching velocity is low. Thus, Fig. 17 is the low-viscosity

analog of Fig. 8, which pertains to high-viscosity liquids.

Figure 17 makes plain that not only does the presence of

surfactant enhance bridge breakup at low Peclet numbers,

but surfactant that is free to move along the liquidgas in-

terface has a small influence on the limiting lengths of liquid

bridges over the entire range of Peclet numbers considered.



v, and dimensionless pressure pinside a bridge of water-like liquid at the


0.028,L 0/R2, V/R32, Pe0.1, 0.5, and 00.5.



v, and dimensionless pressure pinside a bridge of water-like liquid at the


0.028,L 0/R2, V/R32, Pe105, 0.5, and 00.5.




15/20

C. Effect of Ohnesorge number on limiting length

Figures 1820 summarize the effect of viscosity, or

more precisely the Ohnesorge number, on the limiting

lengths of liquid bridges. In all cases, the bridges may be

thought of as being held captive between two rods of radii

R0.16 cm, have initial slenderness ratiosL o/R2, and the

initial bridge profiles are cylindrical regardless of whether

gravity is present or not. The bridges of Figs. 18 and 19 are

being stretched at a low velocity ofUm0.028 whereas the

bridge of Fig. 20 is being stretched at a high velocity of

Um0.28. The bridge surfaces are covered with surfactants

of either high or low diffusivity such that Pe0.1 or 105. For

comparison, Figs. 1820 also show the variation of the lim-

FIG. 17. Variation of the dimensionless limiting lengthL d/R with Pe of a

bridge of water-like liquid on the surface of which a surfactant is deposited

solid curve. Here Oh0.00293, G0.342, Um0.028, L0/R2, V/R3

2, 0.5, and00.5. Also shown are the limiting lengths of a bridge

of pure water and a water-like liquid on the surface of which surfactant is

uniformly distributed, with other parameters being the same.

FIG. 18. Variation of the dimensionless limiting length Ld/R with the

Ohnesorge number Oh at two extremes of Pe. Here G0, Um0.028,

L0/R2,V/R32, 0.5, and00.5. Also shown is the variation of

Ld/R with Oh for surfactant free bridges.


Ohnesorge number Oh at two extremes of Pe. Here G

0.342, U

m

0.028, L 0/R2, V/R32, 0.5, and 00.5. Also shown is the

variation ofL d/R with Oh for surfactant free bridges.


Ohnesorge number Oh at two extremes of Pe. Here G0, Um0.28,

L0/R2,V/R32, 0.5, and 00.5. Also shown is the variation of

Ld/R with Oh for surfactant free bridges.




16/20

iting lengths of surfactant-free bridges with Oh. Values of all

the dimensionless groups are given in the captions to Figs.

1820.

Figure 18 shows the variation of the limiting length with

the Ohnesorge number at a low stretching velocity in the

absence of gravity. In the limit of low viscosities, or low Oh,

Fig. 18 makes plain that Marangoni stresses that become

prominent at large times during the stretching of low Pe

bridges are destabilizing. By contrast, the Marangoni effect

enhances the limiting bridge length for a high Pe bridge for

all viscosities, or Ohnesorge numbers.Figure 19 shows that gravity makes more pronounced

the destabilizing influence of Marangoni stresses on low vis-

cosity, or Ohnesorge number, bridges characterized by a low

Peclet number being stretched at low velocities. As discussed

earlier, this effect is due to the sharp gradients in interface

shape that arise during the final stages of the deformation

and breakup of such bridges.

Figure 20 shows that at high stretching velocities, a

switch over in limiting length occurs for high and low Peclet

number bridges as viscosity, or Ohnesorge number, in-

creases. Both the limiting bridge length and the length of the

neck increase dramatically as stretching velocity and Ohne-

sorge number increase. At large Ohnesorge numbers, the

long necks are totally depleted of surfactant when the Peclet

number is high. Thus the Marangoni effect is ineffective as a

mechanism to enhance the length of a bridge before it

breaks, and it accords with intuition that L d/R is larger for a

low Peclet number bridge then a high Peclet number one

when Oh is large.

D. Switching of the breakup point

The axial location at which a fluid filament breaks is ofinterest in many applications as it can determine whether any

satellite droplets will be formed and the fate of these satel-

lites if any are formed. Zhang et al.16 have shown from com-

putations that the axial location at which the neck of a

surfactant-free bridge breaks first can switch from its bottom

to its top as the velocity with which bridges of water-like

liquids are stretched is increased. A similar switch in the

breakup point has also been reported by Zhang and

Basaran67 in their experimental study of formation of drops

from capillaries in the presence of an electric field. Since the

occurrence of this phenomenon has been inadequately ex-

plored in the literature, this subsection first provides a more


h/R of and dimensionless axial velocity v, dimensionless radial velocity

and dimensionless pressure p, inside a pure water bridge at an intermediate

time of stretching ( t64.25). Here Oh0.00293, G0.342, Um0.028,

L0/R2, and V/R39.766.


h/R of and dimensionless axial velocity v, dimensionless radial velocity,

and dimensionless pressurep inside a pure water bridge at the incipience of

breakup ( t64.28). Here Oh0.00293, G0.342, Um0.028,L 0/R2,

and V/R 39.766.




17/20

careful look into the underlying physics than that which has

heretofore been provided. This is followed by a discussion of

results on the effect of surfactants on the switch in the

breakup point.

Although the pressure profile and the associated pressure

peak that results in the neck region of a bridge being

stretched at a low velocity is during the early stages of the

necking process virtually symmetric about the axial location

where the neck radius is smallest, insights into the breakupdynamics can be gained by examining the radial velocity in

the bridge in addition to the usual variables of interest. In the

context of the slender-jet theory, the radial velocity is a de-

rived quantity and is obtained from the continuity equation,

viz. u(r/2)( v/z).


axial coordinate of the dimensionless radius of and the axial

velocity, the dimensionless radial velocity evaluated at the

free surface, and the dimensionless pressure inside a water

bridge a few time steps before it ruptures. Figure 21 focuses

on the neck region to emphasize certain salient features of

the breakup process. The bridge is being held captive be-

tween two rods of radii R0.16 cm, has volume

V0.04 cm3, and is being stretched at a velocity

Um0.6 cm/sec. Moreover, gravity is present in this case,

the initial slenderness ratio Lo/R2, and the initial bridge

profile is that of the equilibrium shape. Values of all the

dimensionless groups are given in the caption to Fig. 21.

Although the pressure profile is symmetric about the thinnest

part of the neck, careful examination of the radial velocity

profile in Fig. 21 reveals that the neck is contracting faster at

the bottom than at the top. That monitoring of the radial

velocity profile well before rupture can predict where the

neck will ultimately break is confirmed by Fig. 22, which

shows the same bridge at the incipience of breakup and theneck breaking at the bottom.

When all parameters except the stretching velocity are

held fixed butUm is systematically increased, a switch in the

breakup point is observed when a critical stretching velocity

is reached. Figures 23 and 24 depict, respectively, the same

information as that shown in Fig. 21 albeit at stretching ve-

locities ofUm24.86 and 25.04 cm/sec. These figures dem-

onstrate that although both the bridge profiles and the axial

velocities are virtually indistinguishable, the radial velocities

exhibit important differences. Whereas the radial velocity is

more negative at the bottom in Fig. 23, it is more negative at

the top in Fig. 24. These radial velocity fields computed well





L0/R2, and V/R39.766.


h/R of and dimensionless axial velocity v, dimensionless radial velocity



L0/R2, and V/R39.766.




18/20

before bridge breakup suggest that the bridge stretched at the

lower speed will break at the bottom of the neck whereas that

stretched at the higher speed will break at the top. Figures 25

and 26 show the same bridges at their incipience of breakup,

thereby confirming the assertions made on the basis of the

radial velocity fields of Figs. 23 and 24.

Figure 27 shows the effect of surfactants on the phenom-

enon of switch of the breakup point. The bridges are being

held captive between two rods of radii R0.16 cm, have

volumesV0.04 cm3, and are being stretched at various ve-

locities. Moreover, gravity is present in all these cases, theinitial slenderness ratios L o/R2, and the initial bridge pro-

files are the equilibrium shapes. Of the three cases consid-

ered, the first corresponds to pure water, the second to a

water-like liquid the surface of which is covered with a sur-

factant of high diffusivity such that Pe0.1, and the third to

a water-like liquid the surface of which is covered with a

surfactant of low diffusivity such that Pe105. Values of all

the dimensionless groups are given in the caption to Fig. 27.

Figure 27 shows that the phenomenon of the switch of the

breakup point occurs at lower stretching velocities for

surfactant-laden liquids than surfactant-free ones. This shift

in breakup point occurs at a dimensionless velocity Um

0.85 for a surfactant-laden bridge characterized by a low

Pe of 0.1 as compared to Um1.2 for the surfactant-free

bridge. When the Peclet number is increased to 105, the shift

in the breakup sequence is found to occur at a dimensionless

velocityUm1.05.

V. CONCLUDING REMARKS

According to the foregoing results, the presence of an

insoluble surfactant can drastically change the dynamics of

deformation and breakup of stretching liquid bridges. How-

ever, the manner in which surfactant affects the dynamics is

strongly dependent on the values of certain key dimension-

less groups. Especially noteworthy in this regard is the influ-

ence of the Peclet number on the dynamics. Aside from the

obvious fact that the presence of surfactant reduces the over-

all surface tension, it has been found in this work that two

important effects become evident as the Peclet number is

varied from 0 to . One of these is the dilution of surfactant

along the interface, which is observed when diffusion domi-

nates convection or when Pe is small. The other is the Ma-rangoni effect due to the presence of surface tension gradi-

ents, which is observed when Pe is large. Although the



and dimensionless pressure p inside a pure water bridge at the incipience of

breakup ( t4.04). Here Oh0.00293, G0.342, Um1.16,L 0/R2, and

V/R 39.766.



and dimensionless pressurep inside a pure water bridge at the incipience of

breakup ( t4.07). Here Oh0.00293, G0.342, Um1.17,L 0/R2, and

V/R 39.766.




19/20

occurrence of Marangoni stresses and the flows that they

give rise to are reported for all Pe in this paper, the extent to

and the manner in which surface tension gradients affect the

dynamics have been shown to be drastically different de-pending on the Pe as well as the Oh characterizing the bridge

liquid. For example, a high-viscosity bridge being stretched

at a low velocity is stabilized by the presence of a surfactant

of low surface diffusivityhigh Pe because of the favorable

influence of Marangoni stresses on delaying the rupture of

the neck. This effect, however, can be lessened or even ne-

gated by increasing the stretching velocity, as borne out by

the calculations reported in this work. Therefore, computa-

tional results of the sort presented in this work are essential

for developing a comprehensive understanding of the dy-

namics of liquid bridges.

Although local details of interface rupture are indepen-

dent of global details and rupture phenomena in differentsituations look the same when examined on a fine enough

scale,68,17 global features of interface rupture can be drasti-

cally changed by the operating parameters. A case in point is

the switch in the axial location where the neck breaks which

occurs as the stretching velocity is increased from a low to a

high value. The understanding of the physics of the switch in

the breakup location has been improved in this paper by a

detailed examination of the variation with stretching velocity

of the shape of and the axial and radial velocities and pres-

sure profiles inside stretching liquid bridges.

Several extensions of the present study are noteworthy

and are underway. On the one hand, it is important to relax

certain aspects pertaining to the state of the surfactant and its

transport. Toward this end, theoretical and experimental

work is underway to allow surfactant solubility in the liquid

of the bridge and surfactant exchange between the liquid

gas interface, the bridge liquid, and the solid rods. Three-

dimensional but axisymmetric, or two-dimensional, algo-

rithms that do not rely on the slender-jet approximation have

also been developed. Although early indications are that the

predictions of the one-dimensional theory used in this paperare in excellent agreement with those of the exact two-

dimensional theory, the two-dimensional algorithms can be

generalized to more complex situations including the oscil-

lations and breakup of surfactant-laden drops attached to

capillary tubes.

ACKNOWLEDGMENTS

This research was sponsored by the Chemical Sciences

Program of the Basic Energy Sciences Division of the US

DOE. The authors also thank the Eastman Kodak Company

for partial support through an unrestricted research grant.

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FIG. 27. Computed limiting shapes of bridges held captive between two

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00.5.




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