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Effects of insoluble surfactants on the nonlinear deformation and breakup of stretching
liquid bridges
Bala Ambravaneswaran, and Osman A. Basaran
Citation: Physics of Fluids 11, 997 (1999);
View online: https://doi.org/10.1063/1.869972
View Table of Contents: http://aip.scitation.org/toc/phf/11/5
Published by the American Institute of Physics
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PHYSICS OF FLUIDS
VOLUME 11, NUMBER 5
MAY 1999
Effects of insoluble surfactants on the nonlinear deformation and breakup
of 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 of
liquid 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 Navier–Stokes 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. @S1070-6631~99!02705-1#
I. INTRODUCTION
pending on various parameters. Moreover, one or more satellite 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 interface, 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 idealized but useful model in studying the floating zone technique for crystal growth.4 Anilkumar et al.5 have investigated controlling thermocapillary convection in such a liquid
Studies of long, cylindrical fluid columns and their stability 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 experimentally 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 p.
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 slender liquid thread that develops as time advances and eventually thins and breaks. This process subsequently creates two
large drops whose volumes may nevertheless be unequal dea!
Author to whom correspondence should be addressed. Electronic mail:
obasaran@ecn.purdue.edu
1070-6631/99/11(5)/997/19/$15.00
997
© 1999 American Institute of Physics
998
Phys. Fluids, Vol. 11, No. 5, May 1999
bridge by vibrating one of the supporting rods. Other examples include industrially important processes like spraying
and atomization of liquids where a fundamental understanding of the breakup of liquid columns is essential ~Ref. 6; see
also Refs. 7–9!. 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 Tirtaatmadja and Sridhar12 have exploited the dynamics of liquid
bridges for developing techniques for the measurement of
surface 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 formation from a capillary tube and liquid bridge breakup ~cf.
Refs. 15–17!. The stretching liquid bridge provides a ‘‘ controlled 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 between parallel surfaces, crossed cylinders, and nonparallel
surfaces and those undergoing gyrostatic rotation, as reviewed 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 subsequent theoretical work on liquid bridge dynamics has either 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 until
the 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 coordinate 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 the
breakup of liquid bridges with these models. Schulkes6,7 has
carried out careful studies evaluating the validity and limitations of the one-dimensional approximations. Sanz and
Diez23 have studied the nonaxisymmetric but linearized oscillations 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 viscosity. Sanz26 and Mollot et al.27 have studied experimentally the oscillations of liquid bridges. More recently, Nico-
B. Ambravaneswaran and O. A. Basaran
las and Vega28 and Mancebo et al.29 have studied the
nonlinear dynamics of nearly inviscid liquid bridges undergoing 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 polymeric 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, McKinley and co-workers32,33 have studied both computationally
and experimentally the response of viscoelastic liquid
bridges to uniaxial extension. Shipman et al.34 used the finite
element method to solve the free boundary problem that describes 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 and
eventual 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
Zhang et 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 onedimensional equations whose predictions they showed were
in excellent agreement with their experiments. The set of
one-dimensional equations used by these authors were arrived at from the Navier–Stokes 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 location ~Ref. 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 experimentally 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 of
stretching 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 mechanics 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-
Phys. Fluids, Vol. 11, No. 5, May 1999
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 between the interfacial tension and the surfactant concentration. Milliken et al.44 performed simulations over a wider
range of drop viscosities and subjected the drops to a wider
variety 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 accounting for surface saturation and nonideal interaction among
surfactant molecules for the case of insoluble surfactants.
Much of the experimental work on free boundary problems 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 surfactant concentrations along fluid interfaces, and for understanding dynamic interfacial phenomena due to DST effects.
More recently Hirsa et al.49 have used the optical method of
second harmonic generation ~SHG! to measure instantaneous
profiles 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 described in Sec. III. Section IV presents detailed results of
computations, including ones that highlight the relative importance of surfactant convection to surfactant diffusion
along the liquid–gas 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 m and density r. The bridge is surrounded 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
B. Ambravaneswaran and O. A. Basaran
999
FIG. 1. A stretching bridge of a surfactant-laden liquid held captive between
two rods under gravity.
vector g. 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 surfactant is taken to wholly reside on the liquid–gas 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 U m while the
bottom disk is stationary or else the two disks are taken to
move with velocities U m /2 and 2U m /2, respectively, as
shown in Fig. 1. The case of symmetric stretching—moving
the top and the bottom disks in opposite directions—removes
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 U m ; this is a point which is returned to in
the next section. The surface tension of the liquid–gas interface is spatially nonuniform and depends on the local concentration of the surfactant. In what follows, it is convenient
to define a cylindrical coordinate system $ r, u ,z % whose origin lies at the center of the lower disk surface, where r denotes the radial coordinate, z the axial coordinate measured
in the direction opposite to gravity, and u 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 Navier–Stokes system
and appropriate boundary and initial conditions. The dynamics of the insoluble surfactant along the liquid–gas interface
is 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 onedimensional 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:
v~ r,z,t ! 5 v 0 ~ z,t ! 1 v 2 ~ z,t ! r 2 1¯ ,
~1!
p ~ r,z,t ! 5 p 0 ~ z,t ! 1 p 2 ~ z,t ! r 2 1¯ .
~2!
1000
Phys. Fluids, Vol. 11, No. 5, May 1999
B. Ambravaneswaran and O. A. Basaran
In ~1! and ~2!, v n (z,t) and p n (z,t), where n50,2,4,..., are
unknown functions of the axial coordinate and time t that are
to be determined. Substitution of ~1! into the continuity
equation yields the following expression for the radial velocity u(r,z,t):
r ]v0 r3 ]v2
2
1¯ .
u ~ r,z,t ! 52
2 ]z
4 ]z
~3!
Substitution of ~1!–~3! into the remaining governing equations 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 convectiondiffusion equation, and the kinematic condition:
F
G
1 ]p0 m
]v0
]v0
] 2v 0
1v0
52
1
4 v 2 1 2 2g ,
]t
]z
r ]z
r
]z
p 01 m
]v0
5 s ~ 2H! ,
]z
~4!
~5!
F
G
G ]v0
]G
]G
] 2G 1 ] h ] G
1v0
52
1D s
1
,
]t
]z
2 ]z
]z2 h ]z ]z
~6!
h ]v0
]h
]h
1 v 0 52
,
]t
]z
2 ]z
~7!
where h(z,t) is the bridge profile, G(z,t) is the surface concentration of surfactant, g is the magnitude of the acceleration due to gravity, s (z,t) is the surface tension of the interface, 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 expression for v 2 ,
v 25
3 ] h ] v 0 1 ] 2v 0
1 ]s
1
1
,
2 m h ] z 2h ] z ] z
4 ]z2
~8!
which can be used to eliminate this second-order quantity
from ~4!. 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 velocity ṽ [ ṽ (z̃, t̃ ), the bridge profile h̃[h̃(z̃, t̃ ), and the surface
concentration of surfactant G̃[G̃(z̃, t̃ ), where t̃ is the dimensionless time, are
] ṽ
] t̃
] h̃
] t̃
] G̃
] t̃
52 ṽ
] ṽ
] z̃
1
Pe
] p̃
] z̃
13Oh
1 ]
h̃ 2 ] z̃
S D
h̃ 2
] ṽ
] z̃
12
] h̃
1 ] ṽ
2 h̃
,
] z̃ 2 ] z̃
52 ṽ
5
2Oh
S
] 2 G̃
] z̃ 2
1
1 ] h̃ ] G̃
h̃ ] z̃ ] z̃
]s̃
] z̃
2G, ~9!
~10!
D
2 ṽ
] G̃
] z̃
2
1
2
G̃
] ṽ
] z̃
.
~11!
In this paper, the surface tension s̃ of the liquid–gas interface is related to the surfactant concentration G̃ by the nonlinear Szyskowsky equation of state ~see Ref. 53!
s̃ 511 b ln ~ 12G̃ ! .
~12!
Equations ~9!–~12! are already dimensionless because length
is measured in units of R and time in units of
t 5 Ar R 3 / s o , where s o is the surface tension of the pure
liquid, or the solvent. With these choices for the length and
time scales, the velocity scale is not independent but is given
by U[R/ t 5 As o / r R. In ~9!–~12! and below, variables that
appear with a tilde over them are the dimensionless counterparts of those without the tilde. In Eq. ~9!, Oh[ m / Ar R s o is
the Ohnesorge number, which measures the importance of
viscous forces relative to inertial forces and G[ r R 2 g/ s o is
the gravitational Bond number, which measures the importance of the gravitational forces relative to the surface tension forces. In Eq. ~11!, Pe[R 2 /D s t is the Peclet number
which determines the importance of convection of surfactant
relative to its diffusion along the free surface. The parameter
b [G m RT/ s o , where G m is the maximum packing concentration of the surfactant, R̄ is the universal gas constant, and
T, the temperature, provides a measure of the strength of the
surfactant. Moreover, the modified dimensionless pressure p̃,
which is measured in units of s o /R and whose axial derivative appears in Eq. ~9!, is related to twice the dimensionless
local mean curvature of the interface by
Oh p̃5
s̃
2
h̃ @ 11 ~ ] h̃/ ] z̃ ! 2 # 1/2
s̃] 2 h̃/ ] z̃ 2
@ 11 ~ ] h̃/ ] z̃ ! 2 # 3/2
.
~13!
The dimensionless pressure P̃ inside the liquid bridge to the
leading order is then given by37
P̃5 p̃2Oh
] ṽ
] z̃
~14!
.
As shown by Papageorgiou,8 keeping the full curvature term,
as in Eq. ~13!, in the asymptotically correct slender bridge
equation ~9! is not rational. However, Eggers and Dupont,37
who studied drop formation, and Ruschak,54 Kheshgi,55 and
Johnson et al.,56 who studied the dynamics of thin films over
flat 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 demanded by the slender jet asymptotics.
Equations ~9!–~11! 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, t̃ .0,
h̃ ~ z̃50,t̃ ! 51,
h̃ ~ z̃5L/R, t̃ ! 51,
~15!
Phys. Fluids, Vol. 11, No. 5, May 1999
B. Ambravaneswaran and O. A. Basaran
and the axial velocity at the disk surfaces follow the adherence conditions
ṽ~ z̃50,t̃ ! 50 or 2Ũ m /2,
ṽ~ z̃5L/R, t̃ ! 5Ũ m or Ũ m /2.
~16!
In these equations L is the dimensional instantaneous length
of the liquid bridge and the dimensionless disk velocity Ũ m
[U m /U5U m Ar R/ s o measures the importance of inertial
forces relative to surface tension forces. Moreover, because
the surfactant cannot penetrate the disks, the surfactant concentration must obey
] G̃
] z̃
~ z̃50,t̃ ! 50,
] G̃
] z̃
~ z̃5L/R, t̃ ! 50,
~17!
in the context of the one-dimensional theory being considered here.
Initial conditions must be specified to complete the
mathematical statement of the problem. In this paper, situations are considered in which the bridge is impulsively set
into 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 L o /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 G̃ o along the surface of the static
bridge. The initial conditions are
h̃ ~ z̃, t̃ 50 ! 5h̃ o ~ z̃ ! ,
~18!
ṽ~ z̃, t̃ 50 ! 50,
~19!
G̃ ~ z̃, t̃ 50 ! 5G̃ o ,
~20!
where h̃ o is the interface shape function of the equilibrium
shape. The equilibrium bridge shape is, of course, governed
by the Young–Laplace equation
22H̃5K2Gz̃,
~21!
where H̃ is the dimensionless local mean curvature and K is
the reference pressure, and the constraint that the bridge volume is fixed.
Therefore, the dynamics of stretching and breaking of
surfactant-laden liquid bridges are governed by eight parameters, namely the Ohnesorge number Oh, the gravitational
Bond number G, the dimensionless disk velocity Ũ m , the
dimensionless volume V/R 3 , the slenderness ratio L o /R, the
Peclet number Pe, the so-called strength of the surfactant b,
and the initial surfactant concentration G̃ 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-
1001
ement method57,58 for spatial discretization and finite differences for time integration. The problem is reformulated by
introducing a new variable V, so that the highest-order derivative appearing in the governing equations is of second
order with respect to the spatial coordinate z̃. This reformulation requires that Eqs. ~9!–~11! be augmented by the equation
V2
] h̃
] z̃
50.
~22!
With this reformulation, it is required that the basis functions
which represent the unknowns h̃, V, G̃, and ṽ be continuous
or that they fall into a class of interpolating functions known
as C o basis functions.57 In this work, the domain 0<z̃
<L/R is divided into NE elements. The unknowns are then
expanded in terms of a series of linear basis functions f i (z̃):
N
h̃ ~ z̃, t̃ ! 5
( h i~ t̃ ! f i~ z̃ ! ,
i51
~23!
N
V ~ z̃, t̃ ! 5
( V i~ t̃ ! f i~ z̃ ! ,
i51
~24!
N
G̃ ~ z̃, t̃ ! 5
( G i~ t̃ ! f i~ z̃ ! ,
i51
~25!
N
ṽ~ z̃, t̃ ! 5
( v i~ t̃ ! f i~ z̃ ! ,
i51
~26!
where h i , V i , G i , and v i are unknown coefficients to be
determined and N5NE11 is the number of nodes.
The Galerkin weighted residuals of Eqs. ~9!–~11! and
~22! are constructed by weighting each equation by the basis
functions 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 resulting expressions are simplified through the use of boundary
conditions ~15!–~17!. The residual equations are next cast to
a fixed isoparametric coordinate system 0< j <1 by the isoparametric mapping z̃5 ( Ni51 z i f i ( j ), 57 where the z i ’s 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 bottom one is stationary,
z i ~ t̃ ! 5z i ~ t̃ 50 !
L
,
Lo
i51,.., N.
~27!
The evaluation of the residuals then requires that time derivatives at fixed locations in physical space be cast onto
time derivatives at fixed isoparametric locations by
d
d t̃
5
]
] t̃
1vm
]
] z̃
,
~28!
1002
Phys. Fluids, Vol. 11, No. 5, May 1999
B. Ambravaneswaran and O. A. Basaran
where v m (z̃, t̃ )5Ũ m z̃/(L/R). Extension to the situation
when both disks are moving is straightforward. With these
manipulations, the residual equations become
R iI 5
E HF
1
0
2
d ṽ
d t̃
R iII 5
~ 11V 2 ! 3/2
S
h̃ ] z̃ ] z̃
dG̃
~ 12G̃ ! dz̃
1
0
V2
E HF
1
0
2
R iIV 5
S
2 ] ṽ ] h̃
2b
ES
i
R III
5
] z̃
11 b log ~ 12G̃ !
f i2
h̃ ~ 11V 2 ! 1/2
„11 b log ~ 12G̃ ! …] V/ ] z̃ d f i
23 Oh
1
G F
G
D
DJ
] ṽ
1 ~ ṽ 2 v m !
] z̃
dG̃
D
f2
] z̃ dz̃
1 ~ ṽ 2 v m !
d t̃
f i1
1 ~ ṽ 2 v m !
~30!
] G̃
d t̃
dh̃
~29!
f i z̃ j d j ,
Pe h̃ ] z̃ ] z̃
1
] ṽ d f i
i
1G f i z̃ j d j ,
1 1 ] G̃ ] h̃
EF
0
] h̃
dz̃
] z̃
G
f i1
G̃ ] ṽ
2 ] z̃
1 ] G̃ d f i
Pe ] z̃ dz̃
] h̃
]z
1
h̃ ] ṽ
2 ] z̃
G
J
fi
z̃ j d j ,
~31!
f i z̃ j d j ,
~32!
where z̃ j [dz̃/d j and i51,..., N.
The Galerkin weighted residuals ~29!–~32! are a set of
nonlinear ordinary differential equations in time. In this
work, time derivatives are discretized at the pth time step,
D t̃ p 5 t̃ p 2 t̃ p21 , by either first-order backward differences
or second-order trapezoid rule. With time discretization in
place, the resulting system of 4N nonlinear algebraic equations is solved by Newton’s method. Four backward difference time steps with fixed D t̃ p provide the necessary
smoothing before the trapezoid rule is used.62 Moreover, in
this work a first-order forward difference predictor is used
with the backward difference method and a second-order
Adams–Bashforth predictor is used with the trapezoid rule.
The norm of the correction provided by Newton iterations,
i d p11 i ` , is an estimate of the local time truncation error of
the 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 e so that D t̃ p11
5D t̃ p ( e / i d p11 i ` ) 1/3 ~Ref. 63!. Relative error of 0.1% per
time step, e 51023 , 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 computations are continued until dimensionless minimum radius at
some node falls below a specified value, which is typically
set to 1023 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 literature when the disk velocity ~velocities! 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 U m . The sensitivity of the computed solutions to mesh
refinement was also studied. All results to be reported in the
next section were shown to be insensitive to further systematic 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
U m and in the second of which both disks are in motion, the
top one in the upward direction with a velocity U m /2 and the
bottom one in the downward direction with a velocity
2U m /2, both systems observed from an inertial frame of
reference fixed to the laboratory. The mathematical equivalence 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 velocity equal to one-half of the top disk velocity in the upward
direction. However, the condition of an inertial frame of reference 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 velocity is abruptly changed from 0 to U m at t50. Computations 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 absence of gravity, a bridge that is set in motion by moving the
top disk alone deforms asymmetrically about z5L/2 whereas
the same bridge that is set in motion by moving both disks in
opposite directions deforms symmetrically about z5L/2.
However, the asymmetry in bridge deformation in the first
situation can be removed computationally by artificially imposing an initial velocity distribution that varies linearly
from zero at the bottom disk to U m at the top disk. Hence, in
what follows, it is to be understood that, unless otherwise
Phys. Fluids, Vol. 11, No. 5, May 1999
B. Ambravaneswaran and O. A. Basaran
1003
stated, the top disk is moving and the bottom disk is stationary.
IV. RESULTS AND DISCUSSION
In the experiments of Zhang et al.,16 typical values of
the rod radii were R50.16 cm. For a water-like liquid this
corresponds to Oh52.931023 and for a glycerol-like liquid
this corresponds to Oh54.2. Because these authors did not
have access to a high-speed translation stage, U m
<0.6 cm/sec in their experiments. However, disk velocities
of about 5 cm/sec are achievable.65 In this section, the various 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 1023 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 liquids. The Peclet number Pe is varied from 1022 to 105 and b
is varied from 0—ineffective surfactant—to 1.0—a very
strong surfactant. Zhang et al.16 have shown that initial
bridge 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 considered.
FIG. 3. Evolution in time of the shape of the bridge whose profile at the
incipience of breakup is shown in Fig. 2.
In what follows, situations in which viscous effects are
large are discussed first and the impact of various dimensionless 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 dynamics will be shown to differ drastically from those in the
former situation.
A. High-viscosity liquid bridges
FIG. 2. Variation with axial position of the dimensionless bridge radius h/R
of and dimensionless concentration G/G m , dimensionless axial velocity ṽ
and dimensionless pressure p̃ inside a bridge of glycerol-like liquid at the
incipience of breakup ( t̃ 5125.55). Here Oh54.202, G50, Ũ m 50.028,
L 0 /R52, V/R 3 52 p , Pe50.1, b 50.5, and G̃ 0 50.5.
Figure 2 shows the variation with the dimensionless
axial coordinate of the interface shape of and the dimensionless concentration, dimensionless axial velocity, and the dimensionless 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 R50.16 cm and
stretched at a velocity U m 50.5 cm/sec. Moreover, gravity is
absent, the initial bridge profile is cylindrical, and the initial
slenderness ratio L o /R52. The liquid bridge also has a surfactant of very high surface diffusivity deposited on its surface such that Pe50.1. 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 concentration profiles as the bridge approaches the state shown in Fig.
2. In Figs. 2–4 and certain others to follow only one-half of
the various profiles are shown as the problem is axisymmetric. 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
1004
Phys. Fluids, Vol. 11, No. 5, May 1999
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.
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. However, as the bridge nears breakup, there is rapid flow of fluid
out 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
liquid–gas interface of the surfactant-laden bridge compared
to 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 5–7 depict a situation in which all of the dimensionless groups except Pe are identical to those in Figs. 2–4.
In contrast to Figs. 2–4, the liquid bridge of Figs. 5–7 has a
surfactant of very low surface diffusivity deposited on its
surface such that Pe5105 . Although the distribution of surfactant 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
B. Ambravaneswaran and O. A. Basaran
FIG. 5. Variation with axial position of the dimensionless bridge radius h/R
of and dimensionless concentration G/G m , dimensionless axial velocity ṽ ,
and dimensionless pressure p̃ inside a bridge of glycerol-like liquid at the
incipience of breakup ( t̃ 5145.55). Here Oh54.202, G50, Ũ m 50.028,
L 0 /R52, V/R 3 52 p , Pe5105 , b 50.5, and G̃ 0 50.5.
4 reveals that the evolution in time of concentration profiles
in 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. Indeed, the proper view of the dynamics in this case emerges
if one moves with a frame of reference that is based at
z̃5L̃/2, where L̃ is the dimensionless instantaneous length of
the bridge, and translates upward with a velocity Ũ m /2 relative to the stationary bottom plate: while the capillary pressure gradient-induced flows are symmetrically evacuating
the neck they are opposed by the Marangoni stress-induced
flows. Therefore, at these low stretching velocities the Marangoni 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-
Phys. Fluids, Vol. 11, No. 5, May 1999
FIG. 6. Evolution in time of the shape of the bridge whose profile at the
incipience of breakup is shown in Fig. 5.
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 profile h̃ becomes identical to Eq. ~11! which governs
the surfactant concentration G̃.
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 deposited—indicated by the curve labeled as mobile surfactant—in situations in which the
stretching velocity is low. The bridges are held captive between two rods of radii R50.16 cm and stretched at a velocity of U m 50.5 cm/sec. Here gravity is absent, the initial
bridge profiles are cylindrical, and the initial slenderness ratios L o /R52. Values of corresponding dimensionless groups
are given in the caption to Fig. 8. At low Peclet numbers, L d
increases with increasing Pe because of the role played by
Marangoni stresses in delaying bridge breakup as explained
earlier in the context of Figs. 5–7. However, once the Peclet
number 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 depleted of surfactant ~cf. Figs. 4 and 7!. Therefore, L d does
not continue to increase indefinitely with Pe as Peclet number exceeds Pec '1000, 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
Pe@Pec compared to situations at intermediate Peclet num-
B. Ambravaneswaran and O. A. Basaran
1005
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.
bers when 1!Pe'Pec . The correctness of these predictions
have been verified by demonstrating that they remain unchanged upon doubling the number of mesh points used in
obtaining the results shown in Fig. 8. Figure 8 also shows
that 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 limiting length of a bridge whose surface tension is kept constant at a value equal to that of a bridge having surfactant
uniformly distributed on its surface at the initial concentration G̃ o —the so-called uniform surfactant case—is always
higher than that of a bridge covered with a mobile surfactant.
In other words, in the uniform surfactant case s̃ 51
1 b ln(12G̃o) at each point along the bridge surface for all
time, 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
considered ~cf. 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 Pe!1 to Pe@1 is considered in this
paper to observe these two different effects. In systems that
are easily realizable in the laboratory, however, these opposing effects that surfactants can exhibit can be observed by
1006
Phys. Fluids, Vol. 11, No. 5, May 1999
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 deposited ~solid curve!. Here Oh54.202, G50, Ũ m 50.028, L 0 /R52, V/R 3
52 p , b 50.5, and G̃ 0 50.5. 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.
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 subtle
changes 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 stretching 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 attains
exceeds 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, dimensionless 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
B. Ambravaneswaran and O. A. Basaran
FIG. 9. Variation with axial position of the dimensionless bridge radius h/R
of and dimensionless concentration G/G m , dimensionless axial velocity ṽ ,
and dimensionless pressure p̃ inside a bridge of glycerol-like liquid at the
incipience of breakup ( t̃ 570.36). Here Oh54.202, G50, Ũ m 50.28,
L 0 /R52, V/R 3 52 p , Pe50.1, b 50.5, and G̃ 0 50.5 and the rods are
stretched in the opposite directions with speeds Ũ m /2.
magnitude larger, viz. Ũ m 50.28. Moreover, in order to offset 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 Ũ m /2 and
2Ũ m /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 an
intermediate Peclet number, Pe510, on the dynamics of the
bridge breakup. Figure 10 shows that in this situation a favorable 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 Pe,1 and surfactant concentration is
nearly uniform across the bridge surface. Figure 11, where
Pe5105 , shows that when Pe@1 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
Phys. Fluids, Vol. 11, No. 5, May 1999
FIG. 10. Variation with axial position of the dimensionless bridge radius
h/R of and dimensionless concentration G/G m , dimensionless axial velocity
ṽ , and dimensionless pressure p̃ inside a bridge of glycerol-like liquid at the
incipience of breakup ( t̃ 560.72). Here Oh54.202, G50, Ũ m 50.28,
L 0 /R52, V/R 3 52 p , Pe510, b 50.5, and G̃ 0 50.5 and the rods are
stretched in the opposite directions with speeds Ũ m /2.
too far removed from the middle of the long bridge to produce 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 deposited—indicated by the curve
labeled as mobile surfactant—in 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 depletion of surfactant in the neck that occurs with increasing Pe
dominates the stabilizing influence exerted by the Marangoni
effect. Consequently, the limiting length as Pe→` is lower
than that as Pe→0. The fall in L d /R at Pe'10 and the rise in
L d /R for slightly higher values of Pe are due to the appearance 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 Pe5105 than one with Pe50.1 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
B. Ambravaneswaran and O. A. Basaran
1007
FIG. 11. Variation with axial position of the dimensionless bridge radius
h/R of and dimensionless concentration G/G m , dimensionless axial velocity
ṽ , and dimensionless pressure p̃ inside a bridge of glycerol-like liquid at the
incipience of breakup ( t̃ 562.14). Here Oh54.202, G50, Ũ m 50.28,
L 0 /R52, V/R 3 52 p , Pe5105 , b 50.5, and G̃ 0 50.5 and the rods are
stretched in the opposite directions with speeds Ũ m /2.
Peclet number increases. As shown in Fig. 7, at low stretching 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 velocity, 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 presence of gravity of course breaks the symmetry of the bridge
profile about its midplane z̃5L̃/2 and causes liquid to accumulate near the bottom disk.
1008
Phys. Fluids, Vol. 11, No. 5, May 1999
B. Ambravaneswaran and O. A. Basaran
FIG. 12. 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 deposited ~solid curve!. Here Oh54.202, G50, Ũ m 50.28, L 0 /R52, V/R 3
52 p , b 50.5, and G̃ 0 50.5 and the rods are stretched in the opposite directions with speeds Ũ m /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.
B. Low-viscosity liquid bridges
Figure 15 shows the variation with the dimensionless
axial coordinate of the interface shape of and the dimensionless concentration, dimensionless axial velocity, and dimensionless total pressure inside a bridge of water-like liquid as
it is nearing breakup. The bridge is being held captive be-
FIG. 13. Variation of the dimensionless limiting length L d /R with the dimensionless stretching velocity U m /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.
FIG. 14. Variation with axial position of the dimensionless bridge radius
h/R of and dimensionless concentration G/G m , dimensionless axial velocity
ṽ , and dimensionless pressure p̃ inside a bridge of glycerol-like liquid at the
incipience of breakup ( t̃ 592.15). Here Oh54.202, G50.503, Ũ m
50.028, L 0 /R52, V/R 3 52 p , Pe50.1, b 50.5, and G̃ 0 50.5.
tween two rods of radii R50.16 cm, has volume
V50.04 cm3, and is being stretched at a velocity
U m 50.6 cm/sec. Moreover, gravity is present in this case,
the initial slenderness ratio L o /R52, 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
Pe50.1. 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 surfactantfree drops forming from capillaries66,15,17 and stretching
bridges,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 velocities in the vicinity of the two ends of the neck are the consequences 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
Phys. Fluids, Vol. 11, No. 5, May 1999
FIG. 15. Variation with axial position of the dimensionless bridge radius
h/R of and dimensionless concentration G/G m , dimensionless axial velocity
ṽ , and dimensionless pressure p̃ inside a bridge of water-like liquid at the
incipience of breakup ( t̃ 548.93). Here Oh50.00293, G50.342, Ũ m
50.028, L 0 /R52, V/R 3 52 p , Pe50.1, b 50.5, and G̃ 0 50.5.
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 dimensionless groups except Pe are identical to those in Fig. 15. In
contrast to Fig. 15, the liquid bridge of Fig. 16 has a surfactant of very low surface diffusivity deposited on its surface
such that Pe5105 . Several features distinguish the high Peclet 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
B. Ambravaneswaran and O. A. Basaran
1009
FIG. 16. Variation with axial position of the dimensionless bridge radius
h/R of and dimensionless concentration G/G m , dimensionless axial velocity
ṽ , and dimensionless pressure p̃ inside a bridge of water-like liquid at the
incipience of breakup ( t̃ 565.2). Here Oh50.00293, G50.342, Ũ m
50.028, L 0 /R52, V/R 3 52 p , Pe5105 , b 50.5, and G̃ 0 50.5.
shown in Fig. 16 have similar profiles, as demanded by the
governing equations ~10! and ~11!.
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 paragraphs continue to be observed with the following exception.
For a low Pe bridge, the spikes in the surfactant concentration profile that arise at large times are less effective in accelerating 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 in
surfactant 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 liquid–gas interface has a small influence on the limiting lengths of liquid
bridges over the entire range of Peclet numbers considered.
1010
Phys. Fluids, Vol. 11, No. 5, May 1999
FIG. 17. Variation of the dimensionless limiting length L d /R with Pe of a
bridge of water-like liquid on the surface of which a surfactant is deposited
~solid curve!. Here Oh50.00293, G50.342, Ũ m 50.028, L 0 /R52, V/R 3
52 p , b 50.5, and G̃ 0 50.5. 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.
B. Ambravaneswaran and O. A. Basaran
FIG. 19. Variation of the dimensionless limiting length L d /R with the
Ohnesorge number Oh at two extremes of Pe. Here G50.342, Ũ m
50.028, L 0 /R52, V/R 3 52 p , b 50.5, and G̃ 0 50.5. Also shown is the
variation of L d /R with Oh for surfactant free bridges.
Figures 18–20 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
R50.16 cm, have initial slenderness ratios L o /R52, 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 of Ũ m 50.028 whereas the
bridge of Fig. 20 is being stretched at a high velocity of
Ũ m 50.28. The bridge surfaces are covered with surfactants
of either high or low diffusivity such that Pe50.1 or 105 . For
comparison, Figs. 18–20 also show the variation of the lim-
FIG. 18. Variation of the dimensionless limiting length L d /R with the
Ohnesorge number Oh at two extremes of Pe. Here G50, Ũ m 50.028,
L 0 /R52, V/R 3 52 p , b 50.5, and G̃ 0 50.5. Also shown is the variation of
L d /R with Oh for surfactant free bridges.
FIG. 20. Variation of the dimensionless limiting length L d /R with the
Ohnesorge number Oh at two extremes of Pe. Here G50, Ũ m 50.28,
L 0 /R52, V/R 3 52 p , b 50.5, and G̃ 0 50.5. Also shown is the variation of
L d /R with Oh for surfactant free bridges.
C. Effect of Ohnesorge number on limiting length
Phys. Fluids, Vol. 11, No. 5, May 1999
FIG. 21. Variation with axial position of the dimensionless bridge radius
h/R of and dimensionless axial velocity ṽ , dimensionless radial velocity
and dimensionless pressure p̃, inside a pure water bridge at an intermediate
time of stretching ( t̃ 564.25). Here Oh50.00293, G50.342, Ũ m 50.028,
L 0 /R52, and V/R 3 59.766.
iting lengths of surfactant-free bridges with Oh. Values of all
the dimensionless groups are given in the captions to Figs.
18–20.
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 viscosity, 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, increases. Both the limiting bridge length and the length of the
neck increase dramatically as stretching velocity and Ohne-
B. Ambravaneswaran and O. A. Basaran
1011
FIG. 22. Variation with axial position of the dimensionless bridge radius
h/R of and dimensionless axial velocity ṽ , dimensionless radial velocity,
and dimensionless pressure p̃ inside a pure water bridge at the incipience of
breakup ( t̃ 564.28). Here Oh50.00293, G50.342, Ũ m 50.028, L 0 /R52,
and V/R 3 59.766.
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 of
interest in many applications as it can determine whether any
satellite droplets will be formed and the fate of these satellites if any are formed. Zhang et al.16 have shown from computations 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 explored in the literature, this subsection first provides a more
1012
Phys. Fluids, Vol. 11, No. 5, May 1999
B. Ambravaneswaran and O. A. Basaran
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 breakup
dynamics 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 derived quantity and is obtained from the continuity equation,
viz. u52(r/2)( ] v / ] z).
Figure 21 shows the variation with the dimensionless
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 between two rods of radii R50.16 cm, has volume
V50.04 cm3, and is being stretched at a velocity
FIG. 24. Variation with axial position of the dimensionless bridge radius
h/R of and dimensionless axial velocity ṽ , dimensionless radial velocity
and dimensionless pressure p̃, inside a pure water bridge at an intermediate
time of stretching ( t̃ 53.27). Here Oh50.00293, G50.342, Ũ m 51.17,
L 0 /R52, and V/R 3 59.766.
FIG. 23. Variation with axial position of the dimensionless bridge radius
h/R of and dimensionless axial velocity ṽ , dimensionless radial velocity,
and dimensionless pressure p̃, inside a pure water bridge at an intermediate
time of stretching ( t̃ 53.27). Here Oh50.00293, G50.342, Ũ m 51.16,
L 0 /R52, and V/R 3 59.766.
U m 50.6 cm/sec. Moreover, gravity is present in this case,
the initial slenderness ratio L o /R52, 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 the
neck breaking at the bottom.
When all parameters except the stretching velocity are
held fixed but U m 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 velocities of U m 524.86 and 25.04 cm/sec. These figures demonstrate 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
Phys. Fluids, Vol. 11, No. 5, May 1999
B. Ambravaneswaran and O. A. Basaran
1013
50.85 for a surfactant-laden bridge characterized by a low
Pe of 0.1 as compared to Ũ m 51.2 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
velocity Ũ m 51.05.
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. However, the manner in which surfactant affects the dynamics is
strongly dependent on the values of certain key dimensionless groups. Especially noteworthy in this regard is the influence of the Peclet number on the dynamics. Aside from the
obvious fact that the presence of surfactant reduces the overall 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 dominates convection or when Pe is small. The other is the Marangoni effect due to the presence of surface tension gradients, which is observed when Pe is large. Although the
FIG. 25. Variation with axial position of the dimensionless bridge radius
h/R of and dimensionless axial velocity ṽ , dimensionless radial velocity,
and dimensionless pressure p̃ inside a pure water bridge at the incipience of
breakup ( t̃ 54.04). Here Oh50.00293, G50.342, Ũ m 51.16, L 0 /R52, and
V/R 3 59.766.
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 phenomenon of switch of the breakup point. The bridges are being
held captive between two rods of radii R50.16 cm, have
volumes V50.04 cm3, and are being stretched at various velocities. Moreover, gravity is present in all these cases, the
initial slenderness ratios L o /R52, and the initial bridge profiles are the equilibrium shapes. Of the three cases considered, the first corresponds to pure water, the second to a
water-like liquid the surface of which is covered with a surfactant of high diffusivity such that Pe50.1, and the third to
a water-like liquid the surface of which is covered with a
surfactant of low diffusivity such that Pe5105 . 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 Ũ m
FIG. 26. Variation with axial position of the dimensionless bridge radius
h/R of and dimensionless axial velocity ṽ , dimensionless radial velocity,
and dimensionless pressure p̃ inside a pure water bridge at the incipience of
breakup ( t̃ 54.07). Here Oh50.00293, G50.342, Ũ m 51.17, L 0 /R52, and
V/R 3 59.766.
1014
Phys. Fluids, Vol. 11, No. 5, May 1999
B. Ambravaneswaran and O. A. Basaran
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. Threedimensional but axisymmetric, or two-dimensional, algorithms 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 paper
are in excellent agreement with those of the exact twodimensional theory, the two-dimensional algorithms can be
generalized to more complex situations including the oscillations 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.
FIG. 27. Computed limiting shapes of bridges held captive between two
rods of equal radii R50.16 cm at the instants when they are about to break
as a function of the dimensionless rod velocity Ũ m for a bridge of ~a! pure
water, ~b! surfactant-laden water-like liquid characterized by a low Pe of
0.1, and ~c! surfactant-laden water-like liquid characterized by a high Pe of
105 . Here Oh50.00293, G50.342, L 0 /R52, V/R 3 59.766, b 50.5, and
G̃ 0 50.5.
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 depending 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 diffusivity ~high Pe! because of the favorable
influence of Marangoni stresses on delaying the rupture of
the neck. This effect, however, can be lessened or even negated by increasing the stretching velocity, as borne out by
the calculations reported in this work. Therefore, computational results of the sort presented in this work are essential
for developing a comprehensive understanding of the dynamics of liquid bridges.
Although local details of interface rupture are independent of global details and rupture phenomena in different
situations ‘‘look’’ the same when examined on a fine enough
scale,68,17 global features of interface rupture can be drastically 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 pressure 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
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