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 Articles you may be interested in Pinchoff and satellite formation in surfactant covered viscous threads Physics of Fluids 14, 1364 (2002); 10.1063/1.1449893 Deformation and breakup of a stretching liquid bridge covered with an insoluble surfactant monolayer Physics of Fluids 18, 022101 (2006); 10.1063/1.2166657 Numerical simulation of drop and bubble dynamics with soluble surfactant Physics of Fluids 26, 052102 (2014); 10.1063/1.4872174 Microscale tipstreaming in a microfluidic flow focusing device Physics of Fluids 18, 121512 (2006); 10.1063/1.2397023 Marangoni effects on drop deformation in an extensional flow: The role of surfactant physical chemistry. I. Insoluble surfactants Physics of Fluids 8, 1738 (1998); 10.1063/1.868958 Drop formation from a capillary tube: Comparison of one-dimensional and two-dimensional analyses and occurrence of satellite drops Physics of Fluids 14, 2606 (2002); 10.1063/1.1485077 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 1 J. Plateau, ‘‘Experimental and theoretical researches on the figures of equilibrium of a liquid mass withdrawn from the action of gravity,’’ Annual Report of the Board of Regents of the Smithsonian Institution, 270, Washington D.C. ~1963!. 2 Lord Rayleigh, ‘‘On the instability of jets,’’ Proc. London Math. Soc. 10, 4 ~1879!. 3 G. Mason, ‘‘An experimental determination of the stable length of cylindrical liquid bubbles,’’ J. Colloid Interface Sci. 32, 172 ~1970!. 4 R. A. Brown, ‘‘Theory of transport processes in single crystal growth from the melt,’’ AIChE. J. 34, 881 ~1988!. 5 A. V. Anilkumar, R. N. Grugel, X. F. Shen, C. P. Lee, and T. G. Wang, ‘‘Control of thermocapillary convection in a liquid bridge by vibration,’’ J. Appl. Phys. 73, 4165 ~1993!. 6 R. M. S. M. Schulkes, ‘‘Nonlinear dynamics of liquid columns: a comparative study,’’ Phys. Fluids A 5, 2121 ~1993!. 7 R. M. S. M. Schulkes, ‘‘Dynamics of liquid jets revisited,’’ J. Fluid Mech. 250, 635 ~1993!. 8 D. T. Papageorgiou, ‘‘On the breakup of viscous liquid threads,’’ Phys. Fluids 7, 1529 ~1995!. 9 D. T. Papageorgiou, ‘‘Analytical description of the breakup of liquid jets,’’ J. Fluid Mech. 301, 109 ~1995!. 10 M. M. Denn, ‘‘Drawing of liquids to form fibers,’’ Annu. Rev. Fluid Mech. 12, 365 ~1980!. 11 J. Tsamopoulos, T.-Y. Chen, and A. Borkar, ‘‘Viscous oscillations of capillary bridges,’’ J. Fluid Mech. 235, 579 ~1992!. 12 V. Tirtaatmadja and T. Sridhar, ‘‘A filament stretching device for measurement of extensional viscosity,’’ J. Rheol. 37, 1081 ~1993!. 13 B. J. Ennis, J. Li, G. I. Tardos, and R. Pfeffer, ‘‘The influence of viscosity on the strength of an axially strained pendular liquid bridge,’’ Chem. Eng. Sci. 45, 3071 ~1990!. 14 T.-Y. Chen, J. A. Tsamopoulos, and R. J. Good, ‘‘Capillary bridges between parallel and non-parallel surfaces and their stability,’’ J. Colloid Interface Sci. 151, 49 ~1992!. 15 X. Zhang and O. A. Basaran, ‘‘An experimental study of dynamics of drop formation,’’ Phys. Fluids 7, 1184 ~1995!. 16 X. Zhang, R. S. Padgett, and O. A. Basaran, ‘‘Nonlinear deformation and breakup of stretching liquid bridges,’’ J. Fluid Mech. 329, 207 ~1996!. 17 M. P. Brenner, J. Eggers, K. Joseph, S. R. Nagel, and X. D. Shi, ‘‘Breakdown of scaling in droplet fission at high Reynolds number,’’ Phys. Fluids 9, 1573 ~1997!. 18 A. A. Fowle, C. A. Wang, and P. F. Strong, ‘‘Experiments on the stability of conical and cylindrical liquid columns at low Bond numbers,’’ in Proc. 3rd European Symp. Mat. Sci. Space, 317 ~1979!. 19 H. C. Lee, ‘‘Drop formation in a liquid jet,’’ IBM J. Res. Dev. 18, 364 ~1974!. 20 A. E. Green, ‘‘On the non-linear behaviour of fluid jets,’’ Int. J. Eng. Sci. 14, 49 ~1976!. Phys. Fluids, Vol. 11, No. 5, May 1999 21 J. Meseguer, ‘‘The breaking of axisymmetric slender liquid bridges,’’ J. Fluid Mech. 130, 123 ~1983!. 22 J. Meseguer and A. Sanz, ‘‘Numerical and experimental study of the dynamics of axisymmetric slender liquid bridges,’’ J. Fluid Mech. 153, 83 ~1985!. 23 A. Sanz and J. L. Diez, ‘‘Non-axisymmetric oscillations of liquid bridges,’’ J. Fluid Mech. 205, 503 ~1989!. 24 A. Borkar and J. A. Tsamopoulos, ‘‘Boundary-layer analysis of the dynamics of axisymmetric capillary bridges,’’ Phys. Fluids A 3, 2866 ~1991!. 25 T.-Y. Chen and J. Tsamopoulos, ‘‘Nonlinear dynamics of capillary bridges: theory,’’ J. Fluid Mech. 255, 373 ~1993!. 26 A. Sanz, ‘‘The influence of the outer bath in the dynamics of axisymmetric liquid bridges,’’ J. Fluid Mech. 156, 101 ~1985!. 27 D. J. Mollot, J. A. Tsamopoulos, T.-Y. Chen, and A. Ashgriz, ‘‘Nonlinear dynamics of capillary bridges: experiments,’’ J. Fluid Mech. 255, 411 ~1993!. 28 J. A. Nicolas and J. M. Vega, ‘‘Weakly nonlinear oscillations of nearly inviscid axisymmetric liquid bridges,’’ J. Fluid Mech. 328, 95 ~1996!. 29 F. J. Mancebo, J. A. Nicolas, and J. M. Vega, ‘‘Chaotic oscillations in a nearly inviscid, axisymmetric capillary bridge at 2:1 parametric resonance,’’ Phys. Fluids 10, 1088 ~1998!. 30 T. Sridhar, V. Tirtaatmadja, D. A. Nguyen, and R. K. Gupta, ‘‘Measurement of extensional viscosity of polymer solutions,’’ J. Non-Newtonian Fluid Mech. 40, 271 ~1991!. 31 M. J. Solomon and S. J. Muller, ‘‘The transient extensional behavior of polystyrene-based Boger fluids of varying solvent quality and molecular weight,’’ J. Rheol. 40, 837 ~1996!. 32 S. H. Spiegelberg, D. C. Ables, and G. H. McKinley, ‘‘The role of endeffects on measurements of extensional viscosity in filament stretching rheometers,’’ J. Non-Newtonian Fluid Mech. 64, 229 ~1996!. 33 M. Yao and G. H. McKinley, ‘‘Numerical simulation of extensional deformations of viscoelastic liquid bridges in filament stretching devices,’’ J. Non-Newtonian Fluid Mech. 74, 47 ~1998!. 34 R. W. G. Shipman, M. M. Denn, and R. Keunings, ‘‘Mechanics of the ‘‘falling plate’’ extensional rheometer,’’ J. Non-Newtonian Fluid Mech. 40, 281 ~1991!. 35 R. Kroger, S. Berg, A. Delgado, and H. J. Rath, ‘‘Stretching behavior of large polymeric and Newtonian liquid bridges in plateau simulation,’’ J. Non-Newtonian Fluid Mech. 45, 385 ~1992!. 36 S. Gaudet, G. H. McKinley, and H. A. Stone, ‘‘Extensional deformation of Newtonian liquid bridges,’’ Phys. Fluids 8, 2568 ~1996!. 37 J. Eggers and T. F. Dupont, ‘‘Drop formation in a one-dimensional approximation of the Navier-Stokes equation,’’ J. Fluid Mech. 262, 205 ~1994!. 38 J. F. Padday, G. Petre, C. G. Rusu, J. Gamero, and G. Wozniak, ‘‘The shape, stability and breakage of pendant liquid bridges,’’ J. Fluid Mech. 352, 177 ~1997!. 39 Y.-J. Chen and P. H. Steen, ‘‘Dynamics of inviscid capillary breakup: collapse and pinchoff of a film bridge,’’ J. Fluid Mech. 341, 245 ~1997!. 40 J. A. Holbrook and M. D. LeVan, ‘‘Retardation of droplet motion by surfactant. Part 1. Theoretical development and asymptotic solutions,’’ Chem. Eng. Commun. 20, 191 ~1983!. 41 J. A. Holbrook and M. D. LeVan, ‘‘Retardation of droplet motion by surfactant. Part 2. Numerical solutions for exterior diffusion, surface diffusion, and adsorption kinetics,’’ Chem. Eng. Commun. 20, 273 ~1983!. 42 R. W. Flumerfelt, ‘‘Effects of dynamic interfacial properties on drop deformation and orientation in shear and extensional flow fields,’’ J. Colloid Interface Sci. 76, 330 ~1980!. 43 H. A. Stone and L. G. Leal, ‘‘The effects of surfactants on drop deformation and breakup,’’ J. Fluid Mech. 220, 161 ~1990!. B. Ambravaneswaran and O. A. Basaran 44 1015 W. J. Milliken, H. A. Stone, and L. G. Leal, ‘‘The effect of surfactant on the transient motion of Newtonian drops,’’ Phys. Fluids A 5, 69 ~1993!. 45 W. J. Milliken and L. G. Leal, ‘‘The influence of surfactant on the deformation and breakup of a viscous drop: the effect of surfactant solubility,’’ J. Colloid Interface Sci. 166, 275 ~1994!. 46 D. M. Leppinen, M. Renksizbulut, and R. J. Haywood, ‘‘The effects of surfactants on droplet behaviour at intermediate Reynolds numbers, I & II,’’ Chem. Eng. Sci. 51, 479 ~1996!. 47 Y. Pawar and K. J. Stebe, ‘‘Marangoni effects on drop deformation in an extensional flow: The role of surfactant physical chemistry. I. Insoluble surfactants,’’ Phys. Fluids 8, 1738 ~1996!. 48 E. I. Franses, O. A. Basaran, and C.-H. Chang, ‘‘Techniques to measure dynamic surface tension,’’ Curr. Opin. Colloid Interface Sci. 1, 296 ~1996!. 49 A. Hirsa, G. M. Korenowski, L. M. Logory, and C. D. Judd, ‘‘Velocity field and surfactant concentration measurement techniques for free-surface flows,’’ Exp. Fluids 22, 239 ~1997!. 50 L. E. Scriven, ‘‘Dynamics of a fluid interface,’’ Chem. Eng. Sci. 12, 98 ~1960!. 51 R. Aris, Vectors, Tensors, and the Basic Equations of Fluid Dynamics ~Prentice-Hall, Englewood Cliffs, NJ, 1962!. 52 H. A. Stone, ‘‘A simple derivation of time dependent convective diffusion equation for surfactant transport along a deforming interface,’’ Phys. Fluids A 2, 111 ~1990!. 53 D. A. Edwards, H. Brenner, and D. T. Wasan, Interfacial Transport Processes and Rheology ~Butterworth-Heinemann, Stoneham, MA, 1991!. 54 K. J. Ruschak, ‘‘Flow of a falling film into a pool,’’ AIChE. J. 24, 705 ~1978!. 55 H. S. Kheshgi, ‘‘Profile equations for film flows at moderate Reynolds numbers,’’ AIChE. J. 35, 1719 ~1989!. 56 M. Johnson, R. D. Kamm, L. W. Ho, A. Shapiro, and T. J. Pedley, ‘‘The nonlinear growth of surface-tension-driven instabilities of a thin annular film,’’ J. Fluid Mech. 233, 141 ~1991! 57 G. Strang and G. J. Fix, An analysis of the Finite Element Method ~Prentice–Hall, Englewood Cliffs, NJ, 1973!. 58 L. Lapidus and G. F. Pinder, Numerical Solution of Partial Differential Equations in Science and Engineering ~Wiley, New York, 1982!. 59 S. F. Kistler and L. E. Scriven, ‘‘Coating flows,’’ in Computational Analysis of Polymer Processing ~Applied Science, 1983!, p. 243. 60 H. S. Kheshgi and L. E. Scriven, ‘‘Penalty finite element analysis of unsteady free surface flows,’’ in Finite Elements in Fluids ~Wiley, New York, 1983!, Vol. 5, p. 393. 61 O. A. Basaran, ‘‘Nonlinear oscillations of viscous liquid drops,’’ J. Fluid Mech. 241, 169 ~1992!. 62 M. Luskin and R. Rannacher, ‘‘On the smoothing property of the CrankNicholson scheme,’’ Applicable Anal. 14, 177 ~1982!. 63 P. M. Gresho, R. L. Lee, and R. C. Sani, ‘‘On the time-dependent solution of the incompressible Navier-Stokes equations in two and three dimensions,’’ in Recent Advances in Numerical Methods in Fluids ~Pineridge Press, Swansae, UK, 1979!, Vol. 1, p. 27. 64 S. R. Coriell, S. C. Hardy, and M. R. Cordes, ‘‘Stability of liquid zones,’’ J. Colloid Interface Sci. 60, 126 ~1977!. 65 O. E. Yildirim, Deformation and Breakup of Stretching Bridges of Newtonian and Shear-Thinning Liquids, M.S. Thesis, Purdue University, 1999. 66 X. D. Shi, M. P. Brenner, and S. R. Nagel, ‘‘A cascade of structure in a drop falling from a faucet,’’ Science 265, 219 ~1994!. 67 X. Zhang and O. A. Basaran, ‘‘Dynamics of drop formation from a capillary in the presence of an electric field,’’ J. Fluid Mech. 326, 239 ~1996!. 68 J. Eggers, ‘‘Theory of drop formation,’’ Phys. Fluids 7, 941 ~1995!.

1/--страниц