Drop dynamics on the beads-on-string structure for viscoelastic jets:
A numerical study
Jie Lia)
BP Institute & Engineering Department, University of Cambridge, Madingley Road, Cambridge, CB3 0EZ, United Kingdom
Marco A. Fontelosb)
Departamento de Ciencia e Ingenieria, Universidad Rey Juan Carlos, C/Tulipan S/N, 28933 Mostoles, Madrid, Spain
共Received 15 October 2002; accepted 7 January 2003; published 4 March 2003兲
It is well known that a viscoelastic jet breaks up much more slowly than a Newtonian jet. Typically, it evolves into the so-called beads-on-string structure, where large drops are connected by thin threads. The slow breakup process provides the viscoelastic jet sufficient time to exhibit some new phenomena. The aim of this paper is to investigate the drop dynamics of the beads-on-string structure. This includes drop migration, drop oscillation, drop merging and drop draining. We will use a 1D Oldroyd-B model for the viscoelastic jet, and solve this model numerically by an explicit finite difference method. Close to exponential draining of the filament, we found that the variation of the axial elastic force in the filament is roughly four times larger than the variation of the capillary force with opposite sign. This fact implies that the elastic force is responsible for the drop migration and oscillation. Our study of the drop draining process shows that the elastic force also plays an important role here, allowing the liquid to flow from smaller drops into larger drops through the filament. © 2003 American Institute of Physics. 关DOI: 10.1063/1.1556291兴
I. INTRODUCTION
The liquid jet problem has invoked great interest in the scientific community for more than one century. In his pio-neering work,1Rayleigh studied the capillary stability of an inviscid jet. His findings were later extended to viscous jets. Linear theory is particularly successful in predicting the evo-lution of Newtonian jets, in excellent agreement with experi-ment. Recently interest has been focused on the late stages of jet evolution, prior to breakup. In particular, Eggers found a self-similar solution for a Newtonian jet in the inertial regime,2while Papageorgiou found another in the inertia free regime.3 Research on non-Newtonian jets has followed the same path. Linear stability analysis shows that a liquid jet of a viscoelastic fluid is less stable than a Newtonian jet of the same zero shear viscosity.4 This is surprising, since experiments5,6show that the breakup of the viscoelastic jet is delayed by several orders of magnitude. In fact, elastic stress plays an important role in the jet dynamics, and the vis-coelastic effects are only triggered when the polymers are significantly stretched by the flow. A similar situation can be found in the temperature induced Marangoni effect on jet stability, as has been described in the context of fiber spinning.7Regarding the jet evolution prior to breakup, simi-larity solutions have been found for both the linear Oldroyd-B model8and the nonlinear Giessekus model.9
While all prior works focus on jet stability at early times and jet breakup at late times, we restrict ourselves in this
work to some interesting phenomena of jet evolution at in-termediate times. As we mentioned above, a viscoelastic jet breaks up much more slowly than a Newtonian jet. Typically, it evolves into the so-called beads-on-string structure, where large drops are connected by thin threads. This configuration is extremely robust, in some cases, the viscoelastic jet may not even break up over the entire duration of the experiment. The slow breakup process provides the viscoelastic jet suffi-cient time to exhibit some new phenomena. It was reported that the drops slide along the thread at different velocities and as a result, merging occurs creating drops of larger diameter.5It was also reported that drops move up and down along the thread, following oscillatory trajectories.10 Finally, draining between drops of different sizes occurs in some cir-cumstances. The main reason for this drop draining process is that the capillary pressure is higher in smaller drops, and the threads connecting them serve as channels to transport fluids. This is analogous to the situation in which two air bubbles are connected through a small tube; the smaller bubble will collapse while the larger one grows.
Drop size is a primary concern in applications such as ink jet printing, fire fighting or spraying liquids from the air. The abovementioned drop dynamics may have an important effect in controlling the size of drops into which fluids break up. This paper is dedicated to the investigation of the non-linear dynamics of drops on the beads-on-string structure. We choose the linear Oldroyd-B model for the constitutive equations of the viscoelastic liquid, instead of the nonlinear Giessekus or FENE models. It can be shown that for the Oldroyd-B fluid breakup will never occur. This is in dis-agreement with experiments because the viscoelastic jet does a兲Electronic mail: [email protected]
b兲Electronic mail: [email protected]
PHYSICS OF FLUIDS VOLUME 15, NUMBER 4 APRIL 2003
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always break eventually. However, the Oldroyd-B model is appropriate in the frame of this work, since our primary in-terest here is to study the drop dynamics during intermediate times. Keunings has developed an algorithm for the simula-tion of two-dimensional 共axisymmetric兲 flow with a free surface.11 This approach is computationally expensive for our problem. In order to reduce the coding and computing tasks, we consider a much simpler problem by assuming that the kinematics of the jet motion is essentially one dimen-sional. A 1D approach has been already used by Bousfield
et al.12 They also compared the two approaches and con-cluded the agreement between the two is excellent but the 1D approach is computationally much more efficient. The 1D model we used here is equivalent to that of Bousfield
et al. While they used a Lagrangian formulation, in the
present work we use an Eulerian formulation. We prefer the latter formulation because it is easy to code on a fixed mesh. Finally, the resulting equations are solved numerically by an explicit finite difference scheme on a uniform mesh.
The rest of this paper is organized as follows. In Sec. II, the governing equations and the numerical method are ex-plained. In Sec. III, we will show several examples of drop dynamics. In Sec. IV, we will study the drop migration, os-cillation and merging phenomena. In Sec. V, we will inves-tigate the drop draining process. And finally, a conclusion is given in Sec. VI.
II. GOVERNING EQUATIONS AND NUMERICAL METHOD
We consider an axisymmetric jet which is ‘‘slender’’ in the sense that the velocity and the stress components do not vary much in the cross-section. The kinematics of the motion is therefore essentially one dimensional, and all variables can be regarded as functions of the position z and the time t. It is well known that under the slender jet assumption, the mass and the momentum conservation laws can be considerably simplified.2,13,14 We denote the jet radius as h(z,t) and the velocity asv(z,t). In all the following, we choose the initial jet radius R as the characteristic length, and the correspond-ing capillary time (R3/␥)1/2 共where␥ is the surface tension coefficient兲 as the characteristic time. The governing equa-tions for jet evolution are
h2 t ⫹ 共h2v兲 z ⫽0, 共1兲 v t ⫹v v z⫽⫺ z⫹ 1 h2 z关h 2共zz⫺rr兲兴, 共2兲 where is interface curvature,zz andrr are the axial and radial stresses. Equation 共1兲 expresses mass conservation, and Eq. 共2兲 is the momentum equation in the one-dimensional approximation. In order to close this system of equations, we need to relate the axial and radial stresses to the velocityv. This depends on the constitutive law for the fluid. For the Oldroyd-B liquid, we have
zz⫽2s
v
z⫹zz, rr⫽⫺s
v
z⫹rr,
where s is the solvent viscosity, and zz and rr are the polymer contributions to the stresses. The elastic stresses are governed by the partial differential equations:
zz⫹D
冉
zz t ⫹v zz z ⫺2 v zzz冊
⫽2p v z, 共3兲 rr⫹D冉
rr t ⫹v rr z ⫹ v zrr冊
⫽⫺p v z, 共4兲where p is the polymer viscosity and D the dimensionless
relaxation time, also called the Deborah number.
The above 1D model is consistent as long as the jet shape remains slender. This is problematic in the transitory region between the filaments and the drop. Here we adopt an
ad hoc approach by retaining all the curvature terms in the
computation. This makes spherical drops exact static solu-tions of the equation, and we can expect at least some quali-tative validity for our system of equations, especially in the filaments and the drops regions. We emphasize that this ap-proach has been very successful in the study of Newtonian jets.2,3
We solve the above system by a finite difference method. In general, a conservative scheme is preferable. To this aim, we first write the momentum equation in conservative form. The curvature term in Eq.共2兲can be rewritten as共see Ref. 15兲 ⫺z⫽ 1 h2 z
冉
h 2冉
hzz 共1⫹hz2兲3/2⫹ 1 h共1⫹hz2兲1/2冊冊
⬅ 1 h2 z共h 2K兵h其兲. 共5兲Using Eq. 共1兲, we transform Eq.共2兲into conservative form:
共h2v兲 t ⫹ 共h2v2兲 z ⫽ z
冋
h 2冉
K⫹3 s v z⫹zz⫺rr冊册
. 共6兲 Finally, we use an explicit finite difference scheme to solve the above system of Eqs. 共1兲, 共6兲,共3兲, and 共4兲 on uniform mesh. Both collocated grids where all the variables are de-fined at the cell center and staggered grids where the velocity is defined on the cell face were tested. We found no signifi-cant difference in our computational results. Our method has been also compared with an adaptive implicit method, with excellent agreement between the two methods.III. DYNAMICS OF DROPS ON THE BEADS-ON-STRING STRUCTURE
We have observed a rich and complex drop dynamics from our numerical simulation. These include drop migra-tion, drop oscillamigra-tion, drop merging and drop draining. In this section, we will briefly describe these nonlinear phenomena. This will provide us the background for a deeper analysis in the next two sections. In all the following simulations, we study periodic jets of viscoelastic liquid. The jets are initially at rest so that the viscoelastic stresses zz andrr are both 923 Phys. Fluids, Vol. 15, No. 4, April 2003 Drop dynamics on the beads-on-string structure
zero. The jets are uniform and their radii are R. In order to provoke the instability, we introduce a small sinusoidal per-turbation to the jet profile
h共z,0兲⫽R⫹⑀ cos共2␣z/L兲,
where⑀is the amplitude of perturbation,␣ the wave number and L the period. We start with a simple case where a jet evolves straightforwardly towards the beads-on-string struc-ture. Figure 1 shows the evolution of an interface with period
L⫽20. In order to provide the reader a clear visualization, the profiles are shown for a length of two periods. The vis-cosities and Deborah number for this simulation are s ⫽0.79,p⫽2.37, and D⫽94.9. The initial radius of the jet is
R⫽1, and the amplitude of the sinusoidal wave is ⑀
⫽10⫺3. Initially 关Figs. 1共a兲 and 1共b兲兴, the jet is not very much stretched and the elastic stress has not built up to a significant level; the stress is mainly viscous, derived from the solvent viscosity. At this stage, the viscoelastic jet is es-sentially Newtonian in character. However, at time t⫽63.2
关Fig. 1共c兲兴, the elastic stress has grown so that it is compa-rable to the capillary pressure and viscous stress. We see clearly a uniform filament form at the center 共we call this
phase of the flow the elastic phase兲. This is quite different from a Newtonian jet, whose filament, in the inertia free Stokes regime, is in parabolic form 共Papageorgiou’s similar-ity solution3兲. As the elastic stress continues to build up uni-formly in the filament, the jet evolves into beads-on-string geometry. Figure 1共f兲shows a structure where the drops are almost spherical and the filaments collapse into thin threads whose radii are several orders of magnitude smaller than the drop’s radius.
In the above simulation, the viscous force is large com-pared to the surface tension, in the sense that the capillary driven flow is significantly damped by the viscous stress at time t⫽63.2, and the jet does not evolve into asymmetrical shape before it enters into elastic phase. At low viscosities or high surface tension, the scenario is different. Figure 2 shows one simulation with a surface tension 10 times larger than in Fig. 1. If we renormalize all the physical parameters based on the capillary time, the dimensionless numbers are s
⫽0.25,p⫽0.75, and D⫽300. In this flow, the solvent vis-cosity is too small to damp the capillary driven flow and the polymer viscosity too small for the elastic stress to build up
FIG. 1. s⫽0.79, p⫽2.37, and D⫽94.9. After the
rapid formation of the beads-on-string structure, one observes the slow thinning of the filament. The relative times of each profile are共a兲31.6,共b兲55.6,共c兲63.2,共d兲 94.9,共e兲158.1 and共f兲948.4.
quickly. At time t⫽44.0共Fig. 2兲, we see asymmetrical necks form near the drops. This type of pinching is typical of the behavior of a Newtonian jet in the inertia regime 共Eggers’ similarity solution2兲, which we do not see in the last simula-tion. However, at this stage, the jet is very much stretched at the minimum neck so that the axial elastic stress grows rap-idly in this region. We are now entering the elastic phase. The very high axial elastic stress prevents the jet from further necking down and quickly pushes the liquid away towards the middle 共within six time units兲, Figs. 2共a兲–2共e兲, and a small satellite drop is formed at t⫽50. Figure 2共f兲shows the late beads-on-string profile at time t⫽2000 with two differ-ent drop radii. The phenomenon of the axial elastic stress resisting the jet necking down and pushing the liquid away towards the center has been already reported in Ref. 16, and referred to as ‘‘recoil,’’ though they were only able to nu-merically track the jet evolution at the very early phase.
We have seen the filament draining from the above two simulations. The inverse drop draining is possible for small surface tension. In Fig. 3 the surface tension is only two times larger than in Fig. 1. The evolution of the interface is similar to that in Fig. 2 but the amplitude of the neck is
considerably smaller. At time t⫽58.1, we see the first stages of the formation of a satellite drop, which reaches its maxi-mum size at t⫽67.1. However, this time, the surface tension is not strong enough to squeeze the filament and create a permanent satellite drop. Instead, this drop is drained into the filament and disappears completely at t⫽98.4. The jet re-verts to the mono beads-on-string structure. To the best of our knowledge, such drop draining process has not been re-ported elsewhere, and is mainly due to the capillary pressure difference between drops of different size. We shall discuss this phenomenon in detail in Sec. V.
We wish to study the dynamics of these drops, in other words, the motion of drops on the beads-on-string structure. This includes drop migration, oscillation and coalescence. These phenomena were observed in prior experimental works.5,10 The above simulations were performed on half period with the symmetry condition reinforced. The drops we see are therefore stationary. In order to observe and study the drop motion, we shall relax the enforced symmetry and perform the computation for a whole period. In addition to the sinusoidal perturbation, another small perturbation will be added in order to break the symmetry in the initial
sinu-FIG. 2. s⫽0.25,p⫽0.75, and D⫽300. The relative
times of each profile are共a兲44.0,共b兲45.0,共c兲47.0,共d兲 48.0,共e兲50.0, and共f兲2000.
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soidal perturbation. There are various ways to introduce the asymmetry in the initial condition. One easy way is to slightly increase the amplitude of the sinusoidal wave for the right half of the jet. Figure 4 shows such a simulation for a surface tension five times larger than the one in Fig. 1. The renormalized parameters are s⫽0.35, p⫽1.06 and D
⫽212.1. The amplitude of the sinusoidal perturbation ⑀
⫽10⫺3. To introduce some asymmetry in the initial condi-tion, an additional 0.1% is added to the amplitude of right half of the perturbation, so that ⑀⫽1.001⫻10⫺3 for z
⬎L/2. Initially, the jet evolves into the same kind of
beads-on-string structure as in Fig. 2, with drops of different radii
关Figs. 4共a兲– 4共b兲兴. The formation of main drops and satellite drops are also similar. If the initial condition were exactly symmetrical, since the numerical computation preserves the symmetry, we would expect that this beads-on-string struc-ture will last as in Fig. 2. However, we encounter here an unstable situation: a slight deviation from the symmetry will induce unbalanced forces in the filaments at the sides of the satellite drop, and it will start to migrate in the left direction, Figs. 4共b兲– 4共e兲. Finally the satellite drop merges with the
main drop 关see Fig. 4共f兲兴. We shall discuss the migration process in detail in the next section.
IV. MIGRATION, OSCILLATION AND MERGING
The difficulty in analyzing the stability of beads-on-string structures and drop migration is that we encounter an unsteady situation. The filament continues to drain into drops and becomes thinner and thinner. Hence we do not have a steady basic flow, but we can start our analysis from a basic beads-on-string structure in the ideal exponential draining regime. To this aim, we consider the balance between the forces exerted by the filaments at both sides of a drop. From Eq. 共6兲, we see that the total force at a cross-section is
h2
冉
K⫹3sv
z⫹zz⫹rr
冊
.When the beads-on-string structure is established, the elastic axial stress in the filament has grown so that it is comparable with the capillary pressure. On the other hand, the viscous stress and the radial elastic stress are small and may be
ne-FIG. 3.s⫽0.56,p⫽1.68, and D⫽134.2. The relative
times of each profile are共a兲53.7,共b兲58.1,共c兲67.1,共d兲 85.0,共e兲98.4, and共f兲107.3.
glected. Therefore, the force in the filament can be approxi-mated by the sum of the most important terms:
h2共K⫹zz兲⫽h⫹h2zz.
It is well known that in the elastic phase the radius of an uniform circular filament decays exponentially,8,17,18
h⫽h0e⫺t/3D,
while the axial elastic stress increases exponentially
zz⫽0et/3D,
where h0 and0are two constants. Therefore the total force in the filament is
共h0⫹h0 2
0兲e⫺t/3D.
Let us denote the radius of the left filament at a given time
t⫽0 by h1 and the one at the right by h2. Similarly, we denote the axial elastic stresses in both filaments by1 and
2. The force exerted on the drop is therefore
关共h2⫹h2 2
2兲⫺共h1⫹h1 2
1兲兴e⫺t/3D. 共7兲
If this total force is not zero at some time t, which implies that the coefficient before the exponential term is not zero, then the total force can never be zero and it will always keep the same sign. This force acts on the drop and it will move under the hypothesis of exponential draining once the equi-librium is broken.
There is, however, another factor which must be taken into consideration. When the jet collapses into the beads-on-string structure, a lot of surface tension energy may be trans-ferred to kinetic energy, and the asymmetry in the initial condition may affect the force balance between the two sides of the drop. The consequence of this imbalance is that at the beginning of the beads-on-string structure, the drops may already have gained some nonzero momentum. This can be best illustrated through the example in Fig. 5. The dimen-sionless parameters are again s⫽0.35, p⫽1.06, and D
⫽212.1, as in Fig. 4. We initialize the interface perturbation with one wave whose wave number is, however,␣⫽2. The initial wave amplitude is ⑀⫽10⫺3. An additional 5% is added to the right half of the wave, so that the amplitude is
⑀⫽1.05⫻10⫺3 for z⬎L/2. After an initial evolution, the jet
FIG. 4.s⫽0.35,p⫽1.06, and D⫽212.1. The relative
times of each profile are共a兲56.6,共b兲141.4共c兲212.1,
共d兲353.6,共e兲381.8, and共f兲424.3.
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develops into a beads-on string structure with almost identi-cal size drops. Figure 5 shows the migration of the drops and Fig. 6 a close-up of the drop merging process. The immedi-ate consequence of the asymmetry in initial condition is to make the middle drop共initially at the position z⫽10) move in the left direction and it eventually merges with the drop at its left, at time t⫽742.46关Fig. 6共e兲兴. We notice that in the meantime the filaments continue to drain into drops and be-come thinner and thinner as one would expect.
To shed new light into the process of jet collapse and drop migration, we investigate the velocity profile and the force balance during the migration. Figure 7 shows the ve-locity at times t⫽70.7 and t⫽565.7. These correspond to the interface profiles共a兲and共f兲in Fig. 5, and the corresponding force profiles are shown in Figs. 7共c兲 and 7共d兲. At time t
⫽70.7, the jet has just evolved into the beads-on-string
struc-ture, yet the drop in the center has gained a considerable momentum towards the left. This is shown by the negative velocity in the dashed line box in Fig. 7共a兲. As we mentioned above, this kinetic energy is transferred from the surface ten-sion energy when the jet collapses. Though the difference in the initial wave amplitudes between z⬍L/2 and z⬎L/2 is
quite small 共5%兲, its result on the force balance is consider-able. We can reduce this difference further and still observe the onset of the drop motion, but the whole migration and merging processes will be slower. We will show simulations with smaller difference in initial amplitude for other physical parameters later in this paper.
We have seen that the beads-on-string structure is not stable and the collapse of the jet may transfer surface energy to kinetic energy and set the drop into motion. The next question is how this drop motion will interact the force
bal-FIG. 5.s⫽0.35,p⫽1.06, and D⫽212.1. The relative
times of each profile are共a兲70.7,共b兲141.4,共c兲212.1,
共d兲282.3,共e兲424.3,共f兲565.7, and共g兲707.1.
ance between the two filaments. Will the drop move more like a rigid body, or more like a soliton wave? In the first case, an immediate effect of drop motion is to compress the filament before the drop and stretch the filament behind. In this situation the exponential draining regime will be de-stroyed. We may also expect two competing mechanisms in the force balance:共1兲the capillary pressure force whose ef-fect is usually in Newtonian jets destabilizing and would in principle set the drop into motion, and 共2兲 the elastic force which would restore the beads-on-string structure and make the drop oscillate around its equilibrium position. In other words, the capillary force is a destabilizing factor, while the elastic force acts as an stabilizing factor. In fact, a close scrutiny of the velocity profile shows that the scenario is exactly the opposite. In Fig. 7共a兲, the two straight lines cor-respond to the velocity profile in the two filaments at both
sides of the drop at t⫽70.7. Their slopes are well approxi-mated by the exponential decay value 2/3D. Later, at time
t⫽565.7, the drop has moved considerably关corresponding to Fig. 5共f兲兴, the velocity profile in both filaments are still two straight lines whose slopes are still well approximated by the exponential decay value 2/3D. This is clear evidence that the drop migration affects little the exponential thinning of the filament and the drop moves like a soliton which absorbs the liquid from the filament ahead of the direction of motion and releases the liquid to the one behind.
Now we analyze the force balance in the filaments. Fig-ures 7共c兲 and 7共d兲 show the force profile at t⫽70.7 and t
⫽565.7. The force in the filaments is constant, a fact evi-denced by the horizontal lines in the dashed line boxes. The peaks in these profiles are at the transition zone between the filaments and the drops. These peaks are due to the sudden
FIG. 6.s⫽0.35,p⫽1.06, and D⫽212.1. The relative
times of each profile are 共a兲 714.18, 共b兲 721.25,共c兲 738.32, 共d兲 735.39, 共e兲 742.46, 共f兲 749.53, and 共g兲 756.60.
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change of the velocity in the transition zone. At t⫽70.7, the force in the right filament is higher, which means that the forces are working against the drop motion. However, at t
⫽565.7, the force balance is reversed, and the net force cre-ated by the filaments push the drop to move in the left direc-tion. We see that the velocity did increase on module at t
⫽565.7关shown in the dashed line boxes in Fig. 7共d兲兴 com-pared to t⫽70.7. Notice that the difference between the forces in the two filaments is several orders of magnitude smaller than the force itself, yet this tiny difference is enough to accelerate considerably the drop motion.
It is very revealing to look at the total force change during the drop migration as well as the individual contribu-tions. In Fig. 8共a兲, line A represents the capillary force dif-ference h1⫺h2, line B in the elastic stress force difference
h121⫺h2 2
2, and line C the total force difference. We notice that the difference between the two radii共capillary forces兲is decreasing, starting with a positive value, and ending with a negative value. That means that the radius of the filament on the left is larger than the one on the right at the beginning, but finishes being smaller. On the other hand, the elastic
stress force follows an inverse trend. Furthermore, line A and line B illustrate that the difference in elastic force is roughly four times larger than the difference in capillary force with inverse sign. Consequently, the difference of total force in-creases as the elastic force and is positive during most of the drop migration. We conclude that the elastic force, not the capillary force, plays the decisive role in the drop migration. In order to understand the above ‘‘four times’’ rule of thumb, we investigate the relation between the variation of the elas-tic force and the variation of the capillary force close to the exponential decay regime. In a uniform filament, we have
v
z⫽⫺2 h˙ h.
Because the elastic stress is also uniform in the filament, from Eq. 共3兲, we obtain
zz⫹D
冉
zz t ⫹4 h˙ h ˙ zz冊
⫽⫺4p h˙ h .FIG. 7.s⫽0.35,p⫽1.06, and D⫽212.1.共a兲The velocity profile at t⫽70.7,共b兲the velocity profile at t⫽565.7.共c兲The force profile at t⫽70.7,共d兲the force
profile at t⫽565.7.
In the filament of beads-on-string structure, the viscous term is small and can be neglected. The above equation can then be simplified as zz t ⫽⫺
冉
4 h˙ h ˙⫹D冊
zz.It can be integrated and we obtain
zz⫽h0 4
0
h4et/D,
where h0is the radius of the filament at a given time共that we choose to be zero兲, and0 the elastic stress. The variation in the elastic forcezzis related to the variation in the capillary
force h through
共h2zz兲
h 共0兲⫽⫺
2h040
h3et/D共0兲⫽⫺2h00.
In the exponential draining regime, there are two a priori free parameters, h0 and 0. Asymptotic analysis for high
Deborah number jet shows that they are not completely
un-related and their product must be approximately 2. From our numerical simulation, we observe that h00is, in fact, close to 2. Hence
共h2zz兲
h 共0兲⬇⫺4.
This means that a variation of an amount⌬h in the capillary
force around the exponential regime leads to a variation ap-proximately four times higher in the elastic stress force of opposite sign. Perhaps the most amazing fact in the drop motion phenomenon is that drop moves in the direction where the radius of filament is smaller, and the capillary pressure /h is higher. Counter intuitively, the elastic force in the filament, not the capillary force, plays the role of main driving force. In order to track the motion of the drops with clarity, we plot the local maxima of the interface height as-sociated with the two colliding drops against time in Fig.
8共b兲. The full line corresponds to the left drop, and the dashed line to the right one. It is evident that the drops merge in a very short period of time. The elastic modulus
¯⫽p/D 共8兲
plays an important role in the theory associated with the beads-on-string structure. For DⰇ1, it is related to the radius of stable filament as hfilament⫽
冉
¯ 2冊
1/3 .Numerical simulations show also that the above radius is a good estimation for the filament radius at the beginning of the exponential regime. We have performed a simulation with similar parameters as in Fig. 5, except for the polymer viscosity p and the Deborah number D which are taken to
be 10 times larger. Notice that the elastic modulus¯ is then the same as above. We observe no qualitative difference
re-FIG. 9. The position of the maximum of drops versus time. The full line is the for the left drop and the dashed line the right one.共A兲for parameters
s⫽0.35,p⫽1.06, and D⫽212.1,共B兲s⫽0.35,p⫽10.6, and D⫽2121,
and共C兲s⫽0.14,p⫽4.24, and D⫽2121.
FIG. 8. s⫽0.35,p⫽1.06, and D⫽212.1.共a兲The force balance between the two filaments against time. Line A represents the capillary force difference
h1⫺h2, line B the difference in the elastic stress force h1 2
1⫺h2 2
2, and line C the difference of total force.共b兲The position of the maximum of drops versus
time. The full line is the for the left drop and the dashed line the right one.
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garding the evolution of the jet, the migration and merging of drops. However the total viscosity is almost 10 times larger, due to the polymer contribution, so the migration is much slower. It has been reported in Ref. 10 that drops can sometimes move in a more or less oscillatory manner during the evolution of jets. In the above two simulations, no oscil-lation in the drop motion has been observed. This is typically the case for highly viscous liquids. We found that drops fol-low oscillatory trajectories, for example, for liquid with fol-low viscosities s⫽0.14 and p⫽4.24. Figure 9 shows a com-parison of the trajectories of the two migrating drops for the above three sets of parameters.
We shall study the oscillatory motion for liquid with low viscosity in more detail in the next example. The dimension-less parameters we used are s⫽0.079, p⫽2.37, and D
⫽948.7. The initial radius of the unperturbed jet is R⫽1 as usual, while the period is L⫽15. The initial perturbation to the jet interface consists of a sinusoidal wave with wave number ␣⫽2. The initial wave amplitude is ⑀⫽10⫺3. An additional 1% is added to the right half of the wave, so that
the amplitude is ⑀⫽1.01⫻10⫺3 for z⬎L/2. We plot the
drops trajectories against the time in Fig. 10共b兲. We observe that in this simulation, the two drops have oscillated four times before they merge together. To understand this oscilla-tory motion, we show the force balance between the two filaments against time in Fig. 10共a兲. We notice that during the jet evolution the forces are no longer monotone functions of time but oscillate around zero. Once again, the difference of the elastic force in the two filaments is roughly four times larger than the difference of the capillary force, with an op-posite sign. The oscillation of the drops is therefore driven by the elastic force. In Fig. 11, we plot the individual forces in both filaments, and compare them to the theoretical fits in the exponential decay regime. We observe that the radius of the left filament decreases almost in a monotone way and the decrease is slightly faster than the theoretical one. On the other hand, the radius of the right filament oscillates vio-lently around the theoretical curve. Meanwhile, the elastic stress in both filaments evolves according to the ‘‘four times’’ rule of thumb. From Fig. 10共b兲, we can see how the
FIG. 10. s⫽0.079,p⫽2.37, and D⫽948.7.共a兲The force balance between the two filaments against time. Line A represents the capillary force difference
h1⫺h2, line B the difference in the elastic stress force h1 2
1⫺h2 2
2, and line C the difference of total force.共b兲The position of the maximum of drops versus
time. The full line is the for the left drop and the dashed line the right one.
FIG. 11.s⫽0.079,p⫽2.37, and D⫽948.7.共a兲The capillary force共filament radius兲h against time.共b兲The axial elastic force h2zzagainst time. Line A
represents the quantities in the left filament, line B the quantities in the right one, and line C the quantities in the exponential decay regime.
middle drop 共initially at z⫽7.5) migrates in the right direc-tion. This direction of motion is opposite to what we saw in the three previous examples. Finally, we note that in the os-cillatory migration, the forces evolve in a monotone way in the filament ahead of the drop and oscillate behind the drop.
V. DROP DRAINING
In Fig. 3, we show a simulation where a small drop drained into a larger drop. Intuitively, the capillary pressure inside the smaller drop is high so it drains quickly and only the large drop remains at the end. However, if the filament radius is much smaller, then the capillary pressure there is much higher than in the drops and forms an energy barrier. The draining of drops can be possible only when the axial elastic force makes a significant contribution to the force balance and reduces the energy barrier in the filament. This
condition is satisfied until the stretched polymer has time to relax, so that the axial elastic force remains large in the fila-ment, well before the beads-on-string structure enters the ex-ponential draining regime. We observe that for some param-eters the draining of drops and the draining of filaments can occur successively in the evolution of the same jet. In Fig. 12, we have slightly increased the surface tension. The di-mensionless parameters are s⫽0.46, p⫽1.37, and D
⫽164.3. At t⫽16.4, the beads-on-string structure is estab-lished with two different radii. The satellite drop reaches its maximum size at this time 关Fig. 12共a兲兴. From t⫽16.4 关Fig. 12共b兲兴, the satellite drop starts to drain slowly. However, the drop draining stops at t⫽273.9关Fig. 12共e兲兴. The axial elastic stress has relaxed for enough time and inverse filament draining starts. The final result is a beads-on-string structure with two different drop sizes.
FIG. 12. s⫽0.46,p⫽1.37, and D⫽164.3. The
rela-tive times of each profile are 共a兲 16.4, 共b兲54.8, 共c兲 109.5,共d兲164.3,共e兲273.9,共f兲547.7, and共e兲1040.7.
933
The relaxation of the polymer seems to be the crucial factor in the inversion of the draining process. With this in mind, we performed a simulation for a similar set of param-eters. We kept the elastic modulus¯ constant but increased the polymer viscositypand the Deborah number 10 times.
The evolution of the interface profile is shown in Fig. 13. We find no inversion of the draining process and the satellite drop disappears completely at the end. Until now, we have seen only draining of a satellite drop. Drop draining can occur also between drops of similar size. Figure 14 shows a simulation for high Deborah number. The dimensionless pa-rameters are s⫽0.1, p⫽2⫻103 and D⫽105, and the pe-riod L⫽20. We initialize the interface perturbation by super-posing two sinusoidal waves. Both of them have initial amplitude ⑀⫽10⫺2. One wave has a wave number ␣⫽1 while the other has wave number ␣⫽2. This produces two
drops with similar size on the initial beads-on-string struc-ture. The drop in the middle is slightly smaller at time t
⫽100 关Fig. 14共a兲兴. The middle drop drains gradually and disappears completely at time t⫽550关Fig. 14共g兲兴. We show the interface, velocity and force profiles at time t⫽300 in Fig. 15. The vertical dashed lines correspond to the same position for the different pictures. We use them to separate the drop region and the filament region. We see that the force inside the large drop is stronger than inside the small one, but it is even stronger in the filament due to the contribution of the axial elastic stress. This kind of force arrangement breaks the energy barrier in the filament only if the surface tension force is the dominant factor. Figure 15共c兲 shows that the velocity has a negative value everywhere; the liquid is flow-ing from the right to the left, with the highest and almost constant flux inside the filament. Ultimately, the filament
FIG. 13. s⫽0.46,p⫽13.7, and D⫽1643. The
rela-tive times of each profile are 共a兲 65.7, 共b兲87.6, 共c兲 109.5,共d兲120.5,共e兲131.5,共f兲142.4, and共e兲153.4.
serves as a channel for the fluid to flow from the small drop into the large one.
VI. CONCLUSION
Prior work on both Newtonian and non-Newtonian jet evolution investigated the linear stability analysis at early times or the breakup at late times. In this work, our interest is in the new problem of the drop dynamics of the beads-on-string structure for viscoelastic jets. These phenomena in-clude drop migration, oscillation, merging and draining. They occur at the intermediate times of jet evolution. The first three phenomena have been observed in experimental works,5,10but no analysis has been attempted. In fact, Goldin
et al. explained the different drop velocities by the air
fric-tion dependence on the drop size,5which is a quite simplistic and unlikely explanation. It is the first time that the drop draining phenomenon is reported.
A 1D Oldroyd-B model under the ‘‘slender’’ jet assump-tion was used to investigate these issues and it has been solved by an explicit numerical method. From the numerical analysis point of view, it is much simpler to code the 1D model than a 2D axisymmetric model, and much more effi-cient in terms of computation time. Our previous experience showed that the 1D model provides a good overall picture of the beads-on-string structure, in good agreement with the liquid filament rheometer共LFR兲experiments.19In this work, we took the 1D model to a new stage and showed that it is able to capture the essential features of the more complex drop dynamics observed in the existing experiments. Our
FIG. 14. s⫽0.1,p⫽2⫻10
3
, and D⫽105. The rela-tive times of each profile are共a兲100.0,共b兲200.0,共c兲 300.0,共d兲400.0,共e兲450.0,共f兲500.0, and共e兲550.0.
935
work provides, as far as we know, the first study on these complex nonlinear phenomena. We observed two modes of migration prior to drop merging. For a viscous liquid drop migration is monotonic, while for a less viscous liquid, drop migration is oscillatory. By analyzing the force variation close to the exponential draining regime, we found that the variation of the axial elastic force in the filament is roughly four times larger than the variation of the capillary force with opposite sign. The elastic force is therefore the dominant factor in the force balance created by the two filaments at both sides of a given drop. The surprising conclusion from this ‘‘four times’’ law is that the axial elastic force, not the capillary force, is the driving force in the migration process. Finally, the drop draining process from a small drop into a larger one was also investigated from a force balance analy-sis. The capillary pressure inside the smaller drop is higher, and it tends to drain into the larger drop. However, if the filament radius is small enough, the capillary pressure there is high and it forms an energy barrier. This energy barrier can be relaxed if the polymer in the filament is sufficiently stretched and has not had enough time to relax. Most of the
drop draining processes we observed involved drops with very different size in the case of moderate Deborah number. These drops are typically the satellite drops formed during jet collapse. Draining between drops with similar size is only possible for a very high Deborah number jet.
‘‘Real’’ jets break in finite time, roughly estimated as 3 to 10 relaxation times of the fluid. It is therefore interesting to ask whether the physical pattern studied can be observed in the real world, that is, before the jet breakup. The answer to this question is yes, and this work was indeed motivated by the drop migration and oscillation observed in the experi-ment. In general, the drop motion is dictated by asymmetric effects in the filaments, which are dependent on the pertur-bation in the experiment, not the relaxation time of the liq-uid. If the asymmetric effect is strong enough, drop migra-tion and oscillamigra-tion will be observed. Regarding drop drainage, its characteristic time is much smaller than the ten-tative breakup time in our examples.
Due to the complexity of the phenomena involved in drop dynamics of the beads-on-string structure, many ques-tions remain unanswered. For example, 共1兲what is the
con-FIG. 15. s⫽0.1,p⫽2⫻10
3
, and D⫽105.共a兲Interface profile at t⫽300,共b兲force profile, and共c兲velocity profile at t⫽300. The vertical dashed lines correspond to the position in different pictures.
dition separating monotonic and oscillatory migration? 共2兲 What is the condition for drop draining? 共3兲How does the migration process couple with the drop draining process? By no means does this work attempt to be comprehensive. For this reason we have not done the parametric study. Consid-erably more work is necessary before the complex drop dy-namics on the beads-on-string structure will be fully under-stood. In particular, we believe that the classical theory on solitary waves could possibly be applied to the present prob-lem to provide a fully satisfactory theoretical picture of the migration and merging phenomena reported.
Note added in proof: The movies of the drop
dy-namics of the beads-on-string are available at http:// www2.eng.cam.ac.uk/⬃jl305/visjet.html/
ACKNOWLEDGMENTS
The code we used in this work was mainly developed during our visit to Essen University, Germany. We thank Dr. J. Eggers for the invitation, the hospitality and some fruitful discussions. We also thank Professor J. Hinch for his useful suggestions and Dr. D. Lyness for proofreading the manu-script. M.F.’s research is partially supported by MCYT Grant No. BFM2002-02042.
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937 Phys. Fluids, Vol. 15, No. 4, April 2003 Drop dynamics on the beads-on-string structure