the deformation of a newtonian drop in the uniaxial extensional flow of a viscoelastic liquid

The deformation of a Newtonian drop in the uniaxialextensional flow of a viscoelastic liquid

S. Ramaswamy, L.G. Leal*

Department of Chemical Engineering, University of California at Santa Barbara,

Santa Barbara, CA 93106, USA

Received 22 September 1998; received in revised form 18 January 1999

Abstract

The deformation of a Newtonian drop in uniaxial flow of a viscoelastic fluid is studied via numerical computations. The

viscoelastic fluid is modeled using the FENE-CR constitutive theory. The computations are performed for two values of the

Chilcott±Rallison extensibility parameter L2 � 144 and L2 � 600, for Deborah numbers between 0 and 2, and for several

values of the concentration parameter c in the range appropriate for dilute (Boger) solutions. It is found that the deformation is

always increased for the viscoelastic fluid compared to the shape in a Newtonian fluid, and that there is a viscoelastic

mechanism (an `instability' of shape) that tends to produce a high curvature at the tips of the drop. The magnitude of stress

gradients immediately downstream from the tip of the drop are reduced as the viscosity ratio (interior/exterior) is reduced,

though the viscoelastic contribution to the drop deformation is actually increased for small viscosity ratios. # 1999 Elsevier

Science B.V. All rights reserved.

Keywords: Newtonian drop; Reynolds number; Chilcott±Rallison model

1. Introduction

In this paper, we report on numerical solutions for the deformation of a Newtonian drop in uniaxialextensional flow of a viscoelastic fluid. We have, in a previous paper [1], studied the inverse problemof a viscoelastic drop in a Newtonian fluid. Perhaps the most significant result of that study was anunderstanding of the complexity of the role of viscoelastic stresses in determining the drop shape.We found that the direct tensile stress contribution of viscoelasticity at the interface resulted in adecrease in the deformation of the drops, and especially a reduction of the curvature at the ends of thedrop. The bulk viscoelastic stresses, however, changed the flow and this modification of the flow tendedto increase the deformation. For a given set of parameters, the final drop shape was determinedby a balance between these two effects. In the present case of a Newtonian drop in a viscoelastic

J. Non-Newtonian Fluid Mech. 88 (1999) 149±172

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* Corresponding author. Fax: +1-805-893-5458.

0377-0257/99/$ ± see front matter # 1999 Elsevier Science B.V. All rights reserved.

PII: S 0 3 7 7 - 0 2 5 7 ( 9 9 ) 0 0 0 1 0 - 5

fluid, we would expect that the direct contributions of the viscoelastic stresses would enhance thedeformation. The influence of viscoelasticity via changes in the bulk flow is however more difficult topredict.

Relatively little theoretical or numerical work has been done on problems with viscoelastic flowswhich involve a free boundary. The problem most closely related to that reported here was thenumerical study due to Bousfield and co-workers [2] of the transient deformation of a bubble in aviscoelastic extensional flow. A related problem that has been more heavily investigated is thebuoyancy-driven motion of a gas bubble rising through a non-Newtonian liquid. Experiments in thiscase often show the formation of a cusp at the rear of the bubble [3,4]. The small deformation limit for aspherical bubble rising in a viscoelastic fluid was studied by Ajayi [5], Hassager [6], Tiefenbruck andLeaf [7], Wagner and Slattery [8] using perturbation techniques. Of course, small deformation theorycannot predict phenomena such as cusp formation at the rear of the bubbles. Though Chilcott andRallison [9] obtained numerical solutions for viscoelastic flow around a cylindrical bubble, only a fewstudies have been done since the earlier work of Bousefield et al. [2] to investigate the influence ofviscoelastic stresses on the deformation of bubbles or drops. The two most recent that we are aware of,in addition to our companion study to this paper [1], are the numerical investigations of the risingbubble problem due to Noh, Kang and Leal [3], and Tang et al. [10]. Of these, Noh et al. [3] utilized theFENE model of Chilcott and Rallison, and were successful in illustrating the qualitative mechanismwhich leads to points or cusps at the rear of a bubble rising in a viscoelastic liquid. One recent studymore closely related to the present work is due to Delaby et al. [11] who used small deformation theoryto focus on drop deformation during elongational flow involving two Maxwell model, viscoelasticfluids. These authors concluded that the drop could be either more or less deformed than a Newtoniandrop in a Newtonian fluid, depending on the parameters characterizing the two fluids. This result isqualitatively consistent with the behavior reported here and in [1].

In contrast to bubble and drop deformation, a lot of the recent research has concentrated on problemsinvolving viscoelastic flows around solid cylinders and spheres [12±27] for both steady and unsteadyflow conditions. These problems exhibit interesting features pertaining to the viscoelastic stressdistributions for both steady and unsteady flow fields, but to date the main interest is that numericalsolutions have only been obtained over a moderate range of Deborah numbers. Most of the difficultiesassociated with the numerical solution of these problems arise due to the no-slip condition at thesurface of the sphere/cylinder coupled with the extensional flow emanating from the rear stagnationpoint. This combination causes large stress gradients that require a very high degree of spatialresolution near the no-slip surfaces, and in the downstream `wake'. The wake structure found in recentstudies confirm many of the conjectures made earlier by Chilcott and Rallison [9].

The present study of the steady deformation of a Newtonian drop in a uniaxial extensional flow of aviscoelastic fluid is a companion to our recent study [1] of the same problem for viscoelastic drops in aNewtonian fluid. Though our study is primarily focused on the drop deformation, we also have theopportunity to investigate how the deformation and change in boundary conditions with viscosity ratioinfluences the evolution of the viscoelastic stresses in the near-wake and along the outgoingstreamlines. The presence of an interface in our problem makes it somewhat simpler than the solidsphere or cylinder problems, since the finite velocity at the surface has the effect of reducing thegradients in the viscoelastic stress field (relative to the case of the solid sphere) [15,16]. Theconstitutive model used for the outer viscoelastic fluid in this study is the Chilcott±Rallison version ofthe FENE dumbbell model for polymeric fluids [9], which we shall denote as FENE-CR. In the limit of

150 S. Ramaswamy, L.G. Leal / J. Non-Newtonian Fluid Mech. 88 (1999) 149±172

infinite extensibility, it is well known that the FENE-CR model reduces to a particular form of theOldroyd-B equation.

We seek to analyze the deformation of a Newtonian drop in a viscoelastic uniaxial extensional flow.We are not looking at this point for numerical results that fit specific experiments (since there are notmany) but will seek to understand the results from a `physical' viewpoint based upon our understandingof the FENE-CR model. The results, when contrasted with the results for the case of flow aroundspheres and bubbles, might help us in understanding limitations of the model and thus perhaps suggestways in which it might be improved. They will also give us some qualitative insight about deformeddrop shapes, and thus point to some likely routes to droplet breakup.

2. Problem statement

We consider the axisymmetric steady deformation of a Newtonian drop in a viscoelastic fluid(Fig. 1). The viscoelastic outer fluid is undergoing uniaxial extensional flow. The densities of the twofluids are assumed to be the same and the interface is fully characterized by a surface tensioncoefficient . The drop phase is characterized by a constant viscosity � and density �. The outer fluid islikewise characterized by � and � (note that the shear viscosity for the Chilcott±Rallison model is aconstant). This axisymmetric system can be represented by a cylindrical coordinate system (z,�,�). Allthe quantities are independent of �. Due to the symmetry of the problem, we solve for the flow in onlyone octant of the drop.

3. Governing equations and numerical method

In order to utilize the finite-difference method that we have developed for solution of free boundaryproblems, it is necessary to define a coordinate grid upon which the governing equations can be

Fig. 1. Drop in axisymmetric flow.

S. Ramaswamy, L.G. Leal / J. Non-Newtonian Fluid Mech. 88 (1999) 149±172 151

discretized. We have chosen to use a numerically generated orthogonal boundary-fitted coordinatesystem. The method used to develop this coordinate system was originally developed by Ryskin andLeal [28], then extended by Dandy and Leal [29]. Its use here is the same as described in our earlierpaper [1] on the viscoelastic drop, and the reader is referred there for details. We only note here that the(�,�) coordinate system used in our numerical work has the drop surface as � � 1 with � coordinatelines perpendicular to the interface.

The governing equations are also identical to those we used in our earlier study, though of course, thepolymer stress now contributes directly in the exterior fluid.

The equations of motion and continuity then take the following dimensionless form. For the drop,

r � u � 0; (1)

ÿrp� �r2u � 0; (2)

and for the suspending fluid,

r � u � 0; (3)

ÿrp�r2u� c

Der � �f �R�A� � 0: (4)

The pressures p and p are dynamic pressures. We assume that the Reynolds number, Re, is zero forboth the internal and external flows. The density ratio is � � �=� (equal to one in our study) and theratio of the drop viscosity to the viscosity of the solvent in the exterior fluid is denoted as� � �=�solvent. The parameters c, De and the tensor A relate to the viscoelastic fluid model (FENE-CR)and are defined in the following paragraph.

For the Chilcott±Rallison model, the conformation of the polymer is specified in terms of the averageof the dyadic product hRRi of the end-to-end vector, R. This product is denoted as A, and referred to asthe configuration tensor. For the FENE-CR dumbbell, the evolution equation in terms of the velocityfield u is

@A

@t� u � rA � A � ru� �ru�T � Aÿ f �R�

De�Aÿ I�: (5)

The function f(R) specifies the form of the nonlinear spring tension and is given by

f �R� � 1

1ÿ �Tr A�=L2: (6)

The three viscoelastic parameters, c, De and L are the same as used in our earlier work. The so-called`extensibility' parameter L represents the maximum average length of a polymer molecule relative tothe equilibrium end-to-end dimension. The parameter c is a measure of the concentration of dumbbellsdefined in terms of the incremental change in the viscosity of the solution relative to that of the solvent,

c � �ÿ �solvent

�solvent

;

and De is the Deborah number defined as De � E� where � is the polymer relaxation time. Theboundary conditions are identical to those used in [1], and will not be repeated here.


As in the previous study of the viscoelastic drop, we have used third-order differencing for theupwind term in the evolution equation for the configuration tensor. With third-order upwinding, wechecked for numerical convergence with increasing grid discretization by comparing the dropdeformation and the dumbbell configuration for the different grids. For this outer problem, the dropshape proves to be quite sensitive to the presence of viscoelastic effects and hence the shape served as agood metric for comparing the results. Based on the results of our previous study on viscoelastic drops,we chose a grid of 80 � 80 points and compared the solution with the solution for the baseline case of100 � 100 for parameter values of Ca � 0.05, L2 � 600, De � 1, � � 1, c � 0.01. With this set ofparameters, the drop deformation based on the Taylor deformation measure agreed to within 0.003 forthe two grids and the infinity norm of the error of Tr A was less than 9. It should be noted here that thegrid was biased to increase angular spatial resolution in the regions where large gradients of curvaturewere expected. Also due to the fact that we use an inverted coordinate map (see [30]), we were able toachieve a resolution finer or comparable to that used in other studies close to the surface of the drop,enabling us to adequately resolve the gradients in A.

4. Results and discussion

4.1. Results for c � 0

4.1.1. Viscosity ratio � � 1We begin with the limiting case, c � 0, where the flow does not change. As in the previously studied

case of a viscoelastic drop in a Newtonian fluid, knowledge of the evolution of A with the parametersDe, L2 and Ca for c � 0, will enable us to better understand the complex viscoelastic phenomena forc > 0. Physically, the limit c � 0 corresponds closely to the limiting case of very small polymerconcentrations (say, 0 (10 ppm)) where the flow is unchanged from its form for a Newtonian fluid.Solutions were obtained and are available in Ramaswamy's thesis [30] for Ca � 0.05 and 0.1, forL2 � 144 and 600, and for a number of values of � and De. We consider only a few cases, selected toillustrate the key physical phenomena.

Let us begin with the case Ca � 0.05, � � 1, L2 � 600, De � 0.5. In a homogeneous extensional flowwith this small value of De, one would expect very little polymer extension. Indeed, the asymptoticvalue of Tr A far from the drop is only approximately 26. Contours of Tr A in the vicinity of the dropare shown in Fig. 2. In fact, it can be seen that A is quite small everywhere except for anarrow region close to the (outflow) axis of symmetry of the drop. An enlarged close-up of this regionis shown in Fig. 2(c). Note that the scale of the X and Y axes are not the same in this latter figure.Surprisingly, in view of the fact that De � 0.5, the maximum fractional dumbbell extension isapproximately 70%, with a maximum value which occurs at a location that is slightly downstream fromthe tip of the drop (Fig. 2(c) and Fig. 3), rather than at the stagnation point. It may also be noted thatthe large values of Tr A near the drop surface result (through a combination of convection and highstrain rates) in values of A that remain fairly large along the outflow axis for distances that are manytimes the radius of the drop. Hence, though the Deborah number associated with the strain rate of thehomogeneous flow field (at infinity) is not high enough to cause large changes in the dumbbellconfiguration, we find very significant fractional extensions (i.e. �Tr A�=L2� �1=2

close to anddownstream of the drop.


Fig. 2. A outside the drop: Ca � 0.05, � � 1, De � 0.5, L2 � 600, c � 0. The variation of Tr A, A�� and A�� at the surface of

the drop is also shown. The contour plots show the distribution of Tr A outside the drop. Contours are drawn at intervals of 50.

Fig. 3. Variation of A along the axis of symmetry for Ca � 0.05, � � 1, De � 0.5, L2 � 600, c � 0.


Elsewhere in the domain, however, the dumbbells either remain quite close to their steady stateconfiguration for the undisturbed flow, or actually exhibit significantly smaller values of Tr A, e.g.,along the inflow plane of symmetry at � � 0 as we approach the drop surface.

The dumbbell configuration is a consequence of spatial variations in the flow. The magnitude of thestrain rate in the region of the flow external to the drop is shown in Fig. 4 for Ca � 0.05 and a viscosityratio of � � 1. This magnitude is defined as

jEj ��Tr�EET�

p;

where

E � ru� �ru�T2

:

Along the inflow symmetry plane, the strain rate increases very slightly as we approach the dropsurface and then falls quickly near the surface of the drop. This variation in the strain rate near thesurface of the drop changes the local Deborah number to a value below the critical value for onset ofdumbbell extension. More importantly, as we approach the drop, the flow near the inflow symmetryplane switches from uniaxial with the dumbbells aligned parallel to the principal axis of positive strainto biaxial with the inflow symmetry plane now corresponding to the `compression' plane. This changein flow-type causes a sharp decrease in the average dumbbell extension to a value very near theunstretched equilibrium value. A similar relationship exists between the strain rate and the averagedumbbell extension near the outflow axis, except in this case the principle axis of positive extension isin the direction of dumbbell alignment. The resulting change in the local Deborah number is reflectedin the variation in dumbbell extension (Fig. 3) as we move along the outflow axis and away from thedrop. Since the degree of dumbbell extension is a function of both the strain rate and the residence time(or strain), we find the maximum of Tr A, not at the tip of the drop, but at some distance downstreamfrom the tip where the velocities are still relatively small (corresponding to relatively long residencetimes) and the strain rate reaches its maximum value.

Fig. 4. Strain-rate E magnitude outside the drop: Ca � 0.05, � � 1. Contours are drawn at 0.25 intervals.


Having seen the result for the distribution of A for what would classically be considered a small Devalue, let us examine the distribution of A as we increase the value De to 1. At this Deborah number, wefind significant (though spatially uniform) dumbbell extension in the homogeneous flow field far awayfrom the drop. Close to the drop surface, the dumbbell configuration is again dominated by the localstrain and strain rates. We shall only discuss the result for a fixed value of Ca � 0.05 since we havefound that the choice of Ca does not change the basic nature of the results. The only discernibledifference is that the dumbbell extensions near the drop surface for Ca � 0.1 are somewhat largerbecause the drop is slightly more deformed, and this produces slightly larger strain rates in the vicinityof the drop surface. We also note that the distribution of A is similar for both values of L2 (144, 600)considered in [19] and we shall primarily focus on L2 � 600 in this section.

Fig. 5 shows the results for L2 � 600. With increase in the value of Deborah number to De � 1, theoverall maximum value of Tr A increases to 450 (L2 � 600). Other than that, this case looks remarkablysimilar to the lower De results shown in Fig. 2. One difference, however, is that the dumbbell extensionnow also becomes significant on the surface of the drop at � � 08 (�50% of the maximum possibleextension of the dumbbell). As seen in the case when De � 0.5 (Fig. 2), the maximum value of Tr A is

Fig. 5. A outside the drop: Ca � 0.05, � � 1, De � 1, L2 � 600, c � 0. The variation of Tr A, A�� and A�� at the surface of the

drop is also shown. The contour plots show the distribution of Tr A outside the drop. Contours are drawn at intervals of 50.


found at a location downstream from the tip of the drop on the outflow symmetry axis. The point alongthe axis where this maximum occurs appears to be independent of De and L2, at least for this fixedvalue of the viscosity ratio.

Finally, if we now examine the effect of a further increase in the value of De to 2, we find (Fig. 6)that the configuration tensor exhibits significant dumbbell extension on the drop surface at both � � 0and � � �/2. The overall maximum is, as we found earlier, at a point downstream from the tip.

4.1.2. Viscosity ratios � 6� 1

To this point, our discussion of results has focused on the problem when the viscosity ratio is equal to1. In this section, we consider � � 0.1 and � � 10, respectively. The former represents a partialtransition toward the case of a gas bubble (or inviscid drop), while the latter shows the effects oftransition toward the case of a no-slip (albeit deformable) body.

We shall first examine the case when � � 0.1. We illustrate our results via the specific caseCa � 0.05, De � 1, L2 � 600 and c � 0, shown in Fig. 7. Other cases can be found in [31]. Comparingwith the corresponding results for � � 1 in Fig. 5, we see that the dumbbells are more extended

Fig. 6. A outside the drop: Ca � 0.05, � � 1, De � 2, L2 � 600, c � 0. The variation of Tr A, A�� and A�� at the surface of the

drop is also shown. The contour plots show the distribution of Tr A outside the drop. Contours are drawn at intervals of 50.


everywhere near the drop surface for the smaller value of �. In addition, the location of the maximumvalue of Tr A has moved closer to the surface of the drop. This is a consequence of the relatively weakinfluence of the flow inside the drop on the external flow, with the result that the strain rates close to thesurface of the drop are higher and the flow is more extensional than for the case when � � 1. This isparticularly noticeable in the region near � � 0, where the average dumbbell extension is much higherthan for the case when � � 1. Another important point is that the gradients in Tr A are smaller near thestagnation point at � � � for the smaller viscosity ratio, � � 0.1, even though the values of Tr A arelarger. Thus, direct bulk-phase contributions of the polymer stress to flow will be weaker in this region(when c 6� 0), and we may also expect that numerical solutions should be obtained more easily than forthe larger � values.

We have also obtained results for � � 10. As the value of � increases, the velocity at the interface ofthe drop decreases. In the limit of very high � we will essentially have a no-slip condition at theinterface corresponding to the case of a solid `sphere'. Our focus here is primarily on showing thenature of the transition in the dumbbell configuration tensor, A, with increasing viscosity ratio in thevicinity of the stagnation point at the intersection of the outflow axis of symmetry and the drop surface.

Fig. 7. A outside the drop: Ca � 0.05, � � 0.1, De � 1, L2 � 600, c � 0. The variation of Tr A, A�� and A�� at the surface of



The primary change in the Newtonian flow for � � 10, compared to the cases � � 0.1 and 1considered above, is that the strain rate is notably diminished close to the surface of the drop in thelatter case, and approaches very small values near the stagnation points on the drop surface at � � 0 and�. This can be seen more clearly in Fig. 8, where we plot the magnitude of the dimensionless strain rateas a function of distance along the outflow symmetry axis for ��1 and ��10. Away from the drop, thestrain rate profiles are nearly identical. However, as � is increased, the peak in the strain rate occursfurther from the tip of the drop, and the magnitude decreases much more sharply and to smaller valuesat the drop surface. It may also be noted that as � increases, the flow in the immediate vicinity of thedrop surface undergoes a transition from being dominantly extensional for �� 1 to dominantly shearfor �� 1.

The impact of these changes in the flow is felt very strongly in the transition from � � 1 to � � 10.Indeed, if we compare the results for A in Fig. 9, for the case Ca � 0.05, De � 1, L2 � 600 and � � 10with the corresponding case for � � 1 in Fig. 5, there is clearly a qualitative change in the immediatevicinity of the tip of the drop. At the drop surface, for � � 10, there is hardly any dumbbell extension.Further the maximum of Tr A on the surface occurs not at the tip of the drop but rather at � � 1.2 andmore importantly the dominant dumbbell orientation is parallel to the surface of the drop, i.e., A�� is thelarge component and A�� is the small one. This is important because it is primarily A�� that leads to drop

Fig. 8. Variation of strain-rate magnitude along the axis of symmetry for Ca � 0.05: (a) � � 1; (b) � � 10.


deformation when c 6� 0. It is a consequence of the fact that the flow is essentially a shear flow. Finally,as we move away from the drop surface, we find sharp gradients in Tr A near the tip of the drop alongthe outflow axis, as the dumbbell extension goes from negligible to 80% extension over an extremelysmall distance. This corresponds to the sharp gradients in |E| that were shown in Fig. 8. Finally, weshow similar results for � � 10, De � 2 in Fig. 10. The results are qualitatively similar to thosediscussed above for De � 1. However, an important point is that the increase from De � 1 to 2produces a strong increase (or sharpening) of the conformation gradients for A just downstream of thetip of the drop. A similar increase in the gradients of A with increase of De was not found for either� � 0.1 or 1 (see [30]).

We have presented the case of � � 10 separately because of the close connection this case has withthe problem of flow around a solid sphere. Most of the difficulties encountered in numerical solution ofviscoelastic flow for this latter problem can be directly traced to the extremely high gradients in thedumbbell configuration that occur close to the outflow tip of the sphere (no-slip surface). When c 6� 0,these large gradients in turn give rise to very large viscoelastic stresses at moderate Deborah number.

Fig. 9. A outside the drop: Ca � 0.05, � � 10, De � 2, L2 � 600, c � 0. The variation of Tr A, A�� and A�� at the surface

of the drop is also shown. The contour plots show the distribution of Tr A outside the drop. Contours are drawn at intervals of

50.


Inadequate resolution of these stresses near the rear stagnation point for large values of De can result infailure of the numerical scheme or at best, inaccurate results. We noted above that the gradients of Abecome increasingly large with increase of De for � � 10, but that the gradients were both smaller andmuch less sensitive to De for smaller � � 0.1 or 1. Due to our inverted coordinate map, we have theadvantage of good spatial resolution close to the surface of the drop and our resolution is finer orcomparable to that used in other studies [12±27] enabling us to properly resolve the gradients in A (atleast for the case when c � 0). However, we have not attempted to push solutions to higher De for� � 10 because the behavior of the flow in this local region is not our primary objective.

All in all, these three � cases partially cover the wider range of boundary conditions from zero shearstress to no-slip boundary conditions and illuminate the influence of the conditions on the qualitativenature of the distribution of A in the external flow field. In the next section however, we shall consideronly the case when � � 1 since our primary intention is to demonstrate the effect of the viscoelasticstresses on the drop deformation. The case � � 1 lies conveniently between the zero shear and no-slipcases.

Fig. 10. A outside the drop: Ca � 0.05, � � 10, De � 1, L2 � 600, c � 0. The variation of Tr A, A�� and A�� at the surface

of the drop is also shown. The contour plots show the distribution of Tr A outside the drop. Contours are drawn at intervals

of 50.


4.1.3. Drop deformation based on estimates of polymer contributions to interface stress based on theNewtonian velocity field and drop shape

Having examined the influence of the Newtonian and viscoelastic parameters on the polymerconfiguration for the Newtonian velocity field and drop shape (i.e. c � 0), we now turn to the effects ofviscoelastic stresses. In general, these will play two roles. First, and most obviously, the presence ofpolymer will modify the flow outside the drop due to the presence of bulk viscoelastic stresses. Second,there is also a direct contribution of the viscoelastic stresses to the normal stress balance at the dropsurface. The magnitude of the changes in the flow is directly influenced by the magnitude of gradients

in the components of A, whereas the direct contribution to the normal stress balance is influencedprimarily by the magnitude of the A�� component of A.

Of course, in the full solution of the problem (as presented in the next section), the drop deformationwill be simultaneously influenced by viscoelastic changes in the flow, and by the viscoelastic normalstresses at the boundary, and, in turn, these two effects are intimately coupled.

As a preliminary to exploring the complex interdependence between the flow, viscoelasticity anddrop shape, we conclude this section by calculating the contribution to changes in shape due toviscoelastic stresses at the drop surface, estimated using the distributions of A calculated above usingthe Newtonian velocity field and the Newtonian drop shape. Since the problem for A is itself nonlinear,even for c � 0, we must also solve for the perturbation to the drop shape numerically. In order tomagnify the effect of the direct viscoelastic stress contributions to the shape, we show results for valuesof c up to 0.1.

We have calculated the approximate drop deformation described above over a full range ofparameters considered in [30], namely De � (1, 2); L2 � (144, 600); � � (0.1, 1, 10) and Ca � 0.05,but for c � (0.01, 0.05, 0.1). In this section, we present only a few selected results. The remainder canbe found in Ramaswamy's thesis [30]. First we examine the case when � � 10. We have already seenthat the average degree of dumbbell extension on the surface is very small for this value of � and theorientation of the dumbbell is mostly parallel to the interface. Since the direct viscoelastic contributionsto the normal stress balance are quite small, we find only very small changes in the drop shape for therange of parameters considered here. We show only the case (L2 � 600, De � 2) with the largestdeformation in Fig. 11. As expected, the deformation is small. Recall, however, that for this case, thegradients in A close to the drop surface are quite large. Thus, in the full problem, we would expect tosee a considerable change in the flow and it is likely that these changes in the flow could produce amore significant change in the drop shape.

Fig. 11. Drop shapes from the small c theory for different values of c. The shape changes very little when going from c � 0.05

to c � 0.1.


Two representative cases for � � 1 are shown in Fig. 12(a and b). We now find a modest butincreased influence of the direct viscoelastic stresses on the drop shape. In fact, the deformationincreases with increase in both De and L2. In addition, the drop shape is seen to have regions ofsignificantly higher curvature near the tip. If we refer back to the dumbbell configuration forcorresponding parameter values, e.g., Fig. 5, we see that there is significant extension at the surface,and the dumbbell orientation in the region of maximum extension near the outflow-axis is dominantlynormal to the drop surface. This leads to a significant viscoelastic normal stress contribution whichresults in these regions of sharper curvature.

Finally, moving on to the case when � � 0.1, we recall that the degree of dumbbell extension at theinterface is larger than for the case when � � 1, and we would thus expect to see a much strongerviscoelastic effect on the deformation. The results for the drop deformation shown in Fig. 13 confirmthis assumption. Again, we see that viscoelastic stresses lead to increased curvature at the ends of thedrop.

The figures above give a preview of the evolution of the drop shape in the presence of directviscoelastic stresses, essentially as an ad hoc approximation with no account taken of modifications inthe flow and/or changes in the shape of the drop from its Newtonian form. One interesting result is thelocal increase in curvature at the tips of the drop. In the absence of any flow modification, this is whatone would anticipate intuitively as the likely consequence of viscoelastic stresses. Of course, thedeformations here are small and it remains to be seen whether the inference of sharply pointed ends willactually occur when finite deformation and viscoelastic modifications of the flow are taken intoaccount. In the next section, we consider results for the full viscoelastic flow problem where the shape,the flow, and the dumbbell configuration all undergo coupled changes.

Fig. 12. Drop shapes from the small c theory for different values of c for (a) De � 1 and (b) De � 2.


4.2. Results for c > 0

We shall now examine some selective results taking full account of viscoelasticity for small but finitec where we expect the FENE-CR model to be at least approximately valid. Our previous study ofviscoelastic drops [1] has given us considerable insight into the influence of the various parameters ofthe problem, and we restrict ourselves here to a smaller, relatively compact set of cases that capture theessential features of the problem. Specifically, most of the results in this section deal with a fixedcapillary number, Ca � 0.05. For this problem, involving an external viscoelastic fluid, we expect theviscoelastic forces to enhance the deformation and the choice of Ca � 0.05 is dictated by therequirement that the extra deformation produced due to viscoelastic effects be within the maximumpermissible for stable drop deformation. Based on the results of the previous section and the earlierstudy [1] of viscoelastic drop deformation, we expect the results to be qualitatively independent of theexact value of the capillary number for any case where a steady drop shape actually exists.

Our aim is to investigate the qualitative changes in drop deformation and the flow field with changein the viscoelastic parameters De, L2 and c. The two values of the Deborah number, considered here areDe � 1 and De � 2, and for those Deborah numbers, we examine two values of the extensibilityparameter, L2 � 144 and L2 � 600. For each case, the range in the values of c is dictated by the desire toobserve significant change in both the deformation and the flow field, without violating the basic dilutesolution assumption of the FENE-CR model.

We begin with the case L2 � 144. For this value of L2, there is virtually no change in the flow foreither c � 0.01 or c � 0.03. At c � 0.05. However, we find significant and initially surprising changes

Fig. 13. Drop shapes from the small c theory for different values of c for (a) De � 1 and (b) De � 2. Note: There is little

change in shape when going from c � 0 to c � 0.01 in both cases.


in the shape, in the degree of dumbbell extension and in the external flow field. Fig. 14 shows thedistribution of A for this value of c � 0.05. The surprising feature, on comparing this figure with thecorresponding results for c � 0 in Fig. 15, is that the maximum value of Tr A on the surface of the dropis actually increased. In the majority of flows that we have previously studied, a common feature is thatthe viscoelastic effect of polymer is to suppress the strongest extensional regime of the flow, thusdecreasing both the strain rate and the degree of dumbbell extension. Although this was, in someinstances, accompanied by increases in the shear rates elsewhere in the flow, the dominant effect was tosuppress the extensional flow. Here, however, the trend is reversed, and there is a distinct increase in theamount of dumbbell extension near the tip of the drop. Furthermore, the maximum dumbbell extensionis now found at the tip of the drop rather than some distance downstream as was the case for c � 0.comparing the corresponding strain rates (Fig. 16) with those for the Newtonian field (c�0) in Fig. 4,we find that the maximum value of the local strain rate has also increased (from �2.5 to �4.0) and theincrease is in the region near the tip of the drop. This rise in the local strain rate causes the value of Tr Ato increase. The fundamentally `new' feature in this problem, relative to most of those studied earlier, isthat the boundary of the flow domain is the drop interface, and this can deform due to the action of

Fig. 14. A outside the drop: Ca � 0.05, � � 1, De � 1, L2 � 144, c � 0.05. The variation of Tr A, A�� and A�� at the surface of



viscoelastic stresses in the flow. When this deformation leads to a local increase of the interfacecurvature (i.e., a decrease in the local radius of curvature), the local length scale which characterizeschanges in the velocity gradient is decreased. Hence, in a Lagrangian framework, the time scalecharacteristic of changes in the flow is decreased compared to the characteristic relaxation time of thepolymer so that the `local' Deborah number is increased. This leads to increased dumbbell deformation,increased viscoelastic stresses, and then a further increase of the interface deformation. Hence, as c isincreased, rather than suppressing extensional regions of the flow, the increase of viscoelastic stressesproduces larger interface deformation (all else being equal) and both the local strain rate and degree ofdumbbell stretch are increased. This basic mechanism by which viscoelastic stresses can lead tolocalized regions of very large curvature (and hence, presumably to cusping) was first identified by Nohet al. [3] in the context of buoyancy-driven motion of a gas bubble through a FENE-CR model fluid. Asimilar phenomenon was not observed in our previous study of a viscoelastic drop in a Newtoniansuspending fluid (where polymer/dumbbell stretch was inhibited in the usual way for a viscoelasticfluid), because the interface deformation induced in that case by viscoelastic stresses actually decreasedthe curvature at the ends of the drop.




If we increase the value of c still further, the maximum value of A (Fig. 17) at the surface of the dropagain increases. The maximum of Tr A is still found at the tip of the drop though the region along theoutflow axis with large values of the average dumbbell extension is somewhat increased (compared tothe case when c � 0.05). Along the surface of the drop, and near the central plane of symmetry, theaverage dumbbell extension changes very little as c is increased, primarily due to the lack of significantchanges in the flow in that region. Some of the change in the strain rate near the tip of the drop can beattributed to the greater overall deformation of the drop under the action of the viscoelastic forces.Close to the tip of the drop, however, the curvature becomes sharper and a consequence of this is asharp rise in local strain rate as explained earlier, in this case to a maximum value of 5±7.

Fig. 18 shows the distribution of A when L2 � 600, Ca � 0.05, De � 1 and c � 0.03. Even at this`low' value of c, the viscoelastic stresses have a significant influence on both the drop shape and(consequently) on the distribution of A. Comparing these results with the results for c � 0 as shown inFig. 5, we find that the drop deformation has increased and this is again accompanied by an increase inthe degree of dumbbell extension close to the tip of the drop. If we compare the corresponding strainrates, we also find that there is a large increase in strain rate primarily in the region close to the tip ofthe drop on the outflow axis, and this increase is responsible for the increase in the value of Tr A in thesame region. We may also note that large gradients of A are to be found primarily in two regions of the

Fig. 16. (a) Strain-rate E magnitude outside the drop: Ca� 0.05, � � 1, De� 1, L2 � 144, c � 0.05. Contours are drawn at

0.25 intervals; (b) E along the outflow axis.


flow. One is at the top of the drop (� � 0) and the other near the axis of symmetry. There are severalinteresting features associated with the distribution of A which become apparent when we compare thisfigure with Fig. 5, which shows the distribution of A for the corresponding case when c � 0. First, wefind that the maximum value of Tr A has again increased with increasing value of c. Second, as seen forL2 � 144, the maximum is no longer at a small distance downstream from the tip of the drop, but isinstead, at the tip of the drop. Looking at the distribution of A along the outgoing streamline (i.e., theaxis of symmetry), we find, on moving away from the tip of the drop, that the maximum value of Tr Ais not on the axis but slightly offset from it. This effect seems to begin close to the drop surface andpersists for about one drop radius downstream.

The effect of nonzero c on the deformation is quite significant, with the deformation of the dropincreasing quite strongly under the action of the viscoelastic stresses. This increase is accompanied by aregion of sharp curvature close to the axis of symmetry where the extension of the polymer andconsequently, the direct normal stress contribution due to A�� pulls the surface of the drop outward. Theeffect of this on the shape can be seen in Fig. 19. It is clear that the effect of viscoelasticity is strongenough to overcome the surface tension forces at this value of the capillary number (0.05) which, for a




Newtonian problem is roughly half of the critical capillary number needed for breakup. As alluded toearlier, in addition to the overall increase in the deformation, the local curvature of the drop increasesclose to the axis and the drop seems to exhibit incipient cusping under the action of the highviscoelastic stresses. The relative balance between these effects can be seen in Fig. 20, which shows thevalues of the flow and direct viscoelastic contributions to the normal stress balance1. The majorcontributions arise from the direct stresses due to A�� at the surface of the drop and the balancing effectof interfacial tension forces. The high values of the capillary force, due to interfacial tension, areindicative of the large curvature in this region. Comparing this case with that corresponding toFig. 12(a) (which considers just the effect of the unperturbed direct viscoelastic normal stress



1Here, the terms `internal' and `external flow' refer to the Newtonian stress and pressure contributions (evaluated with the

viscoelastic velocity field), the terms A�� and A�� refer to the contributions to the dynamic pressure associated with gradients

of A�� and A�� along the surface of the drop, (due to A��) while A�� is the direct viscoelastic stress corresponding to the ��component of A along the surface of the drop. The interfacial tension term represents the net effect after subtracting thecapillary pressure contribution for a spherical drop.


contributions to the normal stress balance on the shape), we find that both the overall deformation andthe local increase in curvature at the tips of the drop are enhanced when c > 0 and viscoelasticmodifications of both the flow and the viscoelastic stresses are taken into account. This is different fromthe case of a viscoelastic drop in a Newtonian fluid. In that case, we found that the unperturbed directviscoelastic stress contributions to the drop shape (due to A��) were opposite in direction to theadditional contributions due to viscoelastic changes in the flow. The drop deformation was then eitherincreased or decreased due to viscoelastic effects, depending upon the specific values of the parameters.

Though our primary focus has been on the effect of viscoelastic forces on the drop shape, the resultsfor Tr A obtained in the course of this study are unique in the sense that they highlight the complexdependence of the viscoelastic stresses both on the mobility of the interface (i.e., �) and the extent ofdeformation. The results for the enhanced curvature near the tip of the drop are qualitatively similar tothe results obtained in the case of a streaming flow past a deformable bubble [3]. We, therefore, feelthat the mechanism highlighted here should, in general, hold true for flow problems involvingdeformable interfaces and regions of extensional flow.

Associated with this change in Tr A due to enhanced deformation is the change in the wake structuredownstream from the drop. The study of wake structures behind solid spheres and cylinders hasreceived considerable attention of late [12±27] and the common features of these studies are the no-slip

Fig. 19. Drop shape for the case when Ca � 0.05, � � 1, De � 1,L2 � 600, c � 0.03 ps compared to the case when c � 0.

Fig. 20. Components of the normal stress balance at the surface of the drop. Ca� 0.05, � � 1, De � 1, L2 � 600, c � 0.03.


condition at the interface, the non-deformability of the interface and streaming flow past the interface(solid sphere, cylinder). These problems also have regions of strong extensional flow particularly nearthe trailing (outflow) edge of the solid objects and thus share some features with this study. Our results,though preliminary in nature, indicate that having a deformable interface significantly affects the wakestructures behind the objects and could be the basis for a future study.

5. Conclusions

In summary, the combination of enhanced deformation and the large gradients in the curvature of thedrop effectively increase the local strain rate causing the enhanced dumbbell extensions that we seehere. The presence of a finite c does not reduce the local strain rate and consequently, in the presence ofviscoelasticity, the dumbbells become highly extended. The effect of viscoelasticity on the shape nearthe ends of the drop is such that the curvature of the interface becomes very large. The reduction in thelocal length scales in the vicinity of the drop tip effectively increases the local strain rate and furtherenhances dumbbell extension. The complex feedback mechanism between the closely related shape,flow and viscoelastic stresses makes this problem very different from the previous studies onviscoelastic fluids. In view of the results here, we believe that this interdependence will show up inmany problems that will involve a deformable interface and viscoelastic effects.

Acknowledgements

This research was supported by a grant from the Fluid Dynamics program of the National ScienceFoundation.

References

[1] S. Ramaswamy, L.G. Leal, The deformation of a viscoelastic drop subjected to steady uniaxial extensional flow of a

Newtonian fluid, J. Non-Newtonian Fluid Mechanics, 1998, in press.

[2] D.W. Bousfield, R. Keunings, M.M. Denn, Transient deformation of an inviscid inclusion in a viscoelastic extensional

flow, J. Non-Newtonian Fluid Mech. 27 (1988) 205.

[3] D.S. Noh, I.S. Kang, L.G. Leal, Numerical solutions for the deformation of a bubble rising in dilute polymeric fluids,

Phys. Fluids 5(6) (1993) 1315±1332.

[4] L.G. Leal, J. Skoog, A. Acrivos, On the motion of gas bubbles in a viscoelastic liquid, Can. J. Chem. Eng. 49 (1971) 569.

[5] O.O. Ajayi, Slow motion of a bubble in viscoelastic fluid, J. Eng. Math. 9 (1975) 273.

[6] O. Hassager, K. Ostergaard, Aa. Fredenslund, Chemical Engineering with Per Soltoft Chemical Engineering, Teknisk

Forlag, Copenhagen, 1977.

[7] G.F. Tiefenbruck, L.G. Leal, A note on the slow motion of a bubble in a viscoelastic liquid, J. Non-Newtonian Fluid

Mech. 7 (1980) 257.

[8] M.G. Wagner, J.C. Slattery, Slow flow of a non-newtonian fluid past a droplet, AIChE J. 17 (1971) 1198.

[9] M.D. Chilcott, J.M. Rallison, Creeping flow of dilute polymer solutions past cylinders and spheres, J. Non-Newtonian

Fluid Mech. 29 (1988) 381.

[10] Y.N. Tang, Y.S. Chen, W.F. Chen, Numerical simulation of bubble evolution in non-Newtonian fluid, Sci. China Series

A: Math. Phys. Astronomy Tech. Sci. 37(1) (1994) 42±51.


[11] I. Delaby, B. Eenst, R. Muller, Drop deformation during elongational flow in blends of viscoelastic fluids-small

deformation theory and comparison with experimental results, Rheol. Acta 34(6) (1995) 525±533.

[12] G.H. McKinley, R.C. Armstrong, R.A. Brown, The wake instability in viscoelastic flow past confined circular cylinders,

Phil. Trans. R. Soc. London Series A: Phys. Sci. Eng. 344(1671) (1993) 265±304.

[13] R.A. Brown, G.H. McKinley, Report on the VIIth international workshop on numerical methods in viscoelastic flows,

J. Non-Newtonian Fluid Mech. 52 (1994) 407±413.

[14] L.E. Becker, G.H. McKinley, H.K. Rasmusson, O. Hassager, The unsteady motion of a sphere in a viscoelastic fluid,

J. Rheology 38 (1994) 377±403.

[15] M.B. Bush, On the stagnation flow behind a sphere in a shear-thinning viscoelastic liquid, J. Non-Newtonian Fluid

Mech. 55 (1994) 229±247.

[16] M.B. Bush, The stagnation flow behind a sphere, J. Non-Newtonian Fluid Mech. 49 (1993) 103±122.

[17] X.L. Lou, An incremental difference formulation for viscoelastic flows and high resolution FEN solutions at high

Weissenberg numbers, J. Non-Newtonian Fluid Mech. 79 (1998) 57±75.

[18] P. Rameshwaran, P. Townsend, M.F. Webster, Simulation of particle settling in rotating and non-rotating flows of non-

Newtonian fluids, Int. J. Numer. Meth. Fluids 26 (1998) 851±874.

[19] F.P.T. Baaijens, An iterative solver for the DEVSS/DG method with application to smooth and non-smooth flows of the

upper convected Maxwell fluid, J. Non-Newtonian Fluid Mech. 75 (1998) 119±138.

[20] E.P.T. Baaijens, S.H.A. Selen, H.P.W. Baaijens, G.W.M. Peters, H.E.H. Meijer, Viscoelastic flow past a confined cylinder

of a low density polyethylene melt, J. Non-Newtonian Fluid Mech. 68 (1997) 173±203.

[21] R.G. Owens, T.N. Phillips, Steady viscoelastic flow past a sphere using spectral elements, Int. J. Num. Meth. Eng. 39

(1996) 1517±1534.

[22] X.L. Luo, Operator splitting algorithm for viscoelastic flow and numerical analysis for the flow around a sphere in a

tube, J. Non-Newtonian Fluid Mech. 63 (1996) 121±140.

[23] O.G. Harlan, J.M. Rallison, P. Szabo, A split Lagrangian-Eulerian method for simulating transient viscoelastic flows,

J. Non-Newtonian Fluid Mech. 60 (1995) 81±104.

[24] Y. Fan, M.J. Crochet, Higher-order finite element methods for steady viscoelastic flows, J. Non-Newtonian Fluid Mech.

57 (1995) 283±311.

[25] J.V. Satrape, M.J. Crochet, Numerical simulation of the motion of a sphere in a Boger fluid, J. Non-Newtonian Fluid

Mech. 55 (1994) 91±111.

[26] C. Bodart, M.J. Crochet, The time-dependent flow of a viscoelastic fluid around a sphere, J. Non-Newtonian Fluid Mech.

54 (1994) 303±329.

[27] J. Sun, R.I. Tanner, Computation of steady flows past a sphere in a tube using a PTT integral model, J. Non-Newtonian

Fluid Mech. 54 (1994) 379±403.

[28] G. Ryskin, L.G. Leal, Numerical solution of free boundary problems in fluid mechanics Part 3. Bubble deformation in an

axisymmetric straning flow, J. Fluid Mech. 148 (1984) 37±43.

[29] D. Dandy, L.G. Leal, Buoyancy driven motion of a deformable drop through a quiescent liquid at intermediate Reynolds

numbers, J. Fluid Mech. 208 (1989) 161±192.

[30] Ramaswamy, Sitaram, Study of Inertial and Viscoelastic Effects on Drop Deformation Ph.D. Thesis, University of

California, Santa Barbara, 1997.

[31] E. Zana, L.G. Leal, The dynamics and dissolution of gas bubbles in a viscoelastic fluid, Int. J. Multiphase Flow 4 (1978)

237.


the deformation of a newtonian drop in the uniaxial extensional flow of a viscoelastic liquid

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