WALL SLIP AND BOUNDARY EFFECTS IN POLYMER SHEAR FLOWS · 2020-02-06 · Further analysis, employing slip models generalized to include the eﬀects of normal load (i.e. pressure)

WALL SLIP AND BOUNDARY EFFECTS IN

POLYMER SHEAR FLOWS

By

William Brian Black

A dissertation submitted in partial fulfillment

of the requirements for the degree of

Doctor of Philosophy

(Chemical Engineering)

at the

UNIVERSITY OF WISCONSIN – MADISON

2000

i

Abstract

Polymer – surface interactions strongly influence many important industrial and

rheological flows. In particular, polymer melts and solutions slip against the sur-

face; this has long been associated with sharkskin and spurt in extrusion, and

recent experimental observations suggest that slip also plays a role in the formation

of enhanced concentration fluctuations in entangled polymer solutions. Analyses

of melt flow in model geometries, incorporating simple slip models which include

the effects of chain orientation and stretching, and employing common linear and

nonlinear viscoelastic models for the polymer stress, demonstrate that slip can lead

to short-wave hydrodynamic instabilities. The results predict values of the critical

recoverable shear, the critical shear rate, and the frequency of distortion that are

consistent with experimental observations for sharkskin during extrusion of linear

polyethylenes. Further analysis, employing slip models generalized to include the

effects of normal load (i.e. pressure) at the surface, confirms and unifies previ-

ous theoretical work: shear stress dependent slip models cannot predict instabilities

consistent with sharkskin; and pressure dependent slip leads to instability due to ill-

posedness at the boundary. Computational studies performed for polymer solutions

highlight the importance of boundaries on the flow behavior. Computations were

ii

carried out using a two-fluid model in which stress and concentration are coupled.

This coupling gives rise to enhanced fluctuations in the bulk, and the additional

coupling with slip leads to hydrodynamic instability. The length scales for slip

(the extrapolation length) and instability (the reciprocal of the wavenumber for the

disturbance) are the same and are consistent with experiment. Incorporation of

thermal fluctuations shows that the fluctuations near the surface are dramatically

enhanced relative to those in the bulk. The wavevector for enhanced fluctuations

rotates as the shear rate increases, in agreement with bulk experiments and predic-

tions, so that the mechanism for near-surface enhancement is essentially the same

as in the bulk. As opposed to the flow instability, these enhanced fluctuations are

insensitive to the presence of slip. Overall, these analyses highlight the importance

of bounding surfaces on the macroscopic flow behavior and of coupling between

various flow features, such as stress, concentration, and slip.

iii

Acknowledgements

I would like to extend my thanks and appreciation to those who have helped make

this possible: my wife, Lisa; my advisor, Prof. Michael Graham; and my group

mates: Dr. Gretchen Baier, Arun Kumar, Dr. Venkat Ramanan, Dr. John Kasab,

Philip Stone, and Richard Jendrejack, whose help and advice have been most ben-

eficial.

iv

Contents

Abstract i

Acknowledgements iii

List of Tables vii

List of Figures viii

Summary xi

1 Extrusion Instabilities 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Sharkskin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Wall Slip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3.1 Experimental Evidence . . . . . . . . . . . . . . . . . . . . . . 9

1.3.2 Plane Shear Flow Analyses with Slip . . . . . . . . . . . . . . 13

2 Modeling of Wall Slip 18

2.1 Slip Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Connections Between Normal Stresses and Slip . . . . . . . . . . . . . 21

v

2.3 A Slip Model Based on Network Theory . . . . . . . . . . . . . . . . . 24

2.4 Anisotropic Drag Slip Model . . . . . . . . . . . . . . . . . . . . . . . 27

3 Stability of Plane Shear Flow of a Polymer Melt with Slip 30

3.1 Relevance of Viscometric Flow to Die Exit Flow . . . . . . . . . . . . 30

3.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3 Asymptotic Solutions with General Slip Models . . . . . . . . . . . . . 36

3.4 Results with Specific Slip Models . . . . . . . . . . . . . . . . . . . . 44

3.4.1 Analytical Results for the UCM Equation . . . . . . . . . . . . 44

3.4.2 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . 47

3.4.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 51

3.4.4 Comparison with experiment . . . . . . . . . . . . . . . . . . . 57

3.5 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.6 Summary of the Melt Analysis . . . . . . . . . . . . . . . . . . . . . . 63

4 Concentration Fluctuations in Semidilute Polymer Solutions 65

5 Concentration Fluctuations and Flow Instabilities in Sheared Polymer

Solutions 73

5.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.2 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.2.1 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . 78

5.2.2 Stability Results . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.3 Brownian Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

vi

6 Concluding Remarks 99

Nomenclature 101

A Basic Melt Equations 105

A.1 Nondimensionalization . . . . . . . . . . . . . . . . . . . . . . . . . . 105

A.2 Derivation of the General Stability Equation . . . . . . . . . . . . . . 107

A.3 PTT Matrix Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . 110

B 3D Stability Operators 112

Bibliography 114

vii

List of Tables

1 Summary of experimental results for sharkskin . . . . . . . . . . . . 4

2 Summary of measurements of slip velocities. . . . . . . . . . . . . . 10

3 Summary of stability analyses of flow . . . . . . . . . . . . . . . . . 15

4 Comparison of the results for slip and no-slip. . . . . . . . . . . . . 61

viii

List of Figures

1 Extrusion instabilities. . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Flow curve for LLDPE. . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Critical stresses for LLDPE for dies constructed of different metals. 6

4 Effect of DFL coating on the flow curve of HDPE. . . . . . . . . . . 11

5 Effect of Dynamar coating on the flow curve of HDPE. . . . . . . . 12

6 Schematic of the two principal mechanisms for slip. . . . . . . . . . 19

7 Kinetic slip model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

8 Typical die exit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

9 Stress boundary layers in corner flows. . . . . . . . . . . . . . . . . 31

10 Basic parallel shear flow geometry with slip at the solid surfaces. . . 33

11 Snapshot of the destabilizing disturbance at the onset of instability 41

12 Stability diagram for pressure- and normal stress- dependent slip. . 42

13 Critical Weissenberg number as a function of s for several values of

Wes for the network slip model. . . . . . . . . . . . . . . . . . . . . 45

14 Critical Wen as a function of εs for the UCM equation and the network

slip model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

ix

15 Growth rate versus wavenumber for the UCM equation and the N

slip model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

16 Typical eigenvalue spectra obtained analytically and numerically. . 47

17 Semi-analytical neutral curves for the plane Couette flow of the UCM

fluid with the anisotropic drag slip model. . . . . . . . . . . . . . . 48

18 Comparison of the growth rate curves predicted numerically and an-

alytically for the UCM equation and the network slip model. . . . . 52

19 Numerical and analytical neutral curves for the UCM equation and

the network slip model . . . . . . . . . . . . . . . . . . . . . . . . . 53

20 Neutral curves for the PTT constitutive equation with the network

model as the slip relation. . . . . . . . . . . . . . . . . . . . . . . . 53

21 Numerical neutral curves for the PTT constitutive equation with the

anisotropic drag slip model. . . . . . . . . . . . . . . . . . . . . . . 54

22 Master curve of kxb versus Wet for the network model. . . . . . . . . 55

23 Phase shift between the slip velocity and the shear and normal stress

components at the critical point. . . . . . . . . . . . . . . . . . . . 62

24 Enhanced concentration fluctuations in a semidilute polystyrene so-

lution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

25 Schematic of typical light scattering experiments. . . . . . . . . . . 67

26 Scattering intensity as a function of shear rate. . . . . . . . . . . . . 67

27 Scattering pattern as a function of the shear rate. . . . . . . . . . . 68

28 Physical picture of the HF hydrodynamic mechanism for enhanced

concentration fluctuations. . . . . . . . . . . . . . . . . . . . . . . . 70

x

29 Plane Couette geometry showing slip between the solution and solid

surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

30 Eigenvalue spectrum for 2D disturbances with the stress diffusion

term dropped. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

31 Comparison of the standard and SID techniques. . . . . . . . . . . . 85

32 A typical eigenvalue obtained using the SID technique. . . . . . . . 87

33 Neutral curves for kz = 0 and S = 10−2 for various values of b. . . . 88

34 Neutral curves for three dimensional disturbances at the given values

of kx and b. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

35 Unstable eigenfunction for the concentration. The parameters are

kx = 0.4, b = 10, We = 10, S = 0.01, N = 96. . . . . . . . . . . . . . 90

36 The noise correlation function. . . . . . . . . . . . . . . . . . . . . . 92

37 Correlation function at equilibrium. . . . . . . . . . . . . . . . . . . 92

38 Series of concentration correlation functions for increasing We. . . . 96

39 Series of snapshots of typical concentration profiles as We is increased. 97

40 Eigenvalue spectra for the slip and no-slip cases in the bounded flow

domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

xi

Summary

Polymer extrusion processes are severely limited by flow instabilities and product

distortions. The first flow instability observed upon increasing the flow rate from

zero during the extrusion of high molecular weight, linear polymers, such as high

density polyethylene (HDPE) and linear low density polyethylene (LLDPE) is shark-

skin, which is manifested as a short wavelength, periodic distortion of the surface of

the product. This happens at very low Reynolds numbers, due to the large viscosity

of polymer melts, and at flow rates where the shear stress is a monotonic function

of the shear rate. The onset conditions are affected by the die material, indicating

that the instability is interfacial in origin. In addition, the instability is triggered in

the die exit region, where the stresses are largest due to the boundary singularity

and velocity profile rearrangement. As sharkskin is the first instability seen in the

processing of these materials, understanding this phenomenon is key to improving

the productivity of these processes.

There have been many theories put forth to explain the onset of sharkskin distor-

tion, but, clearly, wall slip influences the critical conditions for instability to occur.

Two possible mechanisms for slip exist, desorption of adsorbed molecules from the

surface and disentanglement of anchored chains from the bulk material. In the past,

xii

slip has been modeled macroscopically using slip relations where the slip velocity is

a function of the shear stress at the wall. However, analyses employing these types

of models do not reveal any hydrodynamic instabilities which are consistent with

these distortions, raising the question of whether slip has been modeled correctly.

Physically, the tension in the chains at the surface, or equivalently, the chain ex-

tension at the wall, should influence both possible mechanisms for slip and should

be included in the slip model. This implies that the slip velocity is a function of

the normal stresses, a fact included in few slip models to date. As the die exit is

a region of extensional flow and large normal stresses, due to velocity profile rear-

rangement, these types of models may be particularly relevant for predicting and

modeling sharkskin behavior.

Normal stress-dependent slip models arise generally from simple mesoscopic and

microscopic treatments of polymer-surface interaction. From a mesoscopic point

of view, the interactions between the wall and the polymer can be modeled as

network junction points, giving a slip velocity dependent on the number of network

strands connected to the surface. Successful network theories for bulk constitutive

behavior assume that the lifetime of strands depends upon the extension of the

strands, typically measured by the trace of the extra stress tensor. Application

of this assumption to the surface-tethered strands yields a simple, normal stress-

dependent slip model. Microscopic theories for polymer chains interacting with

solid surfaces also lead to normal stress dependent slip models. Coarse-grained

bead-spring theories begin with a force balance on the beads which expresses the

equality between the drag force on the beads due to flow and the restoring force

due to the springs. Generally, the drag is assumed isotropic, but in reality, the

xiii

drag is highly anisotropic because the chains are distorted and oriented by flow.

Inclusion of anisotropic drag leads directly to a normal stress-dependent slip model.

These models, while simple in formulation, underscore the generality of normal

stress- dependent slip models and the simplicity of incorporating the effects of chain

orientation and stretching into current methodologies for modeling slip.

The slip models mentioned above can be generalized to include pressure effects

and analyzed formally in planar shear flows when the bulk behavior is described by

the upper convected Maxwell (UCM) constitutive equation. The flow is unstable

to short wavelength (i.e. large wavenumber) disturbances at large Weissenberg

numbers. The perturbations are localized near the surfaces and are convected along

the channel at the steady state slip velocity. Growth rates are bounded for all

wavenumbers, so that the model is well-posed, if the slip model does not include

pressure effects. The model becomes ill-posed if pressure effects are considered,

leading to unbounded growth rates for large wavenumbers, consistent with previous

analytical work by other authors. The flow is unconditionally stable if the normal

stress and pressure dependencies are removed so that the slip velocity only depends

on the shear stress.

Examination of the network and anisotropic drag models mentioned above in

plane Couette flow for general wavenumbers reveals several other interesting facets

of the instability. Stability results for both linear (UCM) and nonlinear (Phan-

Thien – Tanner) viscoelastic constitutive equations collapse to one master curve

relating the critical recoverable shear, defined as the critical shear stress divided

by the shear modulus, to the disturbance wavenumber. The critical recoverable

shear is O(10), which is the same order of magnitude measured experimentally.

xiv

The frequency of distortion scales with the bulk polymer relaxation time, a finding

corroborated by several experiments and once argued to conclusively prove that the

mechanism for slip was disentanglement. These results suggest, however, that this

scaling is common to both mechanisms for slip and the processes of reentanglement

and readsorption are both dominated by polymer relaxation. Scaling comparisons

with experiment demonstrate that the predicted temperature and molecular weight

dependence are in agreement with experimental results for LLDPE and HDPE.

These results are encouraging, and suggest that normal stresses play an important

role in sharkskin formation.

Flow instabilities due to surface interactions may also be present in the flow of

entangled polymer solutions as enhanced concentration fluctuations. Experiments

have only recently demonstrated the interfacial nature of this phenomenon – fluctu-

ations were visually observed to initiate near the surfaces and chemically modifying

the surfaces to increase slip delayed the onset of enhancement. Prior to this ex-

periment, all explanations for shear enhanced concentration fluctuations focused

on the coupling between polymer stress and concentration in the bulk of the solu-

tion. Analyses of polymer solution models which employ these ideas show that the

diffusion of random fluctuations is retarded in certain directions giving rise to en-

hanced fluctuations in those directions. These theories are successful in describing

low shear rate behavior in scattering experiments which examine the bulk region,

but are unable to describe the critical shear rates observed in experiments which

sample the near-surface regions.

Stability analysis using a Navier slip boundary condition has revealed that slip

can couple to the stress and concentration, giving rise to flow instability. The

xv

instability is localized near the bounding surfaces and has length scales consistent

with the experimentally observed values. The characteristic length scale is√Dtrλ,

where Dtr is the translational diffusivity of the polymer chain and λ is the relaxation

time, and represents a crossover size from short wavelength fluctuations whose decay

is dominated by diffusion to longer wavelength fluctuations whose decay is influenced

by polymer chain relaxation. For low values of the wavenumber, increasing slip

results in flow stabilization, in agreement with experiment. The orientation of the

destabilizing perturbation is similar to the enhanced random fluctuations that arise

in the bulk as mentioned above, and therefore, the mechanism of instability is clearly

related to the bulk mechanism of enhancement.

Surfaces can enhance fluctuations even when the basic flow is stable. Time in-

tegration of the linearized equations with random forcing revealed the formation of

boundary layers in the polymer concentration profile, even though fluctuations were

imposed uniformly throughout the domain. These boundary layers form on length

scales similar to those in experiments, and are most easily quantified in terms of the

spatial correlation function for the concentration, which is related to the structure

factor and, therefore, contains scattering information. Examination of the correla-

tion functions suggests that random fluctuations are selectively, and dramatically,

enhanced near the surface, even at equilibrium. The scattering intensity near the

surface increases as the shear rate increases. The enhanced fluctuations have a pre-

ferred orientation at low shear rates and appear to rotate clockwise as the shear rate

increases. This prediction is similar to those made for bulk behavior and observed

experimentally in the bulk region. The preferred orientations are confirmed by plot-

ting representative concentration profiles. Interestingly, the enhancement of random

xvi

fluctuations is insensitive to the presence of slip – neither the magnitude nor the

orientation appears to be affected – so that any complete explanation of enhanced

concentration fluctuations must include boundary effects. Overall, these analyses

point out the importance of polymer – surface interactions in understanding the

macroscopic flow behavior of polymer melts and solutions.

1

Chapter 1

Extrusion Instabilities

1.1 Introduction

Flow instabilities and product distortions limit the productivity of many polymer

processing applications, such as extrusion, film blowing, and fiber spinning. The

product becomes deformed upon passing a critical flow rate, thereby hampering its

usefulness and/or marketability. Hence, both industry and academia have expended

a great deal of effort attempting to understand the root causes of these instabilities.

Still, no general consensus has been reached on explanations for many of them.

The general sequence of instabilities for high molecular weight, linear polymers

in extrusion is outlined below. For these polymers, sharkskin is the first distortion

typically seen and a picture of this phenomenon for a high density polyethylene

(HDPE) is shown in Fig. 1(a). It is a short wavelength, wavy distortion confined

to the outside edge of the extrudate. The depth of severe sharkskin is at most

about 10% of the thickness of the product. Increasing the flow rate further results

2

(a)

(b)

(c)

Figure 1: General sequence of extrusion instabilities for linear polymers.From Polymer Processing, Principles and Modeling by Agassant, Avenas, Sergent, and Carreau [2]

in more severe distortion, either stick-slip or bamboo melt fracture, depending on

the experimental setup. Bamboo melt fracture, Fig. 1(b), has alternating regions of

smooth, non-distorted extrudate followed by regions with distortions that resemble

sharkskin, and arises if the volumetric flow rate, or shear rate, is the flow variable

controlled. If, instead, the pressure drop, and hence, the shear stress is controlled,

there is a dramatic increase of the flow rate at a critical shear stress. At higher

shear rates the extrudate becomes grossly distorted, as shown in Fig. 1(c). This is a

large wavelength, large amplitude distortion, and the entire extrudate cross section

3

is deformed. However, the surface is smooth and sharkskin free. Further increases

in shear rate result in more severe gross fracture.

This sequence of instabilities is not unique. Highly branched systems, such as

polystyrene (PS), polypropylene (PP), and low density polyethylene (LDPE) do

not exhibit the sharkskin instability at all. Instead a spiral or corkscrew instability

sets in after a critical flow rate. There is a second critical flow rate after which the

extrudate becomes grossly distorted.

Numerous attempts have been made, both theoretically and experimentally, to

understand these instabilities, generally with mixed results. This body of work is

reviewed below with emphasis on sharkskin. As this is typically the first instability

seen in extrusion operations, an understanding of this phenomenon is crucial for

improving polymer extrusion processes.

1.2 Sharkskin

A partial listing of experimental work on sharkskin is shown in Table 1. By far the

most extensive body of work exists for linear low-density polyethylene (LLDPE)

resins, but other resins, such as high density polyethylene (HDPE), polybutadi-

ene (PB), polydimethylsiloxane (PDMS), and poly(methylmethacrylate) (PMMA)

exhibit sharkskin distortions as well. Sharkskin is first observed as a loss of

gloss of the extrudate surface above a critical shear stress or shear rate. the dis-

tortions become more pronounced as the flow rate is increased, until the surface is

clearly distorted and wavy. The loss of gloss can usually be associated with a slope

change on a plot of the apparent shear stress, τa, versus the apparent shear rate,

4

Author(s) Polymer Die Proposed Mechanism

Tordella [94] PMMA Capillary

Sieglaff [89] PVC Capillary Cohesive failure

Ajji et al. [4] LLDPE Capillary Sporadic loss of adhesionPiau et al. [71] LLDPE Capillary Entrance instabilities

HDPE Capillary ”Kurtz [47, 49] LLDPE Capillary Pre-stressing and critical exit velocityRamamurthy [75] LLDPE Capillary Slip in the die land

HDPE Capillary ”HDPE-LDPE Capillary ”

Kalika and Denn [42] LLDPE Capillary Wall slipLim and Schowalter [52] PB SlitMoynihan et al. [64] LLDPE Capillary Pre-stressing and critical exit velocity

LLDPE Slit ”Piau et al. [72] PDMS Orifice High stresses at the die exitWang et al. [100] LLDPE Capillary Coil-stretch transition at the exitEl Kissi et al. [45] PDMS Capillary Rupture at the die exit

PB Capillary ”LLDPE Capillary ”HDPE Capillary ”

Venet and Vergnes [96] LLDPE CapillaryPerson and Denn [67] LLDPE Capillary Die land slipGhanta et al. [26] LLDPE CapillaryBarone and Wang [7] PB Slit Coil-stretch transition at the die exit

Table 1: Summary of experimental results for melt fracture and sharkskin.1S - sharkskin; MF - melt fracture.

5

γa [75, 42, 47, 100] for polyethylenes. For other polymers, notably PVC [89], this is

not always the case. Fig. 2 shows Ramamurthy’s flow curve for a LLDPE resin and

Figure 2: Flow curve for LLDPE.From Ramamurthy [75]

the point marked OSMF indicates the critical shear stress for the onset of sharkskin

behavior. Below this critical shear stress the extrudate is smooth. At this point,

the flow curve is still monotonically increasing, and the flow curve does not flatten

out until much larger shear rates and stresses are obtained. It should also be noted

that this phenomenon occurs at very small Reynolds numbers, due to the very high

viscosities of the polymer resins.

One of the most striking and suggestive features of sharkskin is the importance

of the polymer melt/die wall interface. Ramamurthy [75] was the first to show

that the material of construction of the die wall affected the onset of sharkskin.

He constructed dies out of many different metals, including steels, brasses, copper,

and chromium, and then extruded LLDPE through these dies. Fig. 3 shows Rama-

murthy’s results for a 1 MI LLDPE resin. The important quantity is τc1, which is

6

the critical shear stress for the onset of sharkskin. This quantity varies from metal

to metal, from a low of 0.104 MPa for beryllium copper, to a high of 0.172 MPa for

Figure 3: Critical stresses for LLDPE for dies constructed of different metals.From Ramamurthy [75]

CDA 360 brass. Ramamurthy also found that the die composition had a profound

effect in film blowing operations, where the use of brass dies entirely removed shark-

skin, which appeared during film blowing with steel dies. This inconsistency was the

subject of recent work on extrusion through brass dies by Person and Denn [67] and

Ghanta et al. [26], which shows clearly the importance of surface preparation with

brass dies, particularly the removal of the oxide layer which forms at the surface

of the die. Removal of this layer in situ using an additive resulted in suppression

of sharkskin distortions. Low surface energy surfaces have been studied by Piau et

al. [71] for LLDPE and HDPE. Dies made from poly(tetrafluoroethylene) (PTFE)

were found to greatly increase the flow rates possible before sharkskin began over

those in dies made of stainless steel.

Placing coatings on the die surface affects the appearance of melt fracture.

Moynihan et al. [64] performed experiments where LLDPE was extruded through

7

a slit die which could be selectively coated with fluoropolymer in the entrance and

exit regions. When the die was uncoated, sharkskin was evident at an apparent

shear rate of 27 s−1. Coating the entrance region or the exit region suppressed

sharkskin up to the maximum flow rate they could obtain; coating both was not

required to suppress the instability. However, the coating at the entrance of the die

may have detached and been transported to the end of the die. Such a mechanism

was suggested by the authors and also reported by Wang et al. [100] in their ex-

periments, in which they reported similar stabilizing effects when their dies were

coated with a fluoropolymer known to discourage surface interactions.

Peeling experiments have also demonstrated the importance of surface interac-

tions. Hill, Hasegawa, and Denn [38] performed peeling tests for solid LLDPE from

a variety of metal surfaces. They measured the propagation speed for the crack as

a function of the force used to peel the sample. In addition to finding differences

in the forces and velocities for various surfaces, they also found that the forces and

velocities changed with aging time. From these results, and some theoretical anal-

ysis, they were able to estimate the critical stress for the onset of sharkskin. These

values compare favorably to those found by Ramamurthy [75] for the same metals.

Other processing conditions, such as the temperature and pressure, as well as

the molecular weight (MW) affect the onset and severity of sharkskin deformations.

During sharkskin, the pressure in the die and the reservoir remains relatively con-

stant, unlike the situation for bamboo distortion, during which both the pressure

and the mass flow rate oscillate. Increasing the L/D ratio of the die decreases the

severity of sharkskin [95, 34] as well as increases the critical shear stress necessary

8

for the instability [42]. Kurtz [47, 49] reported that increased processing temper-

ature had little effect on the critical shear stresses for sharkskin for his LLDPE

samples. Of course, a higher shear rate was necessary to obtain the critical shear

stress at higher temperatures due to lower melt viscosities. Venet and Vergnes [96]

also reported an increase in the critical shear rate with increasing temperature, but

did not report critical shear stresses. Kurtz [48] showed that increasing the molec-

ular weight increases the amplitude of sharkskin and lowers the frequency of the

distortion in LLDPE. There is also evidence that the molecular weight distribution

may have an influence on the flow phenomenon. Ajji et al. [4] partially fractionated

their LLDPE samples to remove the lowest molecular weight components. This

fractionation increased the critical shear rates for instability without significantly

increasing the number or weight average molecular weights. It is unclear whether

the viscosity was affected, precluding statements about the critical stresses for the

instability.

A few general conclusions can be drawn from the experimental evidence. Shark-

skin is a short wavelength, wavy distortion confined to the surface of the extrudate

and observed at low Reynolds numbers. This instability occurs on the increasing

portion of the flow curve, always before the flattening of the flow curve associated

with bamboo, or spurt, distortions. Most importantly, sharkskin is clearly influ-

enced by the interfacial interactions between the polymer melt and the die wall, as

adhesion promoters and coatings which reduce the polymer/die interactions have

been shown to suppress the onset of the instability. In addition, experiments have

clearly established that the surface interactions which lead to sharkskin take place

primarily near the die exit. This is not surprising since the highest stresses, both

9

shear and normal, are at the exit and the flow possesses a strong elongational com-

ponent there. There are few explanations that can account for these observations,

and these generally invoke the notion of wall slip.

1.3 Wall Slip

1.3.1 Experimental Evidence

Wall slip is one of the few explanations for sharkskin that truly depends upon the

interfacial conditions in the die. The idea was proposed as far back as 1931 by

Mooney [62], who also devised a way of calculating slip velocities from capillary

experiments using different sized dies. The critical shear stress for the onset of

sharkskin has generally been observed to coincide with or occur at a slightly higher

shear stress [75, 95] than the critical shear stress for the onset of slip. Mackay

and Henson [53] argued that the critical shear stress for slip is an experimental

artifact because capillary measurements are not sensitive enough to measure slip at

lower stress levels. There have also been arguments that the slip only takes place

at the die exit [47, 49]. Either way, slip relieves some of the shear stress in the

polymer, resulting in what appears as a lower viscosity. Therefore, slip would show

up as increased shear thinning of the sample, resulting in a slope change in the flow

curve. Other explanations, such as cavitation, rupture at the die exit, constitutive

instabilities, and apparent slip generally do not depend upon the interface conditions

and cannot describe sharkskin.

Slip measurement has typically fallen into two broad categories, flow visualiza-

tion and indirect measurement. Table 2 summarizes the experiments which have

10

Author(s) Polymer Apparatus Technique

Atwood and Schowalter [6] HDPE Slit die Hot film anemometry

Lim and Schowalter [52] PB Slit die Hot film anemometry

White et. al. [102] PBR Biconical rheometer

Hatzikiriakos and Dealy [33] HDPE Sliding plate rheometer Mooney analysis

Hatzikiriakos and Dealy [34] HDPE Capillary rheometer Modified Mooney anal-

ysis

Migler, et al. [58] PDMS Simple Shear Evanescent waves

Archer et al. [5] PS Simple Shear Visualization of im-mersed spheres

Wang et al. [100] LLDPE Capillary rheometer Moody analysis

Person and Denn [67] LLDPE Slit die Moody analysis

Table 2: Summary of measurements of slip velocities.

been performed in both areas. These techniques have a variety of limitations

and problems. The traditional Mooney technique fails for capillary dies if either

the viscosity or the slip velocity depends upon the pressure or if there are signifi-

cant temperature gradients. Hatzikiriakos and Dealy [34] have proposed a modified

technique that corrects for the pressure effects. Immersed sphere visualization [5],

while being able to measure slip velocities within microns of the plate surface, is

limited by the requirement of a transparent flow cell as well as the distortion to the

flow field from the finite sized spheres. Evanescent wave techniques [58] also require

a transparent flow cell, however, the technique is non-invasive and slip velocities

can be measured as close as 100 nm from the plate surface. Hot film anemometry

has the distinct advantage of being a real time measuring system for slip, as well as

being non-invasive and not requiring any special properties of the flow cell and fluid.

The probes themselves are usually brittle and have a difficult time withstanding the

11

high stresses at the wall, and the technique also requires complicated data reduc-

tion to back out the slip velocities, as they are not measured directly. Other flow

visualization techniques, such as NMR and X-ray visualization and laser-Doppler

velocimetry (which are not listed in the table) are still quite limited by poor reso-

lution near the wall and are not yet viable for measuring slip velocities.

Several experiments clearly show that the slip velocity and the critical shear

stress for slip depend upon the polymer/die wall interface. Hatzikiriakos and Dealy

[33] have shown that fluorocarbon coatings can both suppress and promote slip in

their sliding plate rheometer. Fig. 4, taken from their paper, shows that the fluo-

Figure 4: Effect of DFL coating on the flow curve of HDPE.

rocarbon coating Dry Film Lube (DFL) reduces the slip velocity without affecting

the critical stress for the onset of slip, while a different fluorocarbon, Dynamar,

illustrated in Fig. 5 promotes slip by reducing the critical shear stress necessary for

slip. Both have been used industrially to suppress sharkskin distortion. White et al.

12

Figure 5: Effect of Dynamar coating on the flow curve of HDPE.

[102] measured slip velocities is a biconical rheometer in which rotors constructed

of different metals were inserted. They found that slip velocities were smaller for

rotors constructed of copper or brass than for steel rotors. Rotors coated with

poly(tetrafluoroethylene) exhibited the highest slip velocities. This ordering is con-

sistent with the observations of Ramamurthy [75]. Ramamurthy [76] argued that

increased adhesion was responsible for the removal of sharkskin in film blowing

when brass dies were used. The observation by Halley and Mackay [30] that the

exit pressure is higher for flow through brass dies than for flow through steel dies is

consistent with the idea that the slip velocity is lower on brass that steel. Halley and

Mackay analyzed the die surface after extrusion and reported dezincification of the

brass and the formation of a pitted, porous surface. This surface roughness would

be expected to hinder slip at the surface [84]. However, Ghanta et al. [26] found

that, with careful preparation of the brass surface, LLDPE actually slips more on

13

brass than on steel and that sharkskin is entirely eliminated. More work is required

to reconcile these experiments.

Several experiments have been done to determine the slip behavior as a function

of the pressure. These experiments generally agree and demonstrate that the slip

velocity decreases as the pressure increases. White et al. [102] performed experi-

ments in a biconical rheometer in which the ambient pressure could be arbitrarily

set. They found that decreasing the pressure caused an increase in the slip velocity

and that at high pressures slip was negligible. They tested a variety of elastomeric

compounds with different coatings and metals for the rotors in the rheometer and

found the same general trends. This is the same general trend as the sharkskin

dependence on pressure.

To summarize, experiments strongly suggest wall slip as a participant in the

sharkskin phenomenon. Slip has been shown to exist, and it occurs at the same

or slightly lower critical stress as does sharkskin. Slip clearly depends on the die

wall/polymer melt interface and has temperature and pressure behavior similar to

sharkskin. It should be possible to take the microscopic mechanisms of slip and

develop macroscopic models which can be used to simulate die flow and analyzed to

determine stability. Until recently, such formalisms have been lacking, but ad hoc

slip models have been proposed and analyzed. The predictions of these models are

described below.

1.3.2 Plane Shear Flow Analyses with Slip

Most of the theoretical evidence is of the negative variety, as very few analyses have

actually demonstrated any hydrodynamic instability which could lead to sharkskin

14

and melt fracture. Attempts have been made with various constitutive equations,

a variety of boundary conditions, and even compressibility, but direct theoretical

evidence of sharkskin is lacking.

Although the study of slip boundary conditions was motivated here by the ob-

servation that slip may be important in sharkskin formation, initial work was moti-

vated by the lack of hydrodynamic instabilities for low Reynolds number flow with

no-slip boundaries. These analyses are summarized in Table 3. The first results

of note are those of Gorodtsov and Leonov [27], who studied the plane Couette flow

of an upper convected Maxwell fluid using a linear stability analysis with no-slip

boundary conditions. They found analytically that the flow is always linearly stable

in the absence of inertia (Re = 0). This result was corroborated by Renardy and

Renardy [82], who also showed that the flow was stable when the Reynolds number

was small. The flow only becomes unstable at large Re, which is unrealistic for

polymer extrusion operations. This result is further supported by the work of Lee

and Finlayson [50]. Several authors have also studied pressure driven flow between

parallel plates, as opposed to the plate driven flow above. Plane Poiseuille flow was

first studied by Porteous and Denn [74] for an upper convected Maxwell fluid and no-

slip boundary conditions. They found that at low Reynolds numbers the flow could

become unstable. However, their analysis failed to account for the existence of spu-

rious eigenvalues, an oversight later corrected by Ho and Denn [40]. These authors

showed that plane Poiseuille flow was stable at zero and low Reynolds numbers and

the flow became unstable only at large Reynolds numbers. Lee and Finlayson [50]

later confirmed this result using a different numerical technique. Finally, Lim and

Schowalter [51] have shown that the plane Poiseuille flow of a Giesekus liquid is also

15

Auth

or(s)

Model

Geo

metry

A/C

B.C

.Result

Goro

dtsov

andLeo

nov

[27]

UCM

PlaneCouette

ANo-slip

Sta

bilitywhenRe=

0.

TlapaandBernstein[93]

UCM

PlanePoiseu

ille

CNo-slip

Instabilityatlarg

eRe.

Hoand

Den

n[40]

UCM

PlanePoiseu

ille

CNo-slip

Sta

bilityatlowRe.

Lee

and

Finlayso

n[50]

UCM

PlanePoiseu

ille

CNo-slip

Flow

isstable

atlowRe.

PlaneCouette

CNo-slip

Flow

isstable

atlowRe.

Ren

ard

yandRen

ard

y[82]

UCM

PlaneCouette

C/A

No-slip

Sta

bilityatzero

andlowRe.

Lim

andSch

owalter

[51]

Giesekus

PlanePoiseu

ille

CNo-slip

Sta

bilityatlowRe.

Ren

ard

y[77]

UCM

PlaneCouette

AM

emory

slip

Instability

whenRe=

0and

slip

ishappen

ing.

Geo

rgiou[25]

UCM

PlanePoiseu

ille

A/C

Non-linea

rslip

Instability

when

theslopeofth

e

flow

curv

eis

neg

ative.

Shoreetal.

[87]

UCM

PlaneCouette

CDynamic

slip

Non-m

onoto

nic

flow

curv

e.

Black

and

Gra

ham

[13]

UCM

PlaneCouette

AKinetic

slip

Instability

whenRe

=0

after

a

criticalflow

rate.

Black

and

Gra

ham

[29]

UCM

,PTT

Planarsh

ear

A/C

Norm

alstress

dep

enden

tslip

Norm

al

stress

dep

enden

cere-

quired

forinstability.

Tab

le3:

Sum

mar

yof

stab

ility

anal

yse

sof

flow

.A

-an

alytica

lre

sults,

C-co

mputa

tion

alre

sults.

16

stable to short waves at low Reynolds numbers. These results, and the results for

plane Couette flow, show clearly that there are no elastic instabilities present when

a reasonable constitutive equation and no-slip boundary conditions are used.

The lack of low Reynolds number instabilities has motivated a number of studies

employing a variety of of slip boundary conditions. Pearson and Petrie [66] were

among the first to analyze flows with slip, and they studied several algebraic slip

models in which the slip velocity depends upon the shear stress. They showed

that the flow could be unstable only if the slip curve, which is a plot of the slip

velocity versus shear stress, was nonmonotonic, i.e. dusdτyx

< 0, for some region

of the curve. Essentially, this is equivalent to a nonmonotonic flow curve, with

the nonlinearity that causes the flow curve to be multi-valued transferred to the

boundary. Renardy [77] used one of the models proposed by Pearson and Petrie,

namely the memory slip model

DusDt

+ λsus = f(τyx), (1)

where us is the slip velocity, τyx is the shear stress, and λs is the relaxation time for

slip. For this dynamic slip model, the slip velocity depends upon the shear stress

and the shear stress history, through the inclusion of a convected derivative term,

and the slip velocity is a monotonically increasing function of the shear stress. The

upper convected Maxwell model was used to describe the bulk behavior, and slip

was assumed to be happening. He found analytically that with Re = 0 the flow was

unstable to short waves. However, the growth rate is proportional to the square

root of the wavenumber, implying that the the growth rate is infinite for infinitesi-

mal waves. Therefore, this instability is a Hadamard-type instability resulting from

17

ill-posedness of the problem at the boundary. Also, the model can predict a fi-

nite slip velocity even when the shear stress is zero. These unphysical properties

bring the model validity into question [13]. Georgiou [25] used a slip model explic-

itly constructed to predict a nonmonotonic region in the slip curve to study the

incompressible plane Couette flow of an Oldroyd-B fluid. He performed both an

analytical stability analysis and numerical simulations of the flow. The results do

predict instability, but only when the slope of the slip curve is negative. Finally,

Shore et al. [87, 88] proposed a rather complex slip model which incorporates a first

order phase transition near the wall. Again, the model only depends upon shear

stresses and instability only occurs on the decreasing branch of the slip curve with

the most unstable wavenumber being zero. Both of the latter two analyses essen-

tially confirm the results of Pearson and Petrie [66]. The obvious conclusion from

these analyses is that shear stress dependent slip models cannot predict instabilities

consistent with experimental observations of sharkskin. The mechanisms for slip

and improved slip models which incorporate more of the essential physics of chain

orientation and stretching are described in the following chapter.

18

Chapter 2

Modeling of Wall Slip

2.1 Slip Mechanisms

Two broad explanations exist for slip at the interface between the melt and the die,

as shown in Fig. 6. First, molecules anchored at the wall can desorb from the surface,

with the rate of desorption dependent upon the strength of interaction between the

die wall and the polymer melt. The second method is disentanglement between a

layer of polymer molecules adsorbed to the wall and the first bulk molecular layer.

As the die surface is changed, the grafting density is changed, thereby changing the

force and hence, the critical stress, for the disentanglement process.

Several observations suggest that desorption is important at the interface. Hal-

ley and Mackay [30] reported dezincification of a brass die and the resultant for-

mation of a porous metal surface during the extrusion of LLDPE. These alter-

ations to the metal surface seemingly cannot be explained by a purely entangle-

ment/disentanglement slip mechanism. Mackay and Henson [53] argued that, based

19

��

��

��

��

��

��

��

��

��

(a)

(b)

Figure 6: Schematic of the two principal mechanisms for slip. The dark line is achain adsorbed to the surface. (a) - the adsorbed chain desorbs and slides alongthe surface, (b) - the adsorbed chain disentangles from the bulk chains, which thenmove along the interface. The shear flow is to the right.

on the activation energy for desorption for PS on steel, the force holding the chains

to the surface was the same order of magnitude as the drag force on the chains, so

that some fraction of the chains was detached from the surface. A similar argument

was put forth by Yarin and Graham [106] with regard to the experimental data of

Piau and El Kissi [70] for LLDPE. Yarin and Graham concluded that, based on

their proposed model for slip, which included competition between desorption and

disentanglement, desorption for LLDPE was a faster process than disentanglement

and desorption can lead to nonmonotonic slip curves.

Several authors, including Brochard and de Gennes [16], Ajdari et al. [3], and

Mhetar and Archer [57, 56], have proposed scaling theories for disentanglement.

These theories are based on the paradigm that an anchored molecule which does

not interact with other anchored molecules can be modeled as a chain pulled by

one end through the bulk material. The conformation of the pulled, or probe, chain

20

changes as the pulling force (F ) increases, and the friction coefficient, ζ , can be

estimated for various regimes. The result is a curve of the pulling force, F , versus

the chain velocity, V = F/ζ. On a macroscopic level, F is related to the shear

stress at the wall and V is related to the slip velocity. There are generally several

low slip regimes, followed by a regime where the probe chain disentangles from the

bulk chains.

The entanglement/disentanglement idea was originally proposed as an expla-

nation for gross melt fracture. At the transition, there is sudden disentanglement

throughout the die and the boundary condition shifts from no-slip to complete slip.

During slip, the stresses relax and the chains reentangle. Hysteresis is expected

due to the resultant chain orientation and stretching. Wang and coworkers [100]

have recently extended the idea in an attempt to explain both the sharkskin be-

havior and the slope change in the flow curve. In their hypothesis, the well-known

stress singularity at the die exit causes the critical stress for disentanglement to be

exceeded locally in the exit region before it is exceeded throughout the die. The

boundary condition oscillates between the stick and slip states at the die exit. Dis-

entanglement releases the singular stress, allowing the chains to reattach and start

the oscillation over again.

The predictions of these theories are in agreement with some experimental ev-

idence [58] but are not conclusive. In particular, Drda and Wang [22] and Wang

and Drda [99] argued that the temperature independent extrapolation length that

they measured for HDPE was direct evidence for the entanglement/disentanglement

transition. Yarin and Graham [106] demonstrated that desorption can lead to the

21

same behavior. Wang et al. [100] showed that the sharkskin oscillation period cor-

related directly with the polymer relaxation time and argued that this also was

conclusive evidence of disentanglement. However, the dynamics of reattachment

and reentanglement are both expected to be dominated by relaxation of the poly-

mer chain so that this observation is inconclusive at best. A final limitation is that

the slip velocity is always associated only with the shear stress at the wall. This

is not consistent with physical intuition regarding slip, where the orientation and

stretching of chains are important.

2.2 Connections Between Normal Stresses and Slip

Several factors and observations motivate the inclusion of normal stresses into slip

models. First, as described in §1.3.2, shear stress dependent slip models have

failed to predict instabilities consistent with sharkskin. Second, from a more meso-

scopic viewpoint, the interactions at the wall can be modeled as junction points,

similar to the junction points in network theories for bulk constitutive behavior.

The most successful network constitutive equations, the Marrucci [1] and Phan-

Thien/Tanner [69, 68] models, were derived with the assumption that the lifetime

of the network strands depends on the stress in the fluid, specifically the normal

stresses through the trace of the extra stress tensor. The situation near the die wall

should be similar, and this was exploited by Hatzikiriakos and Kalogerakis [35],

who developed a stochastic network theory. Third, the normal stresses measure the

elongation of the molecules, or equivalently, the tension in the chains, and higher

22

tension should increase both the rate of desorption and the rate of disentangle-

ment, so that regardless of the mechanism for slip, the higher the normal stresses,

the higher the slip velocity. This idea can also be applied to the scaling theories

for polymer disentanglement. In these scaling theories, the friction force depends

upon the number and lifetime of entanglements surrounding the probe chain. It is

generally assumed that the bulk material is at equilibrium and the probe chain is

being pulled through this quiescent material. However, during extrusion, the bulk

material is stressed, and the lifetime of entanglements depends on the stress. As

a possible first correction the the scaling theories, the friction coefficient should

depend upon the normal stresses to account for the stress in the bulk material.

Experimental observations also suggest a link between normal stresses and shark-

skin. It is widely accepted that the normal stresses are the root cause of die swell.

It has been observed (El Kissi and Piau [44]) that the amount of die swell decreases

at the onset of sharkskin, indicating that the normal stresses are reduced in the

fluid, possibly due to slip. Perhaps the most direct experimental evidence for a

normal stress dependence is the peeling experiments of Hill et al. [38]. Based on

standard theories of adhesion, they derived the following condition for the critical

normal stress difference for the onset of slip and instability:

N1 ∼=2Wa

fδr, (2)

where Wa is the work of adhesion, δr is the thickness of a proposed rubbery region

near the wall (c.f. Vinogradov and Ivanova [97, 98]), and f is the fractional recovery.

f is between zero and one and is used to take into account the fact that the rubbery

region is constrained to remain near the wall by the bulk fluid. The work of adhesion

is related to the surface free energy of the polymer and the die wall. The free energy

23

of the metals used to make dies is typically much larger than the surface free energy

of the polymers, so that Wa basically equals the free energy of the die wall. Several

other approximations give f = 0.2 and δ = R/4, where R is the radius of the

capillary. Therefore, a very simple criterion exists for instability and slip, namely

N1,c ∼=40Wa

R, (3)

For comparison with experimental data, most of which is reported in terms of a

critical shear stress, the normal stress difference can be replaced by the recoverable

shear, sR = N1,w/τyx. The recoverable shear is normally order unity at the onset of

instability so

τyx,c =40Wa

R. (4)

This criterion compares very well to the results of Ramamurthy [75], lending sup-

port to the conclusion that normal stresses influence melt fracture. Peeling tests

performed by Ajji et al. [4] also link the adhesion strength to the onset of sharkskin

melt fracture.

Experiments have shown that slip is relevant to sharkskin, although its role in

sharkskin formation has not been theoretically elucidated. This may be due to the

fact that current slip models have neglected normal stresses in their formulations,

even though experimental evidence and physical intuition suggests that they are

important. Ideally, we would like to understand the molecular dynamics and the

fluid dynamics in a real die geometry during extrusion. As a starting point, and

to highlight the generality with which normal stress effects can be included in slip

models, we derive simple slip models which include more of the essential physics.

The models are introduced below and analyzed in the next chapter.

24

2.3 A Slip Model Based on Network Theory

The network slip law to be used here can be motivated by a few simple, physical

arguments. Consider the coarse grained picture of polymer molecules interacting

with the wall shown in Figure 7. The junction points can be chemical bonds, in-

Figure 7: Kinetic slip model. Left: All polymer molecules are interacting with thewall. Right: A portion of the molecules are no longer interacting with the wall.

termolecular forces, hydrodynamic interactions, physical adherence, or molecular

entanglements, all of which are lumped together in this analysis. X is a structural

parameter that will describe the state of interaction between the polymer and the

wall. At equilibrium, X = 1, and all of the available sites for polymer-wall inter-

action are filled. During flow, some of the strands are lost due to breaking of the

junction points with the wall, 0 < X < 1, and these strands can slide along the

surface. The velocity of the free strands is given by

uf∗ =

ε∗τyx∗

X, (5)

which is simply a rearrangement of Stoke’s law with the drag coefficient proportional

to X. This makes sense, as larger X implies that a moving strand has to travel

25

through more strands still attached to the wall and will experience greater drag.

In Eq. 5, τyx∗ is the shear stress at the wall, ε∗ is a constant, and the superscript

star (∗) implies that the variables have dimension. An overall slip velocity can be

obtained by averaging the velocity of the free segments and the velocity of the bound

segments (which is zero) to get

us∗ = ε∗

(1−X

X

)τw∗. (6)

To complete the slip model, an expression for X in terms of the flow variables is

needed. A kinetic expression is used to describe the evolution of X as a function of

time and the stress,

DX

Dt=

1

λs[(1−X)− sXF (trτ )] , (7)

where λs is the relaxation time for slip, and s is a constant. The first term on the

right hand side describes the attachment kinetics (i.e. the forward reaction), and

hence, λs can be interpreted as the reciprocal of the attachment rate constant. The

attachment kinetics are proportional to the fraction of polymer molecules that are

detached from the wall. The second term describes the detachment kinetics. The

detachment rate is assumed to depend upon the polymer orientation and stretching

at the wall, through the function F . The detachment kinetics are proportional to

the fraction of bonded polymer molecules and s, which essentially is an equilib-

rium constant which gives the ratio between the attachment and detachment rate

constants.

Two main ideas are espoused in this kinetic law. One is that the rearrangement

at the wall is not instantaneous, but rather, has some finite relaxation time. In

addition, attachment and detachment do not necessarily take place at the same

26

rate. The second is that the breakage kinetics depend upon the normal stresses.

For simple kinetic theory models of polymers, specifically Hookean dumbbells, the

trace of the stress tensor is proportional to the mean square end-to-end distance

[10], and the detachment from the wall thus depends upon the conformation of the

molecule. The trace of the stress tensor can also be thought of as a measure of the

elastic stress in the molecules. Higher values of tr τ imply that the molecule feels

more tension in the x- and y-directions.

Several other comments are in order. First, all the F functions which have been

considered are continuous and monotonically increasing. Therefore, the slip velocity

will be a monotonic function of the stresses. Second, the parameters ε, λs, and s

will depend upon the strength and type of interactions between the polymer and the

die wall. In terms of rate constants, λs = 1/ka and s = kd/ka, where ka is the rate

constant for the formation of strands and kd is the rate constant for the destruc-

tion of strands. Within the entanglement/disentanglement mechanism, the rate

constants ka and kd are the inverses of the relaxation times for entanglement and

disentanglement, respectively, at the surface. According to Ajdari et al. [3], these

are the constraint release times, ka ∝ kd ∝ 1/τcr ∝ 1/(τRN5/2). τR is the Rouse

relaxation time, which is proportional to N2 and has an Arrhenius temperature

dependence (because it depends upon the monomer friction coefficient). Clearly,

these kinetic constants only depend upon bulk properties and are independent of

any surface energetics or characteristics. The only place for surface properties to

show up is in the slip coefficient, ε. The slip relation itself is just a statement of

Stokes’ law, so that ε is essentially the friction coefficient. The frictional force will

depend upon the number of adsorbed chains that the bulk chains must slide past,

27

i.e. ε = 1/(neζ), where ζ is the monomeric friction coefficient and ne is the equilib-

rium number density of chains at the surface. The situation is somewhat different

when the mechanism for slip is adsorption/desorption at the wall. In this case, the

kinetic constants essentially follow the Arrhenius relation with activation energies

that depend upon the work of adhesion between the polymer and the surface [37].

The temperature dependence is the same as for the disentanglement case and the

rates of adsorption and desorption increase strongly with temperature. The work

of adhesion is a function of the surface free energies, so that the kinetics of attach-

ment/detachment depend intrinsically on the interactions between the polymer and

the solid surface. In addition, the slip coefficient depends on the surface coverage;

but now, in general, the surface coverage depends on the kinetics at the surface

and is a complicated function of the stress [106]. For the network slip model, this

coefficient should depend only on the equilibrium number of adsorbed chains, ne.

Molecular simulations (Bitsanis and various coworkers [65, 11]) and molecular the-

ories [85, 86] suggest that the number of segment-surface contacts increases as the

square root of the molecular weight. The kinetics of this slip mechanism, however,

do not explicitly depend upon the molecular weight of the polymer, as the kinetic

constants basically describe monomer adsorption and desorption at the surface.

2.4 Anisotropic Drag Slip Model

Similar functional forms for the slip velocity can be obtained from more detailed

arguments. Yarin and Graham [106] recently proposed a slip model obtained from

kinetic theory arguments for a polymer represented by a bead-spring model. The

28

conformation of a molecule anchored to the wall at the origin is given by the balance

between the drag forces due to the bead motions and the restoring spring force,

assumed Hookean,

ζ ·

[us −

dR

dt

]−

3kT

a2R = 0, (8)

where R = (X, Y ) is the end-to-end vector of the dumbbell, ζ is the friction tensor,

us = (us, 0) is the bulk slip velocity in the x-direction, k is the Boltzmann constant,

T is the absolute temperature, and a is the characteristic coil size of the molecule at

equilibrium. The molecular stretching is assumed strong enough to neglect Brow-

nian forces along the chain. Yarin and Graham assume that the friction tensor is

isotropic, i.e., ζ = ζoδ, where δ is the unit tensor. If, instead, the friction tensor is

anisotropic and given by the Giesekus formula [10], ζ−1 = 1ζ0

(δ + ρτ ), then Eq. 8

can be written in component form as

us −dXdt− 3kT

a2ζ0[X + ρτxxX + ρτyxY ] = 0

dYdt

+ 3kTa2ζ0

[Y + ρτyxX + ρτyyY ] = 0. (9)

X is related to the shear stress by Hooke’s law, τyx/n = HX, where H(= 3kT/a2)

is the spring constant and n is the number density of anchored dumbbells. Yarin

and Graham consider the case where the surface coverage depends upon the shear

stress. For simplicity, it is assumed here that anchored molecules are permanently

attached so that n is a constant, Eq. 9 can be solved at steady state to give

us = ε

[(1 + ρτxx)(1 + ρτyy)− ρ2τ2yx

(1 + ρτyy)

]τyx. (10)

where ε = 1/(ζ0n). If ρ = 0, the quasi-steady approximation reduces the anisotropic

drag model to a Navier slip relation. Eq. 10 predicts monotonically increasing slip

29

and flow curves. This contrasts with the full Yarin-Graham model, which predicts

non-monotonic curves. Taking n to be constant eliminates the competition between

surface coverage and molecular stretching responsible for the non-monotonicity.

While the approximations used here are severe, our purpose in deriving this model

is to illustrate that slip models which contain normal stress dependencies can be ex-

pected to arise from more detailed treatments when the relevant physics is included.

Eq. 10 is similar to the static Black-Graham slip model, except that the function F

here depends upon all the stress components, not just the normal stresses.

30

Chapter 3

Stability of Plane Shear Flow of a

Polymer Melt with Slip†

3.1 Relevance of Viscometric Flow to Die Exit Flow

Die exit flow is the problem of primary importance in understanding the inception of

sharkskin deformation during extrusion, however, this problem has several inherent

complications which make analysis difficult. Fig. 8 shows a schematic of a die during

polymer melt extrusion. The melt swells as it leaves the die due to the elasticity of

the fluid, and the boundary is discontinuous, transitioning from solid boundaries to

free surfaces. This leads to velocity profile rearrangement, with the fully developed

pressure driven flow far upstream from the corner evolving to plug flow downstream.

The boundary singularity also causes the development of singular stresses as the

corner is approached. These difficulties make tackling the full die exit flow stability

†The results presented here were published in part in references [13] and [14].

31

Figure 8: Typical die exit showing the geometric singularity where the free surfacebegins and die swell. The fully developed upstream flow is paraboloidal and thedownstream flow is plug.

problem a formidable task.

Fortunately, simple viscometric flows are relevant to die exit flows. This can

be seen by examining flow around sharp corners. Dean and Montagnon [20] and

Moffatt [61] derived similarity solutions for Stokes’ flow around sharp corners, in-

cluding symmetric solutions which are applicable to free surfaces downstream of the

corner. Fig. 9(a) shows the case of a reentrant corner and Fig. 9(b) shows a free

surface downstream of the corner. In both cases, the flow is viscometric near the

333333333333333333333333333333

V

V

CV

Cθ33333

33333B

(a) (b)

θ = 0

θ = 3π2

θ = 0

θ = α

Figure 9: Stress boundary layers in corner flows.

solid surfaces (θ = 0 in (a) and (b) and θ = 3π2

in (a)), with ψ ∼ rρθ2, where ψ is

the stream function, θ is the angular coordinate in cylindrical coordinates, and ρ is

the power law exponent. Near the free surface, however, the stream function adopts

an elongational form, ψ ∼ rρ(α− θ), where θ = α is the position of the free surface.

Renardy [79] and Hinch [39] studied the flow of a upper convected Maxwell fluid

32

around a reentrant corner and found that viscometric boundary layers exist near the

solid surface for both Newtonian and viscoelastic kinematics. These boundary lay-

ers, denoted by V in Fig. 9, are also present for a nonlinear viscoelastic constitutive

equations [81], although the boundary layer thickness is much larger for the Phan-

Thien–Tanner model than the UCM fluid. The presence of this region suggests

that a simplified analysis using model geometries may be useful in understanding

the dynamics of die exit flows.

Therefore, planar shear flows are examined here as a first attempt to under-

stand how normal stresses influence interfacial slip and surface distortions in poly-

mer processing. Such an analysis is useful for extracting the underlying molecular

mechanisms for instability and for determining the qualitative potential of normal

stress dependent slip models to predict instabilities consistent with sharkskin. It is

possible to analyze general slip models in planar, parallel shear flows if a simple con-

stitutive equation, namely the upper convected Maxwell equation, is used. These

results are presented first and, as shown below, several predictions are consistent

with sharkskin deformations. Then, the specific slip models introduced in Ch. 2 are

analyzed in order to relax some of the inherent assumptions in the general analysis.

Analytical solutions are still possible of the UCM equation is used, but to examine

more complex constitutive behavior, numerical techniques must be employed.

3.2 Formulation

The model problem considered here is simple shear flow between two parallel plates,

shown schematically in Fig. 10. The origin is located at the bottom plate and the

33

*u (y)

γt*

.b*

us*

��

��

��

��

��x

yl

Figure 10: Basic parallel shear flow geometry with slip at the solid surfaces. ACouette velocity profile is shown on the left and a Poiseuille profile on the right.For short wavelength perturbations, only the shear rate near the wall is important.

flow can be either a Couette flow driven by the motion of the top plate, a Poiseuille

flow driven by a pressure gradient, or some linear combination of the two. The

constitutive behavior of the fluid is described generally by the Phan-Thien–Tanner

(PTT) network model without nonaffine motion [69],

τ (1) +1

Wen(1 + µ tr τ )τ = γ, (11)

where γ = ∇v + (∇v)T is the strain rate tensor, v is the velocity vector, τ is the

polymer extra stress tensor, µ describes the effect of τ on the creation and destruc-

tion rates of network junctions, and the the convected derivative, τ (1), is given by

τ (1) =∂τ∂t

+v ·∇τ −{τ · (∇v)+ (∇v)T ·τ}. The upper convected Maxwell (UCM)

equation, Eq. 11 with µ = 0, is used for the majority of the results because of its

simplicity even though it does not describe the shear thinning behavior of real poly-

mer melts. The full PTT model, which predicts shear thinning in the viscosity and

first normal stress coefficient, but has a zero second normal stress difference, is used

to gauge the effect of nonlinear viscoelasticity. In Eq. 11, the stresses have been

34

nondimensionalized by the shear modulus, G∗ = ηp/λ, where ηp is the zero-shear

rate viscosity and λ is the bulk polymer relaxation time, time and shear rate by the

nominal shear rate at the wall, and lengths by the gap width, l. More details on

the nondimensionalization can be found in Appendix A.1. The asterisk indicates

dimensional quantities while unmarked quantities are dimensionless. Certain vari-

ables, such as the relaxation time, λ and the viscosity, ηp, always have dimension.

The nominal Weissenberg number, Wen(= λγ∗n), is the dimensionless applied shear

rate at the wall. The shear rate in the fluid is less than the applied shear rate due to

slip, so a second dimensionless shear rate, the true Weissenberg number, Wet = λγ∗t

must be defined, where γ∗t is the actual shear rate in the fluid at the wall. The ratio

of Weissenberg numbers appears frequently and is denoted as γ = Wet/Wen. The

flow is assumed to be incompressible and inertialess (Re = 0), so that the equations

of continuity and motion are, respectively,

∇ · v = 0 (12)

∇ · τ −∇p = 0. (13)

These equations have been nondimensionalized in the same manner as the consti-

tutive equation (c.f. Appendix A.1).

The tangential velocity boundary conditions are provided by slip models relating

the slip velocity to the stress tensor. The slip models considered here have the

general, nondimensional form

us = f(τ , σnn)τyx, (14)

where σ is the total stress tensor, σ = −δp+ τ so that σnn = n · σ ·n is the total

normal stress acting on the wall, where n is the outward unit normal. This general

35

form allows the slip velocity to depend upon molecular stretching and orientation,

through the normal stresses, as well as the isotropic pressure in the system. The

asymptotic results considered below are facilitated by rewriting the slip relation in

terms of the extrapolation length, b∗, defined in general as b∗ = u∗s/γ∗t . As shown

below, the steady state dimensionless shear stress at the wall is τyx = Wet/(1 +

µτxx) = Wenγ/(1 + µτxx), and therefore, b(= b∗/l∗) is, from Eq. 14,

b =fWen

1 + µτxx, (15)

where f = f(τ , σnn) and overbars denote steady state values.

The stability of the flow is determined by a linear stability analysis. Only stabil-

ity with respect to two-dimensional disturbances is examined, as Squire’s theorem

holds directly for the UCM equation [93]. The basic equations are first solved for

the unidirectional flow field, which for plane Couette flow is given by

u = γy + us, (16a)

v = 0, (16b)

τyx =Wet

1 + µτxx, (16c)

τyy = 0, (16d)

the pressure is arbitrary, and τxx is given by the only real solution to µ2τ 3xx+2µτ 2xx+

τxx−2We2t = 0. Stability is determined by examining perturbations to the base flow.

The flow variables, p – pressure, u – x-component of velocity, v – y-component of

velocity, τxx – first normal stress, τyx – shear stress, and τyy – second normal stress,

are written as a = (u, v, τxx, τyx, τyy, p) = a + a, with the perturbations given by

the normal mode form a = a(y)eikx(x−ct)+c.c. The overbar indicates the base state

36

values and the hat denotes the perturbation amplitudes. The basic equations are

linearized around the steady state, resulting in a generalized eigenvalue problem

for the eigenvalues {c} and eigenvectors {a}. This procedure is shown in detail in

Appendix A.2 for planar shear flow of the PTT fluid, where the system of equations

for the perturbations is derived and it is shown that this system can be reduced

to one general stability equation, Eq. 89, which is analogous to the famous Orr-

Sommerfeld equation from Newtonian fluid mechanics [21]. The eigenvalues are in

general complex, with the imaginary part giving the exponential growth or decay

rate of the perturbations. If an eigenvalue satisfies Im(c) > 0, perturbations grow

and the flow is unstable. Analytical expressions for the eigenvalues can be obtained

when the UCM equation (µ = 0) is used as the constitutive equation. For more

complicated constitutive relations, numerical methods must be employed.

3.3 Asymptotic Solutions with General Slip Models

The stability analysis of planar shear flow of the UCM fluid in the asymptotic case

Wen � 1 and b � 1 is discussed first. These assumptions permit examination of

instability without requiring a particular model for slip. These results clarify the

general roles of the elastic normal stresses and pressure in the stability of flow with

slip.

The approach taken here is motivated by the observation that, in the absence

of slip, the growth rates of perturbations approach zero (from below) as We−1n as

Wen →∞ [27]. So, by perturbing 1/Wen from zero, the growth rates are perturbed

from zero, i.e. from marginal stability. The small slip limit (b � 1) allows the

37

approximation Wet = Wen, eliminating the need to specify a particular expression

for us. Finally, the large wavenumber assumption implies that there are boundary

layers localized near the solid surface and it turns out that only the inner, boundary

layer solutions need to be considered to determine the stability. As shown below, the

condition kx � 1/(bWen) is required for the asymptotic solutions to be uniformly

valid and, therefore, the approximation is self-consistent.

Several additional assumptions are made. First, f in Eq. 14 is taken to be an

increasing function of each of the stress components, so that non-monotonicity is

not introduced into the slip behavior or the flow curve, and inertia is neglected. To

find the boundary layer near the bottom plate the y-coordinate is scaled with the

wavenumber as y = kxy. The base state velocity profile is u = us + y/kx + O( 1k2x

),

the steady state slip velocity is us = bγ = b, and the base state stresses are:

τxx = 2We2n+O( 1kx

), τyx = Wen+O( 1kx

), and τyy = 0. When the asymptotic scalings

are applied to the general stability equation, Eq. 89, the result at leading order in

wavenumber is

(Q2d2

dy2− Q2 − 2Q

d

dy+ 2)(

d2

dy2+ 2iWen

d

dy− 1− 2We2n)v = 0 (17)

where Q = y − kxc− i/Wen. The four solutions to this equation are

v = (y − kxc)e±y, e±(iWen+

√1+We2n)y. (18)

Only the two decaying solutions can match the outer solution (v = 0) and are

physically realistic. After application of the no-penetration boundary condition,

v = 0, the solution to Eq. 17 is

v = K1kxce−(iWen+

√1+We2n)y + k1(y − kxc)e

−y, (19)

38

where K1 is a constant of integration which drops out later in the analysis. Eq. 19

suggests a velocity boundary layer thickness of 1/kx, since the second term on the

right hand side dominates at large Wen, and a stress boundary layer of thickness

1/(kxWen), because the first term on the right hand side dominates the expression

for the velocity gradients, and hence the stresses, at large Wen. These boundary

layer scalings are consistent with those reported by Renardy [80] and Graham [28].

The solution given by Eq. 19 is used to derive expressions for the stress components,

which are then substituted into the slip model to obtain the eigenvalues. The slip

equation for the perturbations can be written in the linearized form

us =bB

Wen(τyx + Rτxx + Gτyy −Hσnn), (20)

where the coefficients are

B = 1 + τyxf

(∂f∂τyx

)τ ,σnn

,

R =τyx( ∂f

∂τxx)τ ,σnn

f+(∂f∂τyx

)τ ,σnn

,

G =τyx

(∂f∂τyy

)τ ,σnn

f+(∂f∂τyx

)τ ,σnn

,

H =−τyx( ∂f

∂σnn)τ,σnn

f+(∂f∂τyx

)τ ,σnn

,

(21)

and the negative sign in front of H explicitly takes into account the experimental ob-

servation that increasing pressure decreases the slip velocity(Kalika and Denn [42],

Hatzikiriakos and Dealy [34], White et al. [102]). Examination of the linearized

constitutive equation reveals that τyx = O(Wen), τxx = O(We2n), and τyy = O(1)

as Wen → ∞. Because τyy is relatively small, it is neglected in this analysis. The

coefficients B and R are both positive, as one of the basic assumptions is that the

slip curve is monotonically increasing.

39

The first case of interest is shear stress dependent slip case, i.e., R = H = 0.

To obtain large Weissenberg number expressions, B is assumed to be strictly O(1).

This way, the slip velocity remains nonzero but bounded as Wen → ∞. There are

three discrete eigenvalues, which to leading order in 1/Wen are

c =

(12− i

(1 +

√32

))1

kxWen+ O( 1

We2n)(

12− i

(1−

√32

))1

kxWen+ O( 1

We2n)

(1− i) bB + O( 1kxWen

)

. (22)

Two roots scale as 1/Wen, one root is O(1) and all of the eigenvalues are stable,

i.e. Im(c) < 0. There is also a stable, continuous set of eigenvalues at c = y/kx −

i/(kxWen). In order for the second order terms to be asymptotically smaller than

the first order terms, kx � 1/(bWen). This condition holds for higher order terms

as well, so that the assumption of large wavenumbers was justified. This analysis

confirms the previous literature results [66] that shear stress dependent slip does

not lead to instability when the slip velocity is an increasing function of the wall

shear stress.

To determine the effect of adding normal stress and pressure dependencies, it is

necessary to first determine the scalings necessary for the pressure and normal stress

contributions to be the same order as the shear stress contribution, as this leads

to the most interesting results. The total normal stress on the surface at y = 0 is

σnn = −σyy = p− τyy. An expression for the pressure perturbation can be obtained

from the x-component of the equation of motion as

p = τxx −i

kx

dτyxdy

. (23)

Explicitly introducing the eigenvectors shows that, even though both dτyx/dy andτxx

are O(We3), p is O(We2), due to cancelation, and is therefore, the same order as the

40

shear stress. This suggests the scalings R = R0/Wen and H = O(1), where R0 is

O(1). Since τyy � p, Eq. 20 can be written as

us =bB

Wen

[τyx +

R0Wen

τxx −H

(τxx −

i

kx

∂τyx∂y

)]. (24)

Four discrete eigenvalues are found this time, with three of them being small in

magnitude (O(1/(kxWen))) and one being large (O(1)). If H = 0, we find that

normal stress dependent slip leads to well-posed instability. In this case, the large

root is found to be stable and one of the small roots becomes unstable at R0 =

0.1826 [13]. This criterion is independent of both kx and b. Thus, rewriting in

terms of the unscaled coefficients, instability is predicted to occur if

Wen >0.1826

R, (25)

for small R. The growth rate for the disturbances, kxIm(c) is bounded for all

wavenumbers, so that the model is well-posed for this case, and the real part of c

for this root is O(1/Wen), so that the wave speed, Re(c) = us +Re(c), is essentially

equal to the steady state slip velocity and the instability is convected along with the

base flow. As in the previous case, kx � 1/(bWen). The dimensionless frequency,

defined as ω = kxRe(c) = kxb+kxRe(c) is O(kxb), so that the dimensional frequency,

ω∗ = γ∗nω � kxbγ∗n is proportional to the wavenumber and the critical shear rate

at high Wen. A picture of the destabilizing disturbance is shown in Fig. 11. The

stream function and normal stress perturbations shown would be superimposed on

the base state flow. The streamlines show pairs of counter-rotating vortices, with

the dotted lines denoting clockwise rotation and the solid lines counterclockwise

rotation. The tops of the vortices are cut off on the graph in order to illustrate

the stress boundary layer, which is much thinner than the velocity boundary layer.

41

0 2 4 6 8 10 12x�

0

0.2

0.4

0.6

0.8

1

y�

Figure 11: Snapshot of the destabilizing disturbance at the onset of instability,showing perturbation streamlines overlaid on a density plot of the first normalstress perturbation in a coordinate system moving with the wavespeed (≈ us). Thenominal Weissenberg number is 20. The dashed lines denote clockwise rotation andthe solid lines counterclockwise rotation. The white regions in the density plot areregions of large, positive τxx and the black regions are regions of large, negativestress. The arrow denotes the base flow direction.

These observations comprise the most important result of the analysis: the normal

stress dependence of the slip velocity is necessary for instability, and for B > 0 and

any value of R > 0.1826/Wen there is a finite Wen above which the flow is unstable.

If R0 = 0 but H > 0, then a result similar to the Coulomb friction result of

Renardy [78] is found, as the large root, which has growth rate proportional to

kx, becomes unstable at H = 1. All three of the small roots are stable, and, in

addition, one of them collapses to the end of the continuous spectrum, which is

given by the same relation, c = y/kx − i/(kxWen), as before. In this case, the

instability is the result of ill-posedness at the boundary. Typically, the pressure

dependence is assumed to follow an exponential form, i.e., f ∝ e−β∗p∗, where the

pressure coefficient is very small (β∗ ≈ 10−3 MPa−1 for LLDPE [67]). The stability

42

0.0 0.2 0.4 0.6 0.8 1.0R0

0

1

2

3

4

H

66666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666

11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111

Unbounded growth rate

Bounded growth rate

Figure 12: Stability diagram for pressure- and normal stress- dependent slip. Thesolid curve is the neutral curve for the small (bounded growth rate) root and thecorresponding unstable region in parameter space is indicated by the forward hatch-ing. The dashed curve is the neutral curve for the large (unbounded growth rate)root with its unstable region highlighted by the backward hatching. Both roots areunstable in the cross-hatched region.

criterion for this model is Wen,c = 1/β � 1, where β = β∗G ≈ 10−4 for LLDPE. We

therefore consider it highly unlikely that interactions between elasticity and pressure

dependent slip could be responsible for sharkskin, as this results in extremely large

critical Weissenberg numbers for instability.

Much more complicated dynamics are found if H and R0 are both nonzero. From

the limiting cases described above, increasing the pressure coefficient, H, stabilizes

the small root and destabilizes the large root, while increasing the normal stress

coefficient, R0, has exactly the opposite effect. This leads to regions in parameter

space where both roots are stable, only one is unstable, or both are unstable, as

shown in Fig. 12. Generally, for small values of the pressure coefficient, the small

root is unstable at large enough Wen and the growth rate is independent of kx, while

the large root is unstable for larger values of the pressure coefficient at sufficiently

43

large Wen, and the growth rate is O(kx). There is also an overlap region where both

roots are unstable.

This analysis leads to several stability predictions. These results clearly show

that a slip model where the slip velocity depends solely on shear stresses will not

lead to hydrodynamic instabilities. However, an arbitrarily small, but nonzero,

normal stress slip dependence does lead to short wave instability if Wen is large

enough. In an extrusion die, velocity profile rearrangement at the die exit leads to

an extensional component of the velocity profile there and large normal stresses in

the exit region. The instability predicted here should then be manifest first in the

exit region, which may explain why many investigators have attributed sharkskin

to an exit effect. In fact, similar behavior is expected at an interior boundary

singularity. Barone and Wang [7] coated the downstream half of a slit die with

a fluoropolymer to prevent adhesion and promote slip and left the upstream half

uncoated. They found evidence of instability near the change in surface condition

which decayed downstream. The boundary singularity results in an extensional

velocity component which increases the stresses, particularly the normal stresses,

and triggers the instability. Downstream from the singularity, where fluoropolymer

prevents adhesion and the slip velocity is therefore proportional to the shear stress,

the flow is stable and the instability decays.

Although these results do not depend on the specific form of the slip model,

there are a number of limitations to the analysis. To address these limitations, it is

necessary to perform analytical and numerical analyses incorporating specific slip

models, nonlinear viscoelasticity, and finite wavelength perturbations.

44

3.4 Results with Specific Slip Models

3.4.1 Analytical Results for the UCM Equation

The analytical results discussed below are specialized to the UCM equation in plane

Couette flow, for which the general stability equation simplifies to Eq. 90. The

solutions to this equation were first given by Gorodtsov and Leonov [27] and used to

determine the eigenvalues for no-slip boundaries. Here, the network and anisotropic

drag slip models introduced in Ch. 2 are employed in order to relax the assumptions

inherent in the asymptotic results presented above.

The network model with F = tr τ is, in dimensionless form,

us = ε

(1−X

X

)τyx (26)

DX

Dt=

1

Wes[(1−X)− s tr τ ] (27)

where Wes = λsγ∗n. Black [12] performed an asymptotic analysis for kx � 1, ε� 1,

and Wen � 1 and computed the critical nominal Weissenberg number as a function

of Wes and s, the results of which are reproduced in Fig. 13. Clearly, the critical

values are insensitive to the time dependence of the structural rearrangement at the

wall unless the kinetics are very slow. As the rate of interchange of material at the

surface slows, the flow becomes more stable, since, as Wes becomes large, DXDt→ 0,

X → constant, and us ∝ τyx, which is a stable case, as shown previously. The

remainder of the calculations are presented for Wes = 0, which simplifies the slip

model to us = εs(tr τ )τyx.

Analytical results for the network model with the UCM equation, Eq. 11 with

µ = 0, as the constitutive equation are considered first. Only the spatially decaying

45

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

102

s

10

100

We n

,c

Wes = 0.1Wes = 1Wes = 10Wes = 100

Unstable

Stable

Figure 13: Critical Weissenberg number as a function of s for several values of Wesfor the network slip model.

solutions to Eq. 90 are retained in the derivation of the eigenvalues, in effect ignoring

the presence of the upper boundary. This is a very good approximation for large kx,

due to the localized boundary layers. Application of the boundary conditions results

in a fourth order polynomial for the eigenvalues which can be solved numerically.

Fig. 14 shows neutral curves for plane Couette flow, plotted as the critical applied

Stable

Unstable

Increasing wavelength

101

102

103

104

10-13 10-11 10-9 10-7 10-5 10-3

εs

We n

,c

Figure 14: Critical Wen as a function of εs for the UCM equation and the net-work slip model, keeping only the decaying eigenfunctions. ( ) - kx = 105,(— — —) - kx = 100, and (– – – – –) - kx = 1.

46

flow rate for instability, Wen,c, versus the slip coefficient, εs. As the limit of no-

slip boundaries is approached, εs→ 0, the critical Weissenberg number approaches

infinity as Wen,c ∼ (εs)−1/4. At high enough values of εs, the flow again becomes

unconditionally stable, approaching the singularity as Wen,c ∼ (2.06× 10−4− εs)−1.

When εs � 2.06 × 10−4, a high enough stress to cause the instability cannot be

generated in the fluid, due to the large slip velocities. The different lines in each

graph also show that the flow is more stable for longer wavelength disturbances.

In fact, the most unstable disturbance is that corresponding to kx → ∞, and this

disturbance also has the highest growth rate, as shown in Fig. 15. However, the

102

103

104

105

106

kx

−0.0001

0

0.0001

0.0002

Im(k

x c)

Figure 15: Growth rate versus wavenumber for the UCM equation and the networkslip model with the following parameters: Wen = 11.62, Wet = 11, and εs = 10−5.

growth rate remains bounded as kx → ∞; therefore, the model is mathematically

well-posed. A nonzero, but bounded growth rate as kx → ∞ is also found in the

analysis of interfacial instability of viscoelastic two-layer flow [83]. Fig. 16 shows

a typical analytical eigenvalue spectrum. A numerical spectrum is also shown and

will be discussed below. There are four discrete eigenvalues and a continuous set

along the line c = γy + us − i/(kxWen). Only four discrete eigenvalues are found

analytically, even though one would expect the spectrum to be symmetric across

47

0 0.2 0.4 0.6 0.8 1Re(c)

−0.20

−0.15

−0.10

−0.05

0.00

0.05

Im(c

)

Figure 16: Typical eigenvalue spectra obtained analytically and numerically. The+ symbols and the dashed line indicate the analytical eigenvalues, while the circlesand the solid line indicate the numerical spectrum. The parameters are kx = 5,εs = 10−5, Wen = 17.341, Wet = 15, and for the numerical results, N = 256.

the centerline of the channel, y = 0.5, because only the spatially decaying solutions

to the stream function were kept in deriving the eigenvalues. Replacing F = tr τ

by F = 2 τ 2yx in the slip model leads to the same steady slip behavior but never to

instability, consistent with the asymptotic result that flow with only shear stress

dependent slip is stable. Finally, Fig. 17 shows neutral curves for plane Couette

flow and the anisotropic drag slip model. Comparison of these curves with those for

the network model shows that the stability results are virtually identical for these

two models.

3.4.2 Numerical Method

To gauge the effect of nonlinear viscoelasticity, numerical solutions to the PTT

equation must be obtained. Chebyshev collocation is used to set up a generalized

matrix eigenvalue problem. Chebyshev collocation falls under a general class of

numerical methods for boundary value problems known as the method of weighted

48

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

ερ

10

100

We

Stable

Unstable

n,c

Figure 17: Semi-analytical neutral curves for the plane Couette flow of the UCMfluid with the anisotropic drag slip model. ( ) - kx = 100, (— — —) - kx = 1.For both curves, ρ = 0.1.

residuals [17]. For a general, linear, nonhomogeneous problem with operator L,

Lu(ξ) = f(ξ), (28)

a solution can be proposed in terms of the basis functions, φi(ξ),

uN(ξ) =N∑i=0

aiφi(ξ), (29)

where uN is an approximation to the true solution. The residual, defined by

R(ξ) = LuN(ξ) − f(ξ), is a measure of the error of the approximation and should

be minimized in some fashion. The minimization is accomplished by requiring the

inner product of the residual with another set of functions, the trial functions ψi(ξ),

to be zero, i.e. 〈R(ξ), ψi(ξ)〉 = 0 for all ψi, i = 0, . . . , N . After substituting the

definition of the residual and rearranging, the MWR problem reduces to

〈Lφj , ψi〉aj = 〈f, ψi〉. (30)

49

Everything in this equation is known, except for the vector of coefficients, aj . This

is just a standard system of algebraic equations to solve. For a Chebyshev collo-

cation technique, the basis functions are taken to be Chebyshev polynomials and

the trial functions to be Dirac delta functions. This forces the residual to be zero

at the collocation points. Solution of Eq. 30 then gives the values of the expansion

coefficients.

However, knowledge of the expansion coefficients is not required to discretize

the differential equation. All of the information about the solution is contained in

the values of uN at the collocation points. To see this, let ξi = ξ0, . . . , ξN be the

chosen set of collocation points. Eq. 29 can be evaluated at all of the collocation

points to give a system of equations, written in matrix form as

uN(ξ0)

...

uN(ξN)

= uNi =

φ0(ξ0) · · · φN(ξ0)

......

...

φ0(ξN) · · · φN(ξN)

a0

...

aN

= Aijaj . (31)

The derivative of u can be approximated as (uN)′ =∑N

i=0 aiφ′i(ξ). The derivative

can be evaluated at all of the collocation points and the system of equations written

as (uN)′i = Adijaj. This relation can be combined with Eq. 31 to give (uN)′i =

AdijA−1jk u

Nk . The matrix (DN )ik = AdijA

−1jk is the Chebyshev derivative matrix

operator. It depends only upon the basis functions and collocation points used, not

the expansion coefficients or the differential equation.

Chebyshev collocation is employed in the interval ξ = [−1, 1]. The most common

choice for the collocation points is the Gauss-Lobatto points given by

ξj = cosjπ

N. (32)

50

For these points, the derivative matrix takes on a very simple form

(DN)ik =

ai(−1)i+k

ak(ξi−ξk)i �= k

−ξk2(1−ξk 2)

1 ≤ i = k ≤ N − 1

2N2+16

i = k = 0

−2N2+16

i = k = N

. (33)

The range for the present problem is y = [0, 1] rather than [−1, 1]. The mapping

yj →12(1 + ξj) maps the typical interval onto the computational interval for this

problem. Derivatives with respect to y are replaced by d/dy → (dξ/dy)d/dξ =

2 d/dξ. Fortunately, this mapping only results in the derivative matrix being scaled

by a factor of 2. This choice of collocation points is also beneficial in that the points

are distributed nonuniformly in the region of interest. Specifically, the points are

more densely placed near the walls and more sparsely placed near the center of the

channel. The disturbance eigenvectors are expected to be localized near the wall,

so the points are arranged precisely where they are needed for the calculations.

The formulation of the numerical problem is slightly different than the formu-

lation for the analytical results. Instead of being reduced to one general equation,

the system of equations for the amplitude functions, Eqs. 87a - 87f , is written

in terms of four independent variables: the two normal stresses, the shear stress,

and the stream function. Pressure is eliminated by taking the curl of the equation

of motion. There are two reasons for this choice. The operator M in Eq. 89 is

extremely unwieldy, and more importantly, the general stability equation contains

fourth order derivatives, which leads to an extremely stiff system of equations after

discretization and makes reliable solutions impossible to obtain.

51

The generalized eigenvalue problem for the PTT equation is

ikxWen c Ca = La (34)

where a = (ψ, τxx, τyx, τyy) and the operators L and C are shown in Appendix A.3.

Discretization gives 4(N + 1) unknowns, which are the values of the independent

variables at the collocation points. The discretized eigenvalue problem is solved

using a public domain routine, namely the Transactions on Mathematical Software

(TOMS) routine 535 [23]. This routine returns all of the eigenvalues, allowing

the entire spectrum to be analyzed. To remove spurious eigenvalues, boundary

regularization, as discussed by Graham [28], is used.‡

3.4.3 Numerical Results

The analytical solutions for the UCM fluid presented in §3.4.1 can be used to validate

the numerical technique. Fig. 18 compares growth rate curves obtained numerically

using the Chebyshev technique and analytically using the spatially decaying solu-

tions. The number of collocation points, N , is 256. At large kx, numerical results

are not presented, due to the fact that the continuous spectrum becomes spuriously

unstable above kx ≈ 20. At moderate wavenumbers (kx ≈ 10), the eigenvectors are

localized near the plates, the analytical solutions are still accurate, and the numerics

agree with the analytical solution. At small kx, the curves diverge, as the eigenvec-

tors are no longer localized enough for the analytical solution to be accurate. The

neutral curve for kx = 10 is shown in Fig. 19 for the network model. The points

were obtained using the Chebyshev method while the solid curve is the analytical

‡Boundary regularization was used to remove spurious modes for the results published inRef. [14] and, therefore, was mentioned here. A much better method for eliminating spurious

52

0 5 10 15 20kx

−0.0045

−0.0025

−0.0005

0.0015

Im(k

x c)

Figure 18: Comparison of the growth rate curves predicted numerically and analyt-ically for the UCM equation and the network slip model. Wet = 15, Wen = 17.341,εs = 10−5, and for the numerical solutions, N = 256.

result. As expected, these curves are in excellent agreement, as is the predicted

eigenvalue spectrum, also shown in Fig. 16. The numerical spectrum shows four

pairs of eigenvalues, since the numerical technique obtains all of the solutions to the

general stability equation, and the continuous spectrum appears as a ring since the

eigenvectors corresponding to the continuous eigenvalues are singular and poorly

approximated by polynomials (c.f. Graham [28]). The poor approximation of the

continuous spectrum leads to the spuriously unstable behavior of the continuous

spectrum alluded to earlier for large kx. Other spurious behavior is possible, most

notably, the ring can be large enough in some instances to overlap with and obscure

the discrete eigenvalues.

For the PTT constitutive equation, only numerical solutions are possible. We

chose small values of µ(= 10−4, 10−3) since Phan-Thien and Tanner [69] reported

good fits of experimental data for a low density polyethylene with µ = 10−3 and

pressure modes is to use a primitive variable formulation coupled with a staggered grid for pres-sure (c.f. §5.2.1). Use of a primitive variable formulation similar to the one in §5.2.1 gives identicalresults to the streamfunction formulation with boundary regularization for this problem.

53

10−8

10−7

10−6

10−5

10−4

10−3

ε

10

100

We

s

n,c

Stable

Unstable

Figure 19: Numerical and analytical neutral curves for the UCM equation and thenetwork slip model with kx = 10. For the numerical results N = 192.

we wish to examine small deviations from the UCM equation. Fig. 20 shows the

neutral curves for kx = 1 and 10 when the network model is used as the slip relation.

10−8

10−7

10−6

10−5

10−4

ερ

10

100

1000

We n

,c

Stable

Unstable

εs

Figure 20: Neutral curves for the PTT constitutive equation with the network modelas the slip relation. © - kx = 1, µ = 10−3; 3 - kx = 1, µ = 10−4; � - kx = 10,µ = 10−3; 2 - kx = 10, µ = 10−4. For all curves, N = 192.

Increasing µ increases the slope of the power law region at small εs. At higher εs

the singularity is still present, but the location depends upon µ. In fact, the curves

cross and the curves for higher µ are less stable. At the shear rates required for

54

instability, shear thinning is just beginning to set in, but small changes in the stress

greatly affect the slip velocity, since at steady state us ∝ τ3yx. So, at a given εs

and applied flow rate, the slip velocity for the PTT model is reduced relative to the

slip velocity for the UCM model and this leads to higher shear rates and stresses

in the fluid for the same nominal We. Higher stresses can be maintained in the

fluid allowing the critical stress to be reachable for larger εs and Wen leading to less

stability. Finally, neutral curves for the PTT equation and the anisotropic drag slip

model are shown in Fig. 21. In contrast to the UCM case, comparison of Figs. 20

and 21 reveals that there is a significant difference in the neutral curves when the

10−8

10−7

10−6

10−5

10−4

10−3

10−2

ερ

10

100

We n

,c

Stable

Unstable

Figure 21: Numerical neutral curves for the PTT constitutive equation with theanisotropic drag slip model. © - kx = 1, 3 - kx = 10. For both curves, µ = 10−4,N = 192, and ρ = 0.1.

slip model is changed. In particular, the location of the singularity at large ερ

is shifted to higher value of ερ for the anisotropic drag slip model as opposed to

the network model, dramatically destabilizing the flow at large values of the slip

coefficient.

For comparison with the asymptotic results, as well as experimental evidence

55

and other slip theories, the results here can be recast in terms of the extrapolation

length, b = us/γ. The extrapolation length is given by Eq. 15 with f = εsτxx for

the network model and f = ε(1 + ρτxx − ρ2τ2yx) for the anisotropic drag model.

Interestingly, a master curve can be plotted as the true shear stress, i.e., the true

Weissenberg number, versus the scaled wavenumber kxb, as shown in Fig. 22 for both

the UCM and PTT constitutive equations. The two sets of points are numerical

results for the PTT constitutive equation obtained using the Chebyshev method

10−3

10−2

10−1

100

101

102

103

kx b

10

100

1000

τ yx,c

/G*

UCM; NetworkUCM; ADPTT; NetworkPTT; AD

Unstable

Stable

*

Figure 22: Master curve of kxb versus Wet for the network model.

while the solid curves are the analytical results for the UCM equation on a half-

plane. All of the data fall roughly onto one curve and predict a basically constant

critical shear stress (recoverable shear) of ≈ 11 for kxb � 1. All wavelengths shorter

than the extrapolation length have essentially the same growth rate and wave speed

(= us) and the flow is more stable for disturbance wavelengths greater than the

extrapolation length (kxb < 1). This is in agreement with the asymptotic result

for large wavenumbers where stability only depended upon R0, not kx or b. In an

actual extrusion process, wavelength selection should be strongly influenced by the

fluid mechanics near the die exit, which are not directly addressed here, as well as

56

by molecular dynamics occurring on fast time scales that are not included in the

models. Other possibilities arise when considering more complicated constitutive

behavior( both bulk and interfacial), where physical mechanisms such as reptation

and disentanglement are included (Wang et al. [100], Mhetar and Archer [55], Ajdari

et al. [3]). In any case, the longest wavelength observed would be on the order of

the extrapolation length (i.e., kxb = 1).

The solutions presented in this section qualitatively predict aspects of sharkskin

other than those already predicted by the asymptotic results. Most importantly,

these solutions show that slip can be stabilizing or destabilizing in Couette flow, de-

pending upon the magnitude of the slip velocity. For small slip velocities, increasing

the magnitude of slip destabilizes the flow by decreasing the critical stress for the

instability. At large enough values of the slip coefficient, the critical stress for insta-

bility becomes a constant and further increasing the magnitude of slip stabilizes the

flow by reducing the effective shear rate in the fluid. If the magnitude of slip is large

enough, the critical stress required for instability cannot be obtained and the flow

becomes stable for all We. These results are robust with respect to the exact form

of the slip model and to the constitutive equation used. Care must be exercised

in extrapolating these conclusions to pressure driven flow. The shear stress at the

wall is determined solely by the imposed pressure drop, not the shear rate. It is

always possible to achieve the critical stress, so it is not correct to associate the

substantially increased shear rate with flow stabilization.

3.4.4 Comparison with experiment

57

This analysis leads to several quantitative predictions which can be checked against

experimental evidence. First, the slip models considered here predict that the slip

velocity has a power law dependence on the shear stress in the base state, with a

power law exponent of 3. This value agrees very well with published data for HDPE

(Hatzikiriakos and Dealy [34, 33]). Values for LLDPE range anywhere from 4 to 6

(Hill et al. [38]), so the predicted power law exponent is low. Second, information

about the molecular weight and temperature dependence can be obtained. We

find a critical true Weissenberg number Wet,c for instability that is constant when

kxb � 1. Thus the critical shear rate is

γt,c∗ =

Wet,cλ

. (35)

Generally, the relaxation time strongly decreases with temperature, following an

Arrhenius relation, and strongly increases with molecular weight (MW) [19], e.g.

λ ∼ MW 3.4 for entangled melts, so that the critical shear rate increases sharply

with temperature and decreases with the molecular weight. Venet and Vergnes [96]

have recently shown for LLDPE that the critical shear rate increases as a function

of temperature with a rate much faster than linear, and it is well known that

higher molecular weight polymers are more susceptible to sharkskin distortions.

The predicted critical recoverable shear, τyx,c∗ /G∗ = Wet,c, is a constant, as observed

by Pomar et al. [73], who diluted LLDPE with octadecane to lower the modulus and

found that sharkskin set in at a recoverable shear of 1.73 for a wide range of polymer

weight fractions. Since G∗ ∝ T , we also find that τyx,c∗ /T is a constant at onset, in

agreement with the experiments of Venet and Vergnes [96] and Wang et al. [100]

The longest wavelength observed will be for kxb ∼= 1, for which the frequency of

the distortion is ω∗c = k∗xRe(c∗) ∼= k∗xu

∗s = k∗xb

∗γ∗n = kxbWen/λ ≈ 11/λ, which scales

58

as the reciprocal of the bulk relaxation time, as seen experimentally for LLDPE

by Wang et al. [100] and Barone et al. [8] Finally, the critical recoverable shear

predicted by the model is around 10. Several experiments have reported critical

stresses for sharkskin in LLDPE around 0.1 MPa. For Ramamurthy’s [75] data, in

particular, this gives a recoverable shear of about 8, so the slip results are within

the correct order of magnitude.

3.5 Mechanism

An energy analysis was performed in order to elucidate the cause of the instability. It

is desirable to have a molecular level picture of what is happening at the interface to

destabilize the flow. This has not been fully realized of yet, but the energy analysis

points to the important terms in the equations, and establishes criteria that any

proposed mechanism must fulfill.

In order to do an energy analysis, one wants to follow the evolution of energy or

an energy-like, i.e. a real, positive definite, quantity. The L2 norm of the solution is

used here, and this quantity can be derived starting from the linearized component

equations of the UCM constitutive equation for the asymptotic case, kx ∼ 1/ε� 1,

y = kxy, Wen = Wet. For this scaling, the critical Wen is Wen,c = 10.669 for s = 1.

The scaled stress equations are

∂τxx

∂t+ u

∂τxx

∂x− 2(1 + τxx)

∂u

∂x− 2τyx

∂u

∂y− 2τyx

∂u

∂y+

1

Wenτxx = 0, (36)

∂τyx

∂t+ u

∂τyx

∂x− τyy

∂u

∂y−

∂u

∂y− (τxx + 1)

∂v

∂x+

1

Wenτyx = 0, (37)

∂τyy

∂t+ u

∂τyy

∂x− 2τyx

∂v

∂x− 2

∂v

∂y+

1

Wenτyy = 0, (38)

59

where the barred variables denote the base state and now the unmarked variables

are the perturbations. The perturbations are written in normal mode form as

a = a + a∗, where a = a(y, t)eikxx and the complex conjugate a∗ = a∗(y, t)e−ikxx.

The time dependence has not been explicitly included yet. With these definitions,

Eq. 36 becomes

∂

∂t(τxx + τxx

∗) + ikxu(τxx − τxx∗)− 2(τxx + 1)ikx(u− u∗)− 2τyx(u

′ + u′∗)−

2(τyx + τyx∗) +

1

Wen(τxx + τxx

∗) = 0. (39)

Note that all the terms in this equation are real. The next step is to multiply the

entire equation by (τxx + τxx∗) and then integrate over x and y. This yields

1

2

∂

∂t〈(τxx + τxx

∗)2〉 = −iu〈(τxx − τxx∗)(τxx + τxx

∗)〉+

2i(τxx + 1)〈(u− u∗)(τxx + τxx∗)〉 −

1

Wen〈(τxx + τxx

∗)2〉+

2τyx〈(u′ + u

′∗)(τxx + τxx∗)〉+ 2〈(τyx + τyx

∗)(τxx + τxx∗)〉, (40)

where the brackets denote the double integration. This equation can be further

simplified to give

∂

∂t〈〈τxxτxx

∗〉〉 = 2i(τxx + 1)〈〈uτxx∗ − u∗τxx〉〉 −

2

Wen〈〈τxxτxx

∗〉〉+

2τyx〈〈τxxu′∗ + τxx

∗u′〉〉+ 2〈〈τxxτyx∗ + τxx

∗τyx〉〉, (41)

where the double bracket indicates integration over y, and the integration over x

has been implicitly performed. There are a couple of points to notice. First, since

the goal is to determine the important terms at the onset of instability, Im(c) is

zero, and c is a purely real quantity. Since every term in Eq. 41 contains a variable

multiplied by the conjugate of another, and the perturbations can be written as

60

a(y)e−ikxct, the time dependence will cancel out of every term, and each term will

only be a function of y. Therefore, only the amplitude functions that were calculated

earlier are needed. Second, because the time dependence cancels out of the left hand

side of the equation at the onset of instability, the time derivative will be zero, and

this fact provides a simple check on the calculations. The other two components

of the constitutive law can be reduced the same way. When all three resulting

equations are added together, the result is one equation for the time rate of change

of an energy-like quantity

∂

∂t〈〈τxxτxx

∗ + τyxτyx∗ + τyy τyy

∗〉〉 = 2i(τxx + 1)〈〈uτxx∗ − u∗τxx〉〉+

2τyxγ〈〈τxxu′∗ + τxx

∗u′〉〉+ 2〈〈τxxτyx∗ + τxx

∗τyx〉〉 −

2

Wen〈〈τxxτxx

∗〉〉+ (τxx + 1)i〈〈vτyx∗ − v∗τyx〉〉+

γ〈〈τyxτyy∗ + τyx

∗τyy〉〉+ 〈〈τyxu′∗ + τyx

∗u′〉〉 −

2

Wen〈〈τyxτyx

∗〉〉+ 2iτyx〈〈vτyy∗ − v∗τyy〉〉+

2〈〈τyyv′∗ + v′τyy

∗〉〉 −2

Wen〈〈τyy τyy

∗〉〉. (42)

A very similar equation can be derived from the exact forms of the component

equations so that the analysis can be extended to the no-slip case (Gorodtsov and

Leonov solution). This equation is not shown for brevity. The parameters are chosen

to correspond with the asymptotic result, Wen = 10.7, kx = 100. Table 4 shows the

values of the terms in Eq. 42 obtained for both the destabilizing slip mode and the

Gorodtsov-Leonov mode for the no-slip case as well as the constitutive component

that the terms are from. Note that the terms for the slip case sum to zero,

validating the calculations, but that the terms for the no slip case do not. This is

because the slip case is calculated at the onset of instability, where the Im(c) is zero,

61

τxx

τyx

τyy

Term Slip No-slip (G-L)

2i(τxx + 1)〈〈uτxx∗ − u∗τxx〉〉 11003.4 -3045.31

2τyx〈〈τxxu′∗ + τxx

∗u′〉〉 17078.6 3814.9

2γ〈〈τxxτyx∗ + τxx∗τyx〉〉 44857 45114.2

− 2Den〈〈τxxτxx∗〉〉 -72939 -101921

i(τxx + 1)〈〈vτyx∗ − v∗τyx〉〉 237.622 -23.6812

γ〈〈τyxτyy∗ + τyx∗τyy〉〉 4624.58 62.4942

〈〈τyxu′∗ + τyx

∗u′〉〉 -90.645 6.61066

− 2Den〈〈τyxτyx∗〉〉 -4771.55 -100.899

2iτyx〈〈vτyy∗ − v∗τyy〉〉 20.6294 -.0066018

2〈〈τyyv′∗ + v′τyy

∗〉〉 19.2603 .2046562

Den〈〈τyy τyy∗〉〉 -39.8897 -.439934

Table 4: Comparison of the results for slip and no-slip.

but there is no point where the Im(c) is zero for the no-slip case, since the flow with

no slip is unconditionally stable. This means that the the time exponential does

not cancel out of the terms in the no-slip case. Instead, every term is multiplied

by the factor eikx(c−c∗)t. The calculations were done with the implicit assumption

that t = 0, thereby dropping that factor out. By this method, the largest terms are

the coupling terms between the base state velocity gradients and the perturbation

stresses. However, the effect of this coupling is the same in both cases, even though

destabilizing, and cannot cause a fundamental change in the stability. Also, the

viscous terms (terms containing the reciprocal of the nominal Deborah number) are

very large in absolute value and stabilizing, but, again the behavior is the same in

both cases. The crucial terms are those describing the coupling between the base

state stresses and the perturbation velocity gradients, τ · ∇v. This implies that

62

velocity perturbations further stretch and orient chains, which leads to increased

slip, leading to build up of the instability.

The same coupling behavior is seen in a phase shift between the perturbation

slip velocity and the perturbation stresses, as shown in Fig. 23. Such a phase shift

is impossible in the absence of a τxx dependence of the slip coefficient unless a

0 10 20 30t

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

τ xx, τ

yx, u

τxxτyx u^ ^ ^

^^

^

Figure 23: Phase shift between the slip velocity and the shear and normal stresscomponents at the critical point. ( ) - τxx, (– – – – –) - τyx, and (— — —)- u. The parameter values are kx = 10, εs = 10−5, Wen,c = 14.77, Wet,c = 13.36, andRe(c) = 0.04944, and the amplitudes have been normalized to unity.

“memory slip” model like Eq. 1 is used. Instability is never observed in the absence

of a phase shift (R0 = 0 in the asymptotic analysis). The fact that the memory

slip model also leads to instabilities suggests that the phase shift is an essential

component of the mechanism.

3.6 Summary of the Melt Analysis

Slip models which include a normal stress dependence of the slip velocity lead to

novel shortwave hydrodynamic instabilities in inertialess, viscoelastic parallel shear

63

flows. Analysis of flows with these slip models makes several predictions consistent

with experimental observations of sharkskin. Most significantly, this model pre-

dicts a short wavelength instability at zero Reynolds number that is localized near

the bounding surfaces. This instability is found even though the flow curves are

monotonic increasing and the model is mathematically well-posed at all wavenum-

bers. Both viscoelasticity and normal stress dependent slip are necessary for the

instability to occur. The onset conditions depend upon bulk polymer properties,

in particular the relaxation time, as well as the interfacial ones, although the pre-

dictions are not sensitive to the functional form used to describe slip. The model

predicts that slip can be stabilizing or destabilizing in Couette flow, depending upon

the flow parameters. The neutral curve goes through a minimum as the degree of

slip increases from zero, after which the critical Weissenberg number rapidly in-

creases and the neutral curve becomes singular. Beyond this point the slip velocity

is so large that high enough stresses to cause the instability cannot be generated

in the fluid and Couette flow is unconditionally stable. Replotting the neutral

curves as the critical true Weissenberg number (the dimensionless shear stress) ver-

sus the wavenumber scaled with the extrapolation length yielded a master curve.

The critical Weissenberg number is on the order of 10, which is the same order

of magnitude as observed experimentally, and predictions of the critical shear rate

and molecular weight and temperature dependencies are in qualitative agreement

with experiments. The results are robust with respect to the actual forms of the

interfacial and bulk constitutive relations employed. Finally, the mechanism must

be tied to the phase shift between the perturbation slip velocity and the perturba-

tion shear stress, as this phase shift is only present in normal stress dependent slip

64

models or memory slip models. This analysis is the only analysis which predicts

hydrodynamic instabilities consistent with sharkskin, and highlights the importance

of including chain stretching and orientation effects in models for slip.

A primary limitation to the results presented here is the model geometry used

in the analysis, which does not approximate the real flow situation during extru-

sion. The question naturally arises of whether similar elastic instabilities are seen

in viscometric flows. One possible answer was provided in the context of entangled

polymer solutions by Mhetar and Archer [56], who observed enhanced concentration

fluctuations in a plane Couette cell. The slip behavior of the solution clearly influ-

ences the onset of instability in this system, but the analysis is more complicated

due to coupling between the polymer stress and the concentration in these systems.

65

Chapter 4

Concentration Fluctuations in

Semidilute Polymer Solutions

A recent experimental study by Mhetar and Archer [56] has demonstrated several

novel features of shear enhanced concentration fluctuations in entangled polymer

solutions. They studied flows of semidilute solutions of polystyrene (PS) in di-

ethylphthalate (DEP), to which tracer particles were added to measure the velocity

profile, using a planar Couette cell. Two features of the flow of these solutions stand

out. First, slip occurs at the polymer/solid interface at low shear stresses where no

turbidity was observed. Enhanced concentration fluctuations were observed at large

shear stresses, where the slip velocity was a strong function of the shear stress, as

shown in Fig. 24 for a 20 wt% solution. The extrapolation, or slip, length measured

for these solutions was found to be on the order of 10 µm, which is the same as the

length scale for the fluctuations themselves. Second, the concentration fluctuation

enhancement started near the boundaries and modification of the surface to increase

66

Figure 24: Enhanced concentration fluctuations in a semidilute solution ofpolystyrene in diethylphthalate at a shear rate of 3.5 s−1. Reprinted from Mhetarand Archer [56].

slip delayed the onset on fluctuations to much larger shear stresses. These obser-

vations strongly suggest that slip plays a role in the formation and development of

these fluctuations.

Shear enhanced fluctuations in semidilute polystyrene solutions have been stud-

ied previously, typically via light scattering measurements. There are two typical

experimental configurations, which differ primarily in the direction of illumination.

The two possibilities are shown schematically in Fig. 25. Hashimoto and cowork-

ers [31, 32, 63, 46] have performed a series of experiments using solutions of PS

dissolved in dioctylphthalate (DOP) sheared in a cone and plate rheometer [31]

through which laser light was shined along the velocity gradient direction. The

average scattering intensities in the flow and normal directions were computed and

a typical result is shown in Fig. 26. At low shear rates, γ∗n, the scattering intensity

was low and uniform. At γ∗n > γ∗c , the scattering intensity increased dramatically

67

y

x

z

Flow direction

(a)

(b)

Figure 25: Schematic of typical light scattering experiments. The dashed arrowsshow the two possible directions for illumination: (a), the velocity gradient direction,which samples the near surface regions, and (b), the neutral direction, which onlymeasures scattering in the bulk.

Figure 26: Scattering intensity as a function of shear rate normalized by the equi-librium intensity for a 6wt% PS solution. The molecular weight of the PS was 5.48×106. Reproduced from Kume et al. [46].

68

Figure 27: Scattering pattern as a function of the shear rate for the same solutionas in the previous figure. In (a), the anisotropic peak is just visible. As the shearrate increases, the intensity increases and the peak shape becomes more defined.Reproduced from Hashimoto and Kume [32].

and the scattering pattern became anisotropic, as shown in Fig. 27, with a broad

scattering peak in the flow direction and a narrow dark streak perpendicular to

the flow direction. This “butterfly” pattern indicates the formation of flow struc-

tures which are elongated roughly perpendicular to the flow direction so that the

wavevector of the fluctuations is roughly oriented in the flow direction. The length

scale for these structures can be estimated from the patterns and is on the order of

10 µm. The critical shear rate for the formation of these anisotropic structures is

γ∗c ≈ 1/λ, where λ is the polymer relaxation time. This gives a critical Wen(≡ λγ∗)

of about 1.

Scattering measurements have also been performed with illumination along the

neutral, or vorticity, direction, with somewhat different results. Wu et al. [105] used

a circular Couette device with Pyrex cylinders, and observed anisotropic butterfly

patterns. The angle of maximum scattering at low shear rates was not in the flow

direction; instead, the scattering pattern was tilted at an angle of about 40◦. This

pattern was already visible at a shear rate of 0.4s−1 (Wen ∼ 0.2). The scatter-

ing peak rotated clockwise (towards the flow direction) and the magnitude of the

wavevector decreased as the shear rate was increased. The initial wavenumber for

69

this peak was ∼ 10µm−1, giving a wavelength for the underlying waves of ∼ 1µm.

At high shear rates, the pattern had rotated so that the peak was below the x-axis.

Similar behavior was reported by Wirtz [104] for a slightly more concentrated so-

lution, although the measured wavenumber was much smaller than that observed

by Wu et al. [105]. In addition, Wirtz reported that the peak initially moved away

from the origin at low shear rates. This behavior was not reported by Wu et al.,

nor is it predicted by any of the theories described below.

The seminal theory on a purely hydrodynamic mechanism of enhanced concen-

tration fluctuations was initially proposed by Helfand and Fredrickson [36] (HF)

and later extended by several authors [59, 60, 41, 91]. This mechanism is based on

the idea that variations in polymer stress can drive a flux of polymer molecules

– indeed, several authors [41, 60, 54] have developed simple, two-fluid models

which demonstrate quite generally that the polymer flux, j, for a dilute solution

is j = −Dtr∇n + (Dtr/kBT )∇ · τ , where n is the polymer number density, τ is

the polymer extra stress tensor, Dtr is the polymer translational diffusivity, kB

is the Boltzmann constant, and T is the temperature. Furthermore, a detailed

kinetic theory derivation [18] leads to a very similar expression for the flux. A

physical description of the basic mechanism was elucidated by Milner [59] and Ji

and Helfand [41] and is shown in Fig. 28, which shows a random fluctuation with

wave vector k. The regions of higher concentration (the dark regions) have larger

stresses. If the angle of the wave vector with respect to the x-axis, β, is in the sec-

ond quadrant, as in Fig. 28(a), the shear stresses set up net forces on the molecules

such that the molecules are driven from the regions of high concentration to regions

of low concentration, thereby increasing the effective rate of diffusion. For β in the

70

(a) (b)

kk

Figure 28: Physical picture of the HF hydrodynamic mechanism for enhanced con-centration fluctuations. The wave vector of the fluctuation is such that in: (a)the net force on molecules pushes molecules from regions of high concentration toregions of low concentration; (b) the net force on molecules pulls molecules intoregions of high concentration from regions of low concentration. Adapted from Jiand Helfand [41]

first quadrant, as in Fig. 28(b), the resulting net force on molecules pulls molecules

from low concentration regions into higher concentration regions. This enhances

fluctuations by decreasing the effective rate of diffusion or by overcoming diffusion

altogether. HF predicted the maximum angle of scattering to be ∼ 41◦ at low shear

rates, which is close to the value measured by Wu et al. [105]. If fluctuations are

short enough, diffusion is faster than stress relaxation and the HF mechanism is sup-

pressed [59]. A scattering peak occurs when the diffusion and relaxation times are

equal, which occurs at a wavenumber of k = (Dtrλ)−1/2. This gives a characteristic

length scale for fluctuations of√Dtrλ.

This length scale was observed in the earlier light scattering experiments at

low shear rates. For the experiments of Hashimoto and coworkers, the relaxation

time can be estimated as the reciprocal of the shear rate for the onset of shear

thinning [46] and using the diffusivity measured by Wu et al. [105] for a similar

system gives a predicted wavenumber of k ≈ 0.7µm−1, which is close to the measured

71

value. This length scale was also observed by Wu et al. [105], who found a scattering

peak at kx ≈ 10µm−1, which is in good agreement with their predicted value of

9.4µm−1.

Even though both of these experiments show peaks at the requisite length scale,

qualitative differences exist in the shear rate dependence of the results. In particular,

Hashimoto and coworkers, who measured the intensity essentially averaged over

a volume containing both near surface regions, found that the intensity increased

markedly above a critical shear rate, which was roughly the reciprocal of the polymer

relaxation time. If fact, for a factor of ten increase in the shear rate above the

critical value, the scattering intensity increased one hundred fold. Conversely, Wu

et al., who measured the scattering in the bulk, completely missing the surface

regions, found a much smaller increase in intensity as the shear rate was increased,

approximately an 11-fold increase in scattering intensity for a 25-fold increase in

shear rate.

This difference is not surprising, as fluid mechanics results suggest that the dy-

namics near surfaces should be different than dynamics in the bulk. Gorodtsov and

Leonov [27] analyzed the plane Couette flow of a UCM fluid and found that, while

always stable in the absence of inertia, the slowest decaying modes are localized

near the bounding surfaces. As a result, one would expect the formation of bound-

ary localized structures due to random concentration fluctuations in the polymer

solution. In addition, slip leads to boundary-localized instability in simple shear

flows, as shown in Ch. 3. Taken in whole, the experiments suggest that the basic

HF mechanism, while being able to explain bulk behavior, only partially accounts

for the physics leading to enhancement of fluctuations near surfaces. The goal of

72

the next chapter is to analyze the coupling between concentration, stress, and slip

to elucidate the effects of boundaries on the dynamics.

73

Chapter 5

Concentration Fluctuations and Flow

Instabilities in Sheared Polymer

Solutions†

5.1 Formulation

The experimental results outlined in Ch. 4 clearly point to the importance of

polymer-surface interactions in determining the evolution of the flow of polymer

solutions. In particular, the experiments of Mhetar and Archer [56] explicitly link

wall slip and the formation of enhanced fluctuations in simple shear, by demonstrat-

ing that: 1) substantial slip takes place at shear stresses and rates below those for

which enhanced fluctuations appear; 2) fluctuations are initiated near the surfaces;

and 3) surface treatments can delay the onset of enhanced fluctuations to larger flow

†The results in this chapter have been submitted for publication in Phys. Rev. Lett.

74

z*s�

��

��

�� u

*

��

��

b

*ly

x

γ *.

Figure 29: Plane Couette geometry showing slip between the solution and solidsurface. The flow is in the x-direction, y is the gradient direction and z is theneutral, or vorticity, direction. l∗ is the gap width, u∗s is the slip velocity, γ∗ is thetrue shear rate in the fluid, and b∗ = u∗s/γ

∗ is the extrapolation length.

rates. These facts constitute the primary motivation for the analyses presented in

this chapter.

Given that the majority of the experiments were performed in simple shear

geometries, the link between slip and the formation of enhanced concentration fluc-

tuations is studied in the plane Couette geometry shown in Fig. 29. The x-direction

is the flow direction, the y-direction is the gradient direction, and the z-direction

is the neutral, or vorticity, direction. The constitutive behavior of the solution is

described by a two-fluid theory derived by Beris and Mavrantzas [54] that has been

specialized to Hookean dumbbells [9]. Nondimensionalizing lengths with the charac-

teristic length√Dtrλ, the stresses with n∗0kT , where n∗0 is the average concentration

of the solution, the velocities with√Dtr/λ, time with λ, and concentration with n∗0

75

gives the following model equations

−∇p +∇ · τ + S∇2v = 0, (43a)

τ (1) + τ −∇2 (τ + nδ) + δ

Dn

Dt= nγ, (43b)

Dn

Dt= ∇2n−∇∇ : τ , (43c)

∇ · v = 0, (43d)

where τ is the polymer extra stress tensor, n is the polymer concentration, λ is

the polymer relaxation time, Dtr is the translational diffusivity of a dumbbell, and

S = ηs/ηp = ηs/(n∗0kTλ) is the ratio of the solvent viscosity to the polymer viscosity

at the average concentration of the solution, n∗0. The true shear rate in the fluid has

been used for the nondimensionalization, as opposed to the applied shear rate used

in Ch. 3 for melts, to take advantage of the fact that the master curve reported

in Fig. 22 in §3.4.3 relates the true Weissenberg number to the wavenumber and

to facilitate the solution technique outlined below. Eq. 43b is the polymer con-

stitutive equation and is a generalization of the upper convected Maxwell (UCM)

equation [10] which includes coupling between the concentration and stress as well

as a stress diffusion term. In most cases, this term is negligible [9], but here, the

flow structures of interest have a length scale set by the diffusivity. Eq. 43c de-

scribes the polymer concentration and explicitly includes the coupling of stress and

concentration gradients. Eqs. 43a and 43d describe conservation of momentum and

mass. This model is similar to those studied previously [36, 59, 60, 41], although

only Sun et al. [91] have included the diffusive term in Eq. 43b. Furthermore, all

earlier works have restricted themselves to bulk behavior and We(≡ λγ∗) < 1.

76

Specification of suitable boundary conditions for the velocity, stress, and con-

centration completes the model. The concentration satisfies the typical no-flux

boundary condition at the surfaces,

n · jn =∂n

∂y−

∂τyx∂x−

∂τyy∂y−

∂τzy∂z

= 0, (44)

where n is the outward unit normal, which states that mass cannot be lost through

the boundaries. The presence of the stress diffusion term in the constitutive equation

necessitates a stress boundary condition, which can be derived by rewriting Eq. 43b

can be rewritten in terms of the conformation tensor using the Kramers’ form of

the stress tensor [10], τ = 〈QQ〉 − nδ, as

∂〈QQ〉

∂t= −∇ · jQQ + 〈QQ〉 · ∇v + (∇v)T · 〈QQ〉 − 〈QQ〉+ nδ (45)

where Q is the orientation vector of a dumbbell, jQQ = v · 〈QQ〉 − ∇〈QQ〉 is the

conformation flux, and the remainder of the terms on the right hand side can be

thought of as volume source terms. Assuming that there are no sources or sinks for

the conformation tensor at the boundary (i.e. the walls are neutral) gives a no-flux

boundary condition for the conformation tensor:

∂〈QQ〉

∂y=

∂τ

∂y+

∂n

∂yδ = 0. (46)

All of the computations reported below use this boundary condition. Computations

performed with the classical UCM equation as the stress boundary condition [92]

showed no qualitative difference in the results. Finally, the last pieces to be specified

are the velocity boundary conditions. The normal component of velocity is given

by the no-penetration condition, v = 0, at both surfaces and the tangential velocity

component is given by a slip boundary condition at the interface. In Ch. 3 it

77

was shown that a slip model where the slip velocity depends on normal stresses

is required in order to find instability in polymer melts. Even though the slip

mechanisms in entangled polymer solutions are expected to be similar to those in

melts, a Navier slip boundary condition is used here, where the slip velocity is

proportional to the shear stress, us = ετyx(= b at steady state). This serves as a

point of reference for comparisons to the melt results and a prelude to including

more complicated slip relations.

5.2 Stability Analysis

The dynamics of concentration fluctuations were studied initially by employing a

linear stability analysis similar to that in Ch. 3. The steady state solution to

Eqs. 43a - 43d using Eq. 46 as the boundary condition is one dimensional and given

by

a = {u, v, w, τxx, τyx, τzx, τyy, τzy, τzz, n, p} (47)

={We(y + us), 0, 0, 2We

2,We, 0, 0, 0, 0, 1, 1},

with y ∈ [0,√Pe/We], where the Peclet number Pe = γ∗l∗2/Dtr. Small perturba-

tions are added to this base solution, a = a(y)+a(x, y, z, t), and the basic equations

are linearized with only the terms leading order in the perturbation retained. The

resulting set of 11 equations can be written in matrix notation as

CS∂a

∂t= −LS a (48)

where CS is a constant matrix with zero rows corresponding to the momentum and

constitutive equations and LS is a continuous spatial differential operator. These

78

operators are shown for completeness in Appendix B. The perturbations take the

normal mode form

a(x, y, z, t) = a(y)eik · xe−iσt + c.c. (49)

where k = (kx, 0, kz) is the wavevector of the disturbance in the x-z plane. Substitu-

tion of Eq. 49 into Eq. 48 yields a generalized eigenvalue problem for the eigenvalues

{σ} with associated eigenfunctions {a(y)}. The eigenvalues are, in general, com-

plex and if Im(σ) > 0 then disturbances will grow and the flow is unstable. Three

dimensional disturbances are considered because Squire’s theorem has not been

demonstrated for this model.

5.2.1 Numerical Method

The Chebyshev collocation technique outlined in §3.4.2 is insufficient for this prob-

lem for several reasons. First, the streamfunction formulation becomes extremely

stiff because the solvent term in the equation of motion leads to fourth derivatives

of the stream function (c.f. §3.4.2). Second, the concentration equation leads to

an additional strip of continuous spectrum, when Dtr = 0, which interferes with

the calculation of the discrete modes. For Dtr �= 0, the governing equations are no

longer singular and all of the eigenvalues are discrete. However, there are sets of

discrete eigenvalues which correspond to the continuous modes of the Dtr = 0 case

and are similarly difficult to resolve. This will be more clear below.

A modified Chebyshev collocation scheme, employing a three dimensional, prim-

itive variable formulation, was developed to solve the generalized eigenvalue problem

resulting from Eq. 48. The results in §3.4.3 show that, for the case of the PTT fluid

79

(and, therefore, the UCM fluid), the eigenfunctions are localized near the boundaries

and decay away from the surfaces for large wavenumbers. It was also shown that

accurate approximations were obtained analytically by retaining only the decaying

solutions to the general stability equation. This can be accomplished numerically by

solving the the governing equations on a semi-infinite domain (SID) and requiring

the eigenfunctions to decay to zero as y → ∞. We map y onto the computational

coordinate, ξ, using [17, 15]

y = L

(1 + ξ

1− ξ

), (50)

with {y, ξ |y ∈ [0,∞), ξ ∈ [−1, 1]}. As discussed by Canuto et al. [17], this mapping

is more robust than an exponential mapping and less sensitive to the value of the

map parameter, L. The map parameter is a measure of the degree of stretching of

the coordinates, i.e., it is the value of y which is mapped to the center of the com-

putational domain. Trial and error yielded an optimum value of the map parameter

of L = 0.1 for this problem. All of the results presented below use this value. This

technique is accurate for disturbances with wavenumber much larger than the re-

ciprocal of the gap width, k∗x � 1/l∗, as for the analytical solutions of §3.4.1, a

condition satisfied by the dominant wavenumber measured experimentally [56].

Spurious pressure modes in primitive variable formulations are well known and

described in detail by Canuto et al. [17]. These modes can be eliminated by ex-

panding pressure on a lower order grid. To balance the number of equations and

unknowns, the continuity equation is evaluated on the staggered grid. This requires

extrapolation of the pressure from the staggered to regular grid for evaluation of

the momentum equation and interpolation of the velocity components from the

80

regular to staggered grid for the evaluation of the continuity equation. The extrap-

olation formula is derived from the expansion for pressure in terms of Chebyshev

polynomials is

p(ξ) =N−1∑i=0

biTi(ξ) (51)

with the expansion coefficients given by

bi =2

πci

∫ 1−1

p(ξ)Ti(ξ)w(ξ) dξ, (52)

where

cj =

2 j = 0

1 j ≥ 1. (53)

The staggered grid points, given by, ξsi = cos((i+ 1

2)π

N

), with i = 0, . . . , N−1, are all

in the interior of the domain, so this integral can be evaluated by Gauss quadrature

to give

bi =2

Nci

N−1∑k=0

p(ξsk) cos

(j(k + 1

2)π

N

). (54)

This can be put back into the expansion for p and this polynomial evaluated at the

regular grid points to get

p(ξi) =2

N

N−1∑j=0

N−1∑k=0

1

cjp(ξsk) cos

(j(k + 1

2)π

N

)cos

(ijπ

N

)(55)

This gives the values of pressure at the regular grid points in terms of the pres-

sure values at the staggered grid points. To get the interpolation function for

the velocities, we start with the expansion in terms of Chebyshev polynomials,

81

v =∑N

i=0 aiTi(ξ), and solve for the expansion coefficients using Gauss-Lobatto

quadrature on the regular grid, since the domain includes the boundary points,

ai =2

Nci

N∑j=0

1

cjv(ξj) cos

(jkπ

N

), (56)

where

cj =

2 j = 0, N

1 1 ≤ j ≤ N − 1. (57)

Putting this back into the expansion and evaluating it at the staggered grid points

gives the interpolation function

v(ξsi ) =2

N

N∑k=0

N∑j=0

1

ckcjv(ξj) cos

(k(i + 1

2)π

N

)cos

(kjπ

N

), (58)

Eqs. 55 and 58 can be written in matrix form and are implemented via matrix

multiplication, as was done previously with the Chebyshev derivative operator.

Code Verification

The code was benchmarked against 2D results by setting kz = 0 and dropping the

stress diffusion terms in the constitutive equation. Linearizing around the steady

state solution and introducing the normal mode form given in Eq. 49 gives the

82

following reduced system of equations for 2D disturbances

−ikxp + ikxτxx + τ ′yx + S(−k2xu+ u′′

)= 0 (59a)

−p′ + ikxτyx + τ ′yy + S(−k2xv + v′′

)= 0 (59b)

Qτxx − 2ikx(τxx + 1)u− 2τyxu′ − (iσ − ikxu)n− 2τyx = 0 (59c)

Qτyx − ikx(τxx + 1)v − u′ − n− τyy = 0 (59d)

Qτyy − 2ikxτyxv − 2v′ − (iσ − ikxu)n = 0 (59e)

−n′′ + (−iσ + ikxu + k2x)n+[−k2xτxx + 2ikxτ

′yx + τ ′′yy

]= 0 (59f )

ikxu + v′ = 0 (59g)

where Q = 1− iσ+ ikxu and u = We(y+ us). The constitutive equations, Eqs. 59c -

59e can be solved for the stress components in terms of velocities and n. After elimi-

nating pressure from the momentum equations, Eqs. 59a and 59b, these expressions

for the stresses can be put into the momentum equation to give an equation relating

the velocities and concentration. After using the continuity equation to eliminate

u in favor of v, a general stability equation can be derived, which is

[Q2D2 − k2xQ2 − 2ikxQD − 2k2x] [D

2 + 2ikxτyxD − k2x(τxx + 1)] v+

SQ3(D2 − k2x)2v − ikx [Q2D2 − 2ikxD + k2xQ

2] n = 0.(60)

This equation contains three pieces: 1) the first operator is the polymer (UCM)

term, 2) the second term is the solvent term, and 3) the last term is a concentration

term due to the coupling between stress and concentration. It is not clear from

this equation where the continuous spectra should be found, as the concentration

dependence of the velocities is unknown. So, Eq. 59f , is simplified by substituting

83

for the stresses to give

[1− i

(σ − kxu

Q

)]n′′ +

[i(σ − kxu)− k2x

[1− i

(σ − kxu

Q

)]]n = 0 (61)

This equation is decoupled from the stability equation, due to cancelation of the

stress terms. One set of continuous eigenvalues will be where this equation is sin-

gular, i.e.,

σ = kx(y + us)−i

2(62)

Since n can be determined from Eq. 61, the stability equation is just the Oldroyd-

B stability equation with forcing. Therefore, we also expect strips of continuous

spectra where the Oldroyd-B stability problem is singular [43, 103], specifically

σ =

kx(y + us)− i

kx(y + us)−i(S+1)

S

. (63)

All three strips of continuous spectra are stable. Theoretically, we could also de-

termine some of the discrete eigenvalues by solving Eq. 61 and applying no-flux

boundary conditions. However, there is no fortuitous cancelation after substituting

for the stresses in the boundary condition and consequently, the no-flux bound-

ary condition contains velocities which must be determined by solving the stability

equation. It would appear then that all of the discrete eigenvalues are affected by the

coupling between velocity and concentration and obtaining analytical expressions

for any of them would be difficult. Nevertheless, the locations of the continuous

spectra are useful checks on the code.

Fig. 30 shows the eigenvalue spectrum for 2D disturbances with the stress diffu-

sion terms dropped. Three strips of continuous spectrum exist and their locations

84

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0−3.0

−2.5

−2.0

−1.5

−1.0

−0.5

0.0

0.5

0.0 4.0 8.0 12.0 16.0 20.0 24.0 28.0 32.0Re(σ)

−320.0

−260.0

−200.0

−140.0

−80.0

−20.0

40.0

Im(σ

)

G−L modeSlip mode

Concentration modeSlip mode

Inset

Figure 30: Eigenvalue spectrum for 2D disturbances with the stress diffusion termdropped. The parameters are kx = 0.3467, We = 11.9, b = 0.1442, S = 0.01, N = 96.The eigenvalues which determine the stability are designated by circles and are onthe left side of the spectrum. The “tail” of eigenvalues extending to the right arenearly continuous modes which are not resolved.

are in excellent agreement with the above predictions. There are 4 discrete modes

near the strip of continuous spectra due to the concentration equation: two slip

modes; a mode corresponding to the classical Gorodtsov-Leonov eigenvalue; and a

concentration mode. Note that the continuous strip due to the concentration equa-

tion is less stable than its UCM counterpart, and is therefore expected to interfere

more strongly with the calculation of the discrete modes leading to instability.

The SID technique is very effective at resolving the discrete and continuous

modes, as shown in Fig. 31, which compares the SID technique with the standard

collocation method on a bounded domain, with focus on the portion of the spectrum

near the ends of the UCM and concentration strips of continuous spectra. Only the

slip mode is observed for the standard technique with N = 192; the concentration

mode is absorbed into the continuous strip due to the concentration equation. N

must be increased to 280 before the second mode is resolved. The SID technique

85

2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0Re(σ)

−1.5

−1.0

−0.5

0.0

0.5

Im(σ

)

Slip mode

Concentration mode

Figure 31: Comparison of the standard and SID techniques. The figure does notshow the entire spectrum; instead, the focus is on the least stable portion. Theparameters are: We = 19.64, kx = 10, b = 0.02, Pe = 2000. 3 – Standard:N = 192, © – Standard: N = 280, 2 – SID: N = 192, � – SID: N = 96, � –DD: Nb = 75, Nc = 43. Note that Pe has no meaning for the SID simulations andsimply sets the location of the top plate for the remaining techniques. The stressdiffusion terms have dropped as described in the main text.

with N = 192 clearly resolves the mode and is superior to the standard scheme with

N = 280. In fact, the SID scheme with N = 96 is capable of resolving the mode

as well as the standard scheme with N = 280. This gives an enormous savings in

computer time and storage.

Also shown are the results for a domain decomposition (DD) technique. The

key to improving the resolution of the spectrum near the endpoints is improving

the underlying spatial discretization of the governing equations near the bound-

aries. To do this without increasing the overall number of collocation points, the

domain is broken up into three subdomains, one near each boundary and one in the

center of the channel in which the number of collocation points can be varied in-

dependently. The boundary domains are much smaller than the central domain, in

fact, the boundary domain size scales as 1/kx, since this is the size of the boundary

86

layer that the perturbations are confined to. Matching conditions are applied at the

intersection of each domain. Souvaliotis and Beris [90] reported only needing C0

continuity (piecewise continuity) between domains for their flow calculations, but

here C1 continuity (continuous first derivatives) for each variable had to be imposed

to get meaningful spectra. Fig. 31 also shows the spectrum for domain decompo-

sition for boundary domains with Nb = 75 for the number of collocation points,

and a central domain with Nc = 43 for the number of collocation points, giving a

total number of 193 collocation points (which is the same as a single domain with

N = 192). The resolution of the concentration mode is about the same as for the

single domain problem with N = 280, so this technique is an improvement in that it

does save time and storage over the standard scheme, but not nearly as much as the

SID technique. Note that the domain decomposition technique could be combined

with the SID technique, but the improvement is expected to be slight at best, as

essentially all of the collocation points in the SID technique are already resolving

the boundary layer.

5.2.2 Stability Results

Fig. 32 shows a portion of a typical eigenvalue spectrum obtained using the SID

technique. The “tail” of eigenvalues extending to the right is composed of the

nearly continuous modes mentioned above. These modes are not mesh-resolved at

N = 96, and in fact, part of the tail has Re(σ) > 0, spuriously indicating instability.

The modes on the left designated with circles are the modes which determine the

stability and are resolved for this number of collocation points.

The coupling of slip to concentration and stress leads to novel hydrodynamic

87

0.0 100.0 200.0 300.0 400.0 500.0Re(σ)

−20.0

−15.0

−10.0

−5.0

0.0

5.0

10.0

15.0

Im(σ

)

Figure 32: A typical eigenvalue obtained using the SID technique. The parametersare We = 10, kx = 1, b = 1, S = 0.01, and N = 96.

instabilities in plane Couette flow. Note that if Dtr is set to zero in the dimensional

equations, the model reduces to the Oldroyd-B model, and plane Couette flow with

either the no-slip [27] or Navier slip [66, 13, 14] boundary condition is stable for

all Weissenberg numbers. Also, if concentration variations are permitted but slip

is not, by setting b = 0, the flow is again stable. Fig. 33 shows neutral curves with

slip, plotted as Wec vs. kx for two dimensional disturbances, kz = 0, with b fixed.

At low values of kx, power law behavior is observed. For b(= b∗/√Dtrλ) = 1, the

critical wavenumber is kx ≈ 0.8, so that the length scales for slip and the instability

are the same. Careful examination reveals that at low wavenumbers (those most

easily observed by microscopy), the critical We first decreases and then increases

with increasing b – the increasing section of the curve is in agreement with the

observations of Mhetar and Archer [56] that treating the surface to increase slip

delayed the onset of fluctuations. Note that there is a transition from one mode

of instability to another, leading to multiple minima in the neutral curve, as most

clearly seen in the curve for b = 2. For larger values of b, the neutral curves

88

10−1

100

101

kx

1

10

100

We c

b = 1b = 2b = 4b = 8b = 10

Unstable

Stable

Figure 33: Neutral curves for kz = 0 and S = 10−2 for various values of b. Thesecurves were computed using the SID technique with N = 96.

nearly lie on top of one another, due to localization of the waves with respect to

the slip length. Typical three dimensional neutral curves are shown in Fig. 34 for

both unstable modes, where clearly the most dangerous modes are two dimensional.

Only two dimensional results are considered further.

Fig. 35 shows a density plot of the unstable eigenvector for the concentration.

The concentration perturbation is oriented in the same direction as enhanced fluc-

tuations within the HF mechanism described by Ji and Helfand [41]. In agreement

with the observations of Mhetar and Archer [56], perturbations are localized near

the surfaces and are observed at large We. As a result, one would expect scatter-

ing to be more pronounced in the boundary regions as opposed to the bulk. One

concern is that for Rouse chains the characteristic length√Dtrλ is proportional to

the end-to-end distance of the chain, i.e., this is a molecular length scale. Note

that the validity of the model becomes suspect for fluctuation sizes on the order

of molecular lengths because it does not capture dynamics on these length scales.

However, as the characteristic length is experimentally much larger than molecular

89

0.00 0.10 0.20 0.30 0.40 0.50kz

8.0

9.0

10.0

11.0

We c

b = 2 (Mode 1)b = 6 (Mode 2)

Unstable

Stable

Figure 34: Neutral curves for three dimensional disturbances at the given values ofkx and b. These curves were computed using the SID technique with N = 96, S =10−2.

length scales, the model is valid until kx � 1.

5.3 Brownian Fluctuations

The stability results are suggestive, but do not address the dynamics of random,

thermal fluctuations on the system. Typical theoretical calculations involve deter-

mining the linear response of the system to random forcing throughout the domain,

then computing the concentration correlation function. Instead of introducing fluc-

tuations in multiple directions, the normal mode form of the linearized variables is

used. This is equivalent to Fourier transforming the governing equations in the x-

and z-directions. The correlation function is then determined for a given wavenum-

ber in each direction. In addition, we only consider random concentration fluctua-

tions and do not force the other variables. This assumption is not unprecedented,

as Helfand and Fredrickson [36] only considered velocity and concentration fluctua-

tions and neglected variations in stress, while Ji and Helfand [41] demonstrated that

90

0 Π�kx 2Π�kx 3Π�kx 4Π�kxx

0

1

2

3

4

5

y

Figure 35: Unstable eigenfunction for the concentration. The parameters are kx =0.4, b = 10, We = 10, S = 0.01, N = 96.

velocity is a fast variable as well. The linearized concentration equation becomes

∂n

∂t+ v · ∇n = ∇2n−∇∇ : τ +∇ ·w(x, t) (64)

where w a the random component of the polymer mass flux satisfying [24]

〈w(x, t)w(x′, t′)〉 = 2n∗0(Dtrλ)32δ δ(x− x′)δ(t− t′) (65)

〈w(x, t)〉 = 0. (66)

In order to satisfy conservation of mass, n · w = 0 on the boundaries, where n

is the outward unit normal, and the forcing was weighted appropriately so that

the magnitude was uniform throughout the domain. Time integration is performed

using an implicit Euler scheme

(CS + ∆tLS

)an+1 = CSan + ∆t∇ ·wn(x, t). (67)

with arbitrary initial condition. Here, n(y, 0) = cos(6y), τ (y, 0) = 0. Comparison of

the time integration results without noise with the SID stability results confirmed

convergence as ∆t → 0. For accurate results, the time step had to be small;

∆t = 0.001 for all of the simulations below.

91

Generally, time integration on a bounded domain provided the best results.

The SID technique is very good at approximating the eigenvalues near the surfaces,

however, the price paid is very poor resolution of the continuous modes further away

from the surface. These modes can be spuriously unstable, making time integration

impossible. To use the bounded domain, the location of the top plate must be

specified, and is chosen to satisfy the SID constraint that the wavenumber be much

larger than the reciprocal of the gap width, in dimensionless terms, kx �√We/Pe.

The primary quantity of interest is the spatial correlation function, defined in

general for a given x-wavenumber kx as:

〈g(x; kx)g(x′; kx)〉 = 2Re〈g(y; kx)g

∗(y′; kx)〉 cos(kx(x− x′))−

2 Im〈g(y; kx)g∗(y′; kx)〉 sin(kx(x− x′)). (68)

To confirm that the noise is being generated correctly, we first compute the spatial

correlation for the forcing by setting g(x; kx) = ∇·w(x; kx) = f(x; kx). Figs. 36(a)

and 36(b) show the real and imaginary parts of the correlation, respectively, and

demonstrate clearly that the noise is real and correlated in y as ∇2δ(y − y′), as

anticipated. The jaggedness evident in Fig. 36(a) is a plotting artifact due to the

fact that the noise is evaluated at discrete points in y.

The scattering is determined by the concentration spatial correlation function,

g(x; kx) = n(x; kx). Fig. 37 shows the concentration correlation function at equilib-

rium. Fluctuations are very strongly correlated near the surface and the magnitude

drops off to a value O(1) in the bulk. The imaginary part of the correlation is

zero, so that fluctuations decay without a directional bias, as is intuitively expected

when there is no shear. The length scale given by the width of the correlation is

92

025

5075

0

25

5075

10005�10�19

�5�10�19

2550

75y

y’Im<

f(y;

k x)

f*(y

’;k x

)>

025

5075

0

25

50

75100

00.02.04

2550

75y

y’

Re<

f(y;

k x)

f*(y

’;k x

)>

(a) (b)

Figure 36: The noise correlation function. (a) real part, (b) imaginary part. Thenoise is δ correlated in y and the imaginary part of the correlation is zero, confirmingthat the noise is being generated correctly.

01

23

45 0

1

23

45

02�105�105

12

34y

y’Re<

n(y;

k x)

n(y’

;kx)

>

Figure 37: Correlation function at equilibrium.

O(√Dtrλ). Figure 38 shows a series of correlation functions for increasing Weis-

senberg number. The real part of the correlation remains essentially unchanged, but

the imaginary part is nonzero, indicating that the fluctuations are enhanced in a

particular direction. The direction of enhancement can be determined by rewriting

the correlation function as a single sinusoid

〈n(x; kx)n(x′; kx)〉 =√

Re〈n(y; kx)n∗(y′; kx)〉2 + Im〈n(y; kx)n∗(y′; kx)〉2 ×

sin [(x− x′)− φ] , (69)

93

where φ is the phase angle, defined as

φ = tan−1 (−Im〈n(y; kx)n∗(y′; kx)〉/Re〈n(y; kx)n

∗(y′; kx)〉) , (70)

and computing the phase angles from the correlation functions. In all cases shown

in Fig. 38, the real part of the correlation is positive, so it suffices to consider

only the imaginary part to get the general idea. Consider Fig.38(f), which is the

imaginary part of the concentration correlation for We = 5, and the boundary point

y′ = 0. For points y in the interior, y > y and Im〈n(y; kx)n∗(y′; kx)〉 is negative.

This gives a positive phase shift, φ > 0. Therefore, the angle of positive correlation

is positive. The wavevector of the enhanced fluctuation is normal to the correlation

angle and points into the fourth quadrant. This is indeed the structure observed

if the concentration profile is plotted, as shown in Fig. 39(c). In general, if the

correlation is negative for y′ > y, then the orientation is in the fourth quadrant

and if the correlation is positive, then the orientation is in the first quadrant. For

We = 0.1, the orientation is in the first quadrant, except for perhaps a very small

region near the surface and that for We = 1 the orientation changes away from

the surface, from the first to the fourth quadrant. This interpretation is borne

out looking at the concentration profiles in Fig. 39. Evidently, the wavevector of

the enhanced fluctuation rotates as the shear rate increases. This rotation of the

wavevector has been predicted previously for the bulk by Ji and Helfand [41] and

Milner [60] and observed experimentally by Wu et al. [105], also for the bulk. The

results presented here suggest that fluctuations near the surface are enhanced by

mechanisms analogous to those in the bulk, but with larger magnitudes because

of the vicinity to the no-flux boundary. Results for the no-slip case are virtually

identical – the response of the flow to Brownian noise is quite insensitive to the

94

presence of slip. In other words, the modes excited by the Brownian noise are

those that are common to both the slip and no-slip cases, and in fact, most of the

spectrum is changed only trivially by slip, as shown in Fig. 40. Therefore, even in

the absence of flow instability, (i.e. when We � O(1) and/or slip is absent), the

near surface regions may make a nontrivial contribution to the scattering signal,

particularly its anisotropy under flow.

5.4 Summary

Recent experimental evidence suggests that slip at the polymer/solid interface may

play a role in the formation and development of enhanced concentration fluctuations

in semidilute polymer solutions. In particular, the initiation of enhanced concen-

tration fluctuations was directly observed to happen at the interface and modifying

the surface to increase slip delayed the onset of enhancement to much higher shear

rates. Previous theoretical treatments have focused on the behavior in the x-y plane

and have successfully explained most of the behavior observed. Slip leads to a new

class of viscoelastic flow instabilities which result from the interaction of slip with

stress and concentration. The critical wavenumber agrees with those observed in

experiments so that the relevant length scale for the instability is√Dtrλ. The crit-

ical We is O(10) for b ∼ O(1), which is within the correct order of magnitude. Time

integration of the governing equations with random concentration fluctuations dis-

tributed throughout the domain shows that Brownian fluctuations are selectively

and dramatically enhanced near the surface, with a boundary layer size consistent

with experiment. The local wavevector for the enhanced fluctuation rotates as the

95

shear rate increases, as predicted and observed for the bulk. Overall, these results

are consistent with experimental observations and underscore two points regarding

the flow behavior of polymeric liquids: (1) the distinctness and importance of the

dynamics of flowing polymers near boundaries, even at the continuum level, and

(2) the importance of couplings between various phenomena for the dynamics and

stability of these flows.

96

01

23

45 0

1

2

34

5

�5000

50000

12

34y

y’Im<

n(y;

k x)

n*(y

’;k x

)>

01

23

45 0

1

2

34

5�105

2�105

0

12

34y

y’Re<

n(y;

k x)

n*(y

’;k x

)>

01

23

45 0

1

2

34

501000

�1000

12

34y

y’Im<

n(y;

k x)

n*(y

’;k x

)>

01

23

45 0

1

2

34

5

02�105�105

12

34y

y’Re<

n(y;

k x)

n*(y

’;k x

)>

01

23

45 0

1

2

34

5

�200

200

0

12

34y

y’Im<

n(y;

k x)

n*(y

’;k x

)>

01

23

45 0

1

2

34

5�105

2�105

0

12

34y

y’Re<

n(y;

k x)

n*(y

’;k x

)>

(a)

(b)

(c)

(d)

(e)

(f)

Figure 38: Series of concentration correlation functions for increasing We. Pictures(a) and (d) are for We = 0.1, (b) and (e) are for We = 1, and (c) and (f) arefor We = 5. As the Weissenberg number increases, the wavevector of the enhancedfluctuation rotates from the first quadrant to the fourth. The remaining parametersare kx = 1, S = 0.01, N = 192.

97

0 Π�kx 2Π�kx 3Π�kx 4Π�kx

y

x

0 Π�kx 2Π�kx 3Π�kx 4Π�kx

0

1

2

3

4

5

(a)

(b)

(c)

0 Π�k 2Π�k 3Π�k 4Π�kx

y

Figure 39: Series of snapshots of typical concentration profiles as We is increased:(a) - We = 0.1; (b) - We = 1; and (c) - We = 5. Note change in orientation, fromthe first quadrant to the fourth, as We is increased.

98

0.0 25.0 50.0 75.0 100.0Re(σ) − b

−2.0

−1.5

−1.0

−0.5

0.0

0.5

Im(σ

)

Figure 40: Eigenvalue spectra for the slip and no-slip cases in the bounded flowdomain. This figure only shows the least stable portions of the spectra. 3 - b = 1,© - b = 0. The remaining parameters are We = 10, kx = 1, S = 0.01, N = 128.

99

Chapter 6

Concluding Remarks

The results in Chs. 3 and 5 highlight the importance of polymer – surface inter-

actions, wall slip in particular, on the macroscopic flow behavior of polymer melts

and solutions. It has been demonstrated that well-posed slip models can lead to

instabilities in polymer melt flows. In particular, inclusion of chain orientation and

stretching at the surface in slip models is crucial for predicting instabilities con-

sistent with experimental observations. Slip couples to stress and concentration in

polymer solutions as well, leading to hydrodynamic instability in these systems,

although this is not the only effect of surfaces on solutions. Indeed, random fluc-

tuations are selectively and dramatically enhanced near the surfaces, even in the

absence of slip and even at equilibrium. This type of surface effect may be important

in other systems driven by random fluctuations, particularly phase separating poly-

mer solutions and blends, where the surface may affect the onset of phase separation

as well as the morphology.

To conclude, a recent experimental result for sharkskin is highly suggestive of

100

possible future directions for this work and highlights some of the synergies be-

tween flows of melts and entangled solutions. Barone and Wang [7] studied ex-

trusion of polybutadiene through a transparent quartz slit die. They chemically

treated the downstream half of the die, by coating it with a polysiloxane. This

enhanced wall slip and created an internal boundary singularity. Instability was

observed using birefringence in the vicinity of the singularity which decayed down-

stream. The period of oscillation was similar to the period of oscillation at the

die exit for an uncoated die in which the extrudate exhibited sharkskin. Wang and

Plucktaveesak [101] observed a similar effect due to an internal boundary singularity

during HDPE extrusion. Mhetar and Archer [56] used quartz plates for their plane

Couette cell and a similar siloxane treatment was effective in promoting slip for

polystyrene solutions. The velocity profile measurement technique of Mhetar and

Archer could be adapted to Barone and Wang’s system to reveal the slip behavior

near the internal singularity and/or Mhetar and Archer’s Couette cell could be used

to reproduce Barone and Wang’s experiment. From a theoretical point of view, these

bounded systems are the next step up in complexity from the viscometric analyses

reported here, and have the advantage of being much more amenable to analysis

than the full die exit problem, as there are no free surfaces and the attendant die

swell to consider. Even so, these geometries capture the critical issues of velocity

profile rearrangement downstream of the singularity and extensional velocity com-

ponents upstream. Understanding the role of boundary singularities and velocity

profile rearrangement will prove crucial to understanding sharkskin dynamics.

101

Nomenclature

The following table summarizes the notation used in this work. In general, starred

quantities have dimension and unstarred ones have been nondimensionalized. Cer-

tain variables, namely the ones typically used to nondimensionalize the others, are

always dimensional, even though they are not denoted with the asterisk. These are

noted in the description.

Variable Description Location(s)

b, b∗ Extrapolation length §3.2

B Linearization parameter §3.2

c, c∗ Eigenvalue describing the time dependence §3.2

δr Rubbery region thickness §2.1

Dtr Translational diffusivity §3.2, §5.1

ε, ε∗ Slip coefficient §2.3

f Fractional recovery §2.1

fn Random forcing §5.2

F Drag force §2.1

G Linearization parameter §3.2

Continued on next page

102

Continued from previous page

Variable Description Location

G∗ Shear modulus §3.2

γ Ratio of true and nominal Weissenberg

numbers

§3.2

γ∗n Nominal shear rate §3.2

γ∗t True shear rate §3.2

ηp Polymer viscosity (dimensional) §3.2

ηs Solvent viscosity (dimensional) §5.1

H Linearization parameter §3.2

k Fluctuation orientation vector §4

k Magnitude of k §4

kx Perturbation wavenumber in the flow di-

rection

§3.2, §5.2

kz Perturbation wavenumber in the neutral

direction

§5.2

l Gap width (dimensional) §3.2, §5.1

L Map parameter §5.2

λ Polymer relaxation time (dimensional) §3.2,§5.1

λs Relaxation time for slip (dimensional) §1.3.1

µ Phan Thien - Tanner model parameter §3.2

n Polymer concentration §5.1

n Outward unit normal §5.2


103



N1 First normal stress difference §2.1

Q Dumbbell orientation vector §5.1

R Radius (dimensional) §2.1

R Linearization parameter §3.2

ρ Density (dimensional) §A.1

ρ Anisotropic drag slip model parameter §2.4

Re Reynolds number §1.3.1

s Equilibrium constant for the kinetic ex-

pression

§2.3

S Ratio of solvent viscosity to polymer vis-

cosity

§3.2

sR Recoverable shear §2.1

t, t∗ Time §1.3.1

T Temperature (dimensional) 2.4

τ , τ ∗ Polymer extra stress tensor §3.2, §5.1

τ Amplitude of the extra stress perturbation §3.2, §5.1

τ Extra stress perturbation §3.2, §5.1

u Velocity §3.2

u∗f Velocity of the free segments §2.3

us, u∗s Slip velocity §1.3.1

v Velocity §3.2


104



V Characteristic velocity (dimensional) §2.1

Wa Work of adhesion §2.1

We Weissenberg number §4

Wen Nominal Weissenberg number §3.2

Wet True Weissenberg number §3.2

X Structural parameter/Bonding fraction §2.3

y Scaled y coordinate §3.3

ζ , ζ∗ Friction coefficient §2.1

ζ, ζ∗ Friction tensor §2.4

105

Appendix A

Basic Melt Equations

A.1 Nondimensionalization

The equations to be nondimensionalized are the equation of motion and the consti-

tutive equation. These equations are

ρDv∗

Dt∗= ∇∗ · τ ∗ −∇∗P ∗, (71)

(1 +µ

G∗tr τ ∗)τ ∗ + λτ∗(1) = ηp

(∇∗v∗ + (∇∗v∗)T

). (72)

The following relations are used to nondimensionalize the variables:

v =v∗

γ∗nl, (73)

t = γ∗nt∗, (74)

τ =τ ∗

G∗, (75)

p =P ∗

G∗, (76)

∇ = l∇∗. (77)

106

where γ∗nl is the characteristic velocity, γ∗n is the applied shear rate at the wall, G∗

is the shear modulus and l is the gap width. When these definitions are substituted

into Eqs. 71 and 72, the result is

Dv

Dt=

G∗

ργ∗ 2n l2(∇ · τ −∇P ) , (78)

τ (1) +G∗

ηpγ∗n(1 + µtr τ )τ = ∇v + (∇v)T (79)

Recall that G∗ = ηp/λ. With this definition, the constant on the left hand side of

the equation of motion can be written as:

G

ργ∗ 2n l2=

(ηp

ργ∗ 2n l

)(1

γ∗nλ

). (80)

Since Wen = γ∗nλ and Re = ργ∗nl∗

ηp, the equations can finally be written in dimension-

less form as:

ReWenDv

Dt= (∇ · τ −∇p) , (81)

τ (1) +1

Wen(1 + µ tr τ )τ = ∇v + (∇v)T (82)

(83)

The equation of continuity for an incompressible fluid is simply

∇ · v = 0 (84)

For all of the calculations in this thesis, Re = 0.

A.2 Derivation of the General Stability Equation

107

Equations 81, 82, and 84 can be written in component form for the two dimensional

problem with Re = 0 as:

∂τxx∂t

+ u∂τxx∂x

+ v∂τxx∂y− 2(τxx + 1)

∂u

∂x− 2τyx

∂u

∂y+

1

Wenτxx(1 + µτxx + µτyy) = 0, (85a)

∂τyx∂t

+ u∂τyx∂x

+ v∂τyx∂y− τyy

∂u

∂y−

∂u

∂y− (τxx + 1)

∂v

∂x+

1

Wenτyx(1 + µτxx + µτyy) = 0, (85b)

∂τyy∂t

+ u∂τyy∂x

+ v∂τyy∂y− 2τyy

∂v

∂y− 2τyx

∂v

∂x− 2

∂v

∂y+

1

Wenτyy(1 + µτxx + µτyy) = 0, (85c)

∂τxx

∂x+

∂τyx

∂y−

∂p

∂x= 0, (85d)

∂τyx∂x

+∂τyy∂y−

∂p

∂y= 0, (85e)

∂u

∂x+

∂v

∂y= 0, (85f )

The next step is to linearize this system of equations. This is accomplished by

assuming that each variable is equal to its steady state value plus some small per-

turbation, then retaining only the terms that are linear in the perturbations. Each

variable has the form a = a + δa where the bar denotes the steady state value, the

tilde indicates the perturbation, and δ is a small parameter used to keep track of the

linear terms. Collecting the O(1) terms yields the system to be solved for the steady

state solution. The O(δ) system is the linear system for the perturbations. Note

that the steady state values of the y-component of velocity and the second normal

stress are identically zero, and there are no gradients of the base state variables in

108

the x-direction. Finally, the linearized system of equations is

∂τxx∂t

+ u∂τxx∂x

+ vdτxxdy− 2(τxx + 1)

∂u

∂x− 2τyx

∂u

∂y− 2τyx

∂u

∂y+

1

Wen(1 + 2µτxx)τxx +

µ

Wenτxxτyy = 0, (86a)

∂τyx∂t

+ u∂τyx∂x

+ vdτxxdy− τyy

∂u

∂y−

∂u

∂y− (τxx + 1)

∂v

∂x+

µ

Wenτyxτxx +

1

Wen(1 + µτxx)τyx +

µ

Wenτyxτyy = 0, (86b)

∂τyy∂t

+ u∂τyy∂x− 2τyx

∂v

∂x− 2

∂v

∂y+

1

We(1 + µτxx)τyy = 0, (86c)

∂τxx∂x

+∂τyx∂y−

∂p

∂x= 0, (86d)

∂τyx∂x

+∂τyy∂y−

∂p

∂y= 0, (86e)

∂u

∂x+

∂v

∂y= 0, (86f )

The last step is to introduce a form for the perturbations. For a normal mode

analysis, it is a assumed that each variable can be written as a(y)eikx(x−ct) + c.c.,

where a(y) is the amplitude and is a function only of y, kx is the wavenumber, and

c is the eigenvalue for the time dependence of the disturbance. After substituting

the perturbations into Eqs. 86a - 86f , the system of equations to be solved for the

109

perturbations is

(−ikxc + ikxu +1

Wen+

2µ

Wenτxx)τxx − 2ikx(τxx + 1)u− 2τyxu

′ − 2u′τyx +

µ

Wenτxxτyy = 0,(87a)

(−ikxc + ikxu +1

Wen+

µ

Wenτxx)τyx − ikxv − u′ − u′τyy +

µ

Wen(τxx + τyy) = 0,(87b)

(−ikxc + ikxu +1

Wen+

µ

Wenτxx)τyy − 2ikxτyxv − 2v′ = 0,(87c)

ikxτxx + τ ′yx − ikxp = 0,(87d)

ikxτyx + τ ′yy − p′ = 0,(87e)

ikxu + v′ = 0,(87f )

The ′ denotes differentiation with respect to y.

The system of equations defined by Eqs. 87a through 87f can be reduced to

a single equation for the stream function. Since the stream function perturbation

is assumed to have the same form as the other perturbations, reducing to a single

equation for the stream function is equivalent to reducing to a single equation for

v. The first step in the reduction is to combine Equations 87d and 87e to eliminate

pressure and get a single equation relating just the stress components:

ikxτ′xx + k2xτyx + τ ′′yx − ikxτ

′yy = 0. (88)

The three constitutive relations can then be solved for the stresses and substituted

into Eq. 88. After noting that u = ikxv′ (from the equation of continuity) and

110

doing some substantial rearrangement, the general stability equation is

[(Q2D2 − k2xQ

2 + 2(u′)2 − 2Qu′D)(D2 + 2ikxτyxD − k2x − k2xτxx)+

k2xQu′′(τxx + 1) + ikx(ikxQτ ′xx + 2Qτ ′′yx − 4τyxu′′)(QD − u′)− 3Qu′′ +

4u′u′′D + ikxQ2τ ′′′yx − 2Qu′′′D − 2ikxQτyxu

′′′ + 3ikxQτ ′yx(QD2 − u′′)−

ikxτ′yx(k

2xQ2 + 4u′QD − 2(u′)2)

]v + µM(τ , µ)v = 0, (89)

where Q = −c+ u− ikxWen

, D is the ordinary derivative with respect to y, andM is

a complicated differential operator. For µ = 0, and plane Couette flow (i.e. the base

state gradients of stress with respect to y and second order and higher derivatives

of the base state velocity are zero), this reduces to

(Q2D2 − k2xQ2 + 2(u′)2 − 2Qu′D)(D2 + 2ikxτyxD − k2x − k2xτxx)v, (90)

which is identical to the general equation derived by Gorodtsov and Leonov [27].

A.3 PTT Matrix Eigenvalue Problem

The generalized eigenvalue problem that is obtained when the PTT constitutive

equation is used is

111

L=

0iα

d dy

d2

dy2+α2

−iα

d dy

−2W

e nτ yxd2

dy2−

2iαWe n

(1+τ x

x)d dy

1+iαWe n

d dyψ

+2µ

τ xx

−2W

e nd2

dy2ψ

µτ x

x

−α2We n

(1+τ x

x)−We n

d2

dy2

µτ yx

1+iαWe n

d dyψ

+µτ x

x−We n

d2

dy2ψ

+µτ yx

−2α2We nτ yx+

2iαWe n

d dy

00

1+iαWe n

d dyψ

+µτ x

x

(91)

C=

00

00

01

00

00

10

00

01

(9

2)

112

Appendix B

3D Stability Operators

The three dimensional operators for the polymer solution model are, with ∇2 =

d2

dx2+ d2

dy2+ d2

dz2and v · ∇ = u d

dx+ v d

dy+ w d

dz,

CS =

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0 1 0

0 0 0 0 1 0 0 0 0 0 0

0 0 0 0 0 1 0 0 0 0 0

0 0 0 0 0 0 1 0 0 1 0

0 0 0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 0 0 1 1 0

0 0 0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 0 0 0

(93)

113

LS=

SWe∇2

00

d dx

d dy

0SWe∇2

00

d dx

00

SWe∇2

00

−2We(1−ξ)[ (τxx+n)d dx

+τ yxd dy+τ zxd dz

]We(1−ξ)( dτxx

dy+dndy

)0

We(1−ξ)(v·∇)

−(1−ξ)(∇2−1)

−2We(1−ξ)dudy

−We(1−ξ)[ τ yxd dx+

(τyy+n)d dy+τ zyd dz

]−We(1−ξ)[ (τxx+n)d dx

+τ yxd dy+τ zxd dz−dτyx

dy

]0

0We(1−ξ)(v·∇)

−(1−ξ)(∇2−1)

−We(1−ξ)[ τ zxd dx

+τ zyd dy+(τzz+n)d dz

]We(1−ξ)dτzx

dy

−We(1−ξ)[ (τ xx+n)d dx

+τ yxd dy+τ zxd dz

]0

−We(1−ξ)dwdy

0−2We(1−ξ)[ τ yxd dx+

(τyy+n)d dy+τ zyd dz−1 2

dτyy

dy

]0

00

0−We(1−ξ)[ τ zxd dx+τ zyd dy

+(τzz+n)d dz−dτzy

dy

]−We(1−ξ)[ τ yxd dx

+τ yyd dy+(τzy+n)d dz

]0

0

0We(1−ξ)( dτzz

dy+dndy

)−2We(1−ξ)[ τ zxd dx

+τ zyd dy+(τzz+n)d dz

]0

0

0We(1−ξ)dndy

0(1−ξ)d2

dx2

2(1−ξ)d2

dxdy

d dx

d dy

d dz

00

···

d dz

00

00

−d dx

0d dy

d dz

00

−d dy

d dx

0d dy

d dz

0−d dz

00

00

We(1−ξ)(v·∇)

−(1−ξ)∇2

0

0−We(1−ξ)d dy

00

−We(1−ξ)dudy

0

We(1−ξ)(v·∇)

−(1−ξ)(∇2−1)

0−We(1−ξ)dudy

00

0

0We(1−ξ)(v·∇)

−(1−ξ)(∇2+1)

00

We(1−ξ)(v·∇)

−(1−ξ)(∇2−1)

0

0−We(1−ξ)dwdy

We(1−ξ)(v·∇)−dvdy

−(1−ξ)(∇2−1)

0−We(1−ξ)dwdy

0

00

−2We(1−ξ)dwdy

We(1−ξ)(v·∇)

−(1−ξ)(∇2−1)

We(1−ξ)(v·∇)

−(1−ξ)∇2

0

2(1−ξ)d2

dxdz

(1−ξ)d2

dy2

2(1−ξ)d2

dzdy

(1−ξ)d2

dz2

We(1−ξ)(v·∇)

−(1−ξ)∇2

0

00

00

00

(94)

114

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