Shear Flow Instabilities
in
Visco elastic Fluids
Helen Jane Wilson
Clare College
September
A dissertation submitted to the University of Cambridge
for the degree of Do ctor of Philosophy
Preface
The work describ ed in this dissertation was carried out b etween Octob er
and September while the author was a research student in the
Department of Applied Mathematics and Theoretical Physics University
of Cambridge The dissertation is the result of my own work and includes
nothing which is the outcome of work in collab oration Except where explicit
reference is made to the work of others the contents of this dissertation are
b elieved to b e original No part of this dissertation has b een or is b eing
submitted for a degree diploma or other quali cation at any other University
It is a pleasure to record my thanks to my sup ervisor Dr John Rallison
for all his help No one could wish for a b etter sup ervisor I would also like
to thank Professor John Hinch for useful discussions and Jeremy Bradley
for pro ofreading
This work was funded by a research studentship from the Engineering
and Physical Sciences Research Council
To my parents
Contents
Introduction
Outline
Shear rheology of non Newtonian uids
Dilute p olymer solution Boger uid
Entangled p olymer melt
Constitutive equations theoretical mo dels
Governing equations
Newtonian uid
Oldroyd B mo del
Upp er Convected Maxwell mo del
Nonlinear dumbbell mo dels
Other rate equations
Retarded motion expansions nth order uid
Doi Edwards mo del
Mo dels with viscometric functions of
Summary
Observations of instabilities
CONTENTS
Interfacial instabilities in channel and pip e ows
Viscometric ows having curved streamlines
Instabilities in extensional ows
More complex ows
Extrudate distortions and fracture
Constitutive instability
Theoretical analyses of instability
Instabilities in inertialess parallel shear ows
Interfacial instabilities
Flows with curved streamlines
Stagnation p oint ow
Scop e of this dissertation
Stability of Channel Flows
Introduction
Geometry
Constitutive equations
Base solution
Linear stability
Temporal stability
Weakly nonlinear stability
Perturbation equations for channel ow
Alternative form for the equations
Existence of ro ots
Numerical metho d
CONTENTS
Co extrusion Instability
Introduction
Statement of the problem
Base state
Perturbation equations
The long wave limit k
The short wave limit k
The e ect of layer thickness on short waves
The e ect of p olymer concentration on short waves
The e ect of ow strength on short waves
Numerical results
Conclusions
Continuously Strati ed Fluid
Introduction
Geometry and constitutive mo del
Perturbation ow
Long wave asymptotics
Outer ow
Inner solution
Numerical results
App endix Long wave asymptotics
Outer region
Inner region
Matching
CONTENTS
Concentration Strati cation
Introduction
Base state
Stability problem for two uids
Governing equations
Long wave limit
Numerical results
Single uid with varying concentration
Sp eci c C pro le
Conclusions
App endix Details of the expansion for long waves
Channel Flows of a White Metzner Fluid
Introduction
Governing equations for a channel ow
Statement of the equations
A mathematical di culty
Long waves
Order calculation
Order k calculation
Long waves and low n
Short waves
Analytic scalings for short waves
Numerical results for short waves
Intermediate wavelengths numerical results
Scalings for highly shear thinning limit n
CONTENTS
Governing equation
Physical meaning of the reduced equation
Numerical results for small n
Conclusions
App endix Preliminary asymptotics for n
Outer
Wall layer
Matching
Criterion for Co extrusion Instability
Introduction
A mo del problem
White Metzner uid
Oldroyd B uid
Base state
White Metzner uid
Oldroyd B uid
Choice of parameters
White Metzner uid
Oldroyd B uid
Perturbation equations
White Metzner uid
Oldroyd B uid
Numerical results
Conclusions
CONTENTS
Conclusions
Chapter
Introduction
CHAPTER INTRODUCTION
Outline
Visco elastic liquids for example p olymeric melts and solutions have ow
prop erties that are in part viscous and in part elastic The presence of
elastic stresses can generate instabilities even in inertialess ow that do
not arise in Newtonian liquids These are often describ ed as purely elastic
instabilities The principal aim of this dissertation is to examine the class
of such instabilities that arise in shear ows and the mechanism resp onsible
for them
We start with a review of the principal non Newtonian features of uids
of interest section then consider how these may b e captured in a con
stitutive mo del section We then turn to a review of the exp erimental
literature on purely elastic instabilities section and the current theoret
ical understanding of these phenomena section An outline of the new
instabilities discussed in this thesis is given in section
Fuller descriptions of non Newtonian phenomena are given by Bird et al
and of constitutive mo dels by Larson and Bird Wiest
We select here the items of greatest concern for the understanding of shear
instabilities
Shear rheology of non Newtonian uids
In steady simple shear having a velocity vector u y it may b e
shown that any incompressible simple uid will have a stress tensor with
in general four nonzero comp onents
SHEAR RHEOLOGY OF NON NEWTONIAN FLUIDS
N
C B
C B
C B
A
N
As a result of the incompressibility the stress is undetermined to within
addition of an isotropic pressure term For a Newtonian uid N N
and In a visco elastic uid however these three quantities may
vary with In considering the di erent categories of exp erimentally relevant
uid b elow we de ne N and N
We note in passing that b ecause of the nonzero N the stress tensor for
a planar ow is not entirely two dimensional This may a ect the stability
of the ow to p erturbations in the third dimension For Newtonian uids
Squire s theorem states that the most unstable disturbance is always two
dimensional and hence stability calculations may b e restricted to the two
dimensional domain For a general non Newtonian uid this may not b e
true
Dilute p olymer solution Boger uid
For a dilute p olymer solution the rst normal stress di erence N is found
to b e p ositive and may b e large compared with Both and show
very little shear thinning at mo derate shear rates i e they are more or less
constant However at higher shear rates there is some thinning often with
The second normal stress di erence N is almost zero
Figure sketches the shear b ehaviour of typical dilute p olymeric uids
A basic way to start characterising visco elastic uids is by their zero
shear rate shear viscosity and their longest relaxation time which we
CHAPTER INTRODUCTION
Shear rate
Figure Generic b ehaviour of b oth the viscometric functions and against shear
1
rate for dilute solutions Both axes use a logarithmic scale There is a low shear rate
plateau followed by a region of weak shear thinning For a Boger uid the plateau would
extend across a very large range of shear rates
shall call This time can b e made large by using high molecular weight
p olymers or a very viscous solvent Dilute solutions in which b oth of these
metho ds are used to create a very long relaxation time are called Boger
uids They use as solvent a very low molecular weight melt of the same
p olymer as the solute These uids have a very long plateau of constant
and
Even here the picture is complicated using a Boger uid made of two
p olystyrenes MacDonald and Muller have found a long time relax
ation of the normal stresses which seems to b e an inherent prop erty of the
uid rather than an e ect of degradation instability etc They conclude
that the de nition of a single timescale is uncertain Indeed dynamic linear
visco elastic data indicate that these uids actually p ossess a sp ectrum of
relaxation times for example Ferry
SHEAR RHEOLOGY OF NON NEWTONIAN FLUIDS
Entangled p olymer melt
As for a dilute solution N is large enough to play an imp ortant r ole in
the dynamics of these uids However b oth the viscosity and the rst
normal stress co e cient can decrease by several orders of magnitude as
is increased Laun Figure shows schematically the b ehaviour of
b oth these quantities
The second normal stress co e cient is often signi cant and negative
as shown by several pap ers for example Keentok et al
Exp erimentally for mono disp erse entangled p olystyrene solutions Magda
Baek have shown that shear thinning also a ects the second normal
stress co e cient
Shear rate
Figure Sketch of the generic b ehaviour of b oth the viscometric functions and
1
against shear rate for entangled p olymeric melts Both axes use a logarithmic scale
Both quantities shear thin strongly but still exhibits a low shear rate plateau as it do es
for a dilute solution
CHAPTER INTRODUCTION
Constitutive equations some theoretical
mo dels
Governing equations
For an incompressible uid with velocity u any ow must satisfy the mass
conservation equation
r u
and any continuum ob eys the momentum equation
D u
r F
D t
where is the stress tensor is the density and F represents any b o dy
forces acting on the uid D D t t u r is the material derivative
The inertial term on the left hand side of this equation is often small for
visco elastic ows b ecause of the high viscosities involved In that case
r F
The stress then needs to b e de ned in terms of the ow history by means
Newtonian uid
For a Newtonian uid
pI E s
CONSTITUTIVE EQUATIONS THEORETICAL MODELS
where p is the pressure the viscosity of the uid I the identity matrix
s
and E is the symmetric part of the velocity gradient
fr u ru g E
Its derivation is based on the assumptions that the uid is isotropic and
that it resp onds instantaneously and linearly to the velocity gradient applied
to it
Equation b ecomes the Stokes equation
r p r u F
s
Since p only app ears in the form r p it is determined for prescrib ed velocity
rather than stress on the b oundaries only up to addition of an arbitrary
constant
For a p olymer solution in a Newtonian solvent the obvious extension is
to set
p
pI E
s
p
where is the elastic or p olymeric contribution to the stress tensor
Oldroyd B mo del
The Oldroyd B mo del may b e derived by considering a susp ension of
Ho okean dumbbells see gure that is dumbbells in which the spring
force F is directly prop ortional to the extension of the spring
F GR
1
We shall use the convention ru r u There is no consistent convention in the
ij i j
literature for the ordering of the subscripts i and j
CHAPTER INTRODUCTION
R t
Figure A dumbbell
The b eads which de ne the dumbbell ends move under the action of a
Stokes drag from the solvent Brownian motion and the spring force Because
of the sto chastic nature of Brownian motion the macroscopic prop erties
of the uid are derived from ensemble averages denoted by The
p olymeric stress is given by
p
RF G RR GA
Its evolution may b e expressed as
r
A A I
with I b eing the unit tensor and a relaxation time de ned as G
where is the drag co e cient a acting on an isolated b ead The cases
s
p
A I and G corresp ond to a Newtonian uid and the
case is the Upp er Convected Maxwell mo del section
s
The time derivative in equation is the upp er convected derivative
CONSTITUTIVE EQUATIONS THEORETICAL MODELS
de ned as
r
A A u r A r u A A ru
t
This derivative is co deformational with the line elements dl in the ow
so that
r
dl dl
It is the failure of the molecules to deform with uid elements that generates
elastic stress in the uid
The Oldroyd B mo del is in shear ows a quantitatively go o d mo del
for Boger uids nevertheless its simplicity means that in general it cannot
capture the full nature of a p olymeric uid For example it has only one
relaxation time whereas real uids have a relaxation sp ectrum do es not
shear thin and has a zero value of N More seriously in extensional ows it
can pro duce an unbounded extensional viscosity b ecause of the linear spring
b ehaviour of the dumbbells In a real uid the molecules b ecome fully
extended and the viscosity saturates
Upp er Convected Maxwell mo del
The Upp er Convected Maxwell uid UCM is the high concentration limit
of an Oldroyd B uid section ab ove As such it is used as a mo del
for p olymer melts
It is given by
p
pI
r
2
In particular we note that I E
CHAPTER INTRODUCTION
r
p
p
I
It has all the disadvantages of the Oldroyd B uid unbounded exten
sional viscosity single relaxation time zero N and constant shear viscosity
However b ecause it has one less parameter it is mathematically even more
simple than the Oldroyd B uid In some of the ows discussed in this dis
sertation we will see that the equations of motion are analytically tractable
for a Maxwell uid where they are not for Oldroyd B
Nonlinear dumbbell mo dels
The problem of in nite extensional viscosity for nite extension rate in the
Oldroyd B uid is caused by unbounded extension of the dumbbells Thus
a sensible mo di cation of the mo del is to use a nonlinear force law and limit
the maximum extension of the springs
F GRf R f R
R L
where R R R This leads to the set of Finitely Extensible Nonlinear
Elastic FENE mo dels Because of the nonlinearity in F a closed evolution
equation for RR and hence for the stresses is not available a closure
approximation is needed
FENE P
If the average length of all dumbbells is taken to de ne R the force law f R
is replaced by f R giving a pre averaging approximation
F GRf R
CONSTITUTIVE EQUATIONS THEORETICAL MODELS
which leads Peterlin to the FENE P mo del
p
RF G RR f R
and therefore
pI E Gf R A
s
r
A I A
f R f R
where in this case
R tr A
FENE P improves the b ehaviour of the mo del in extension and gives a shear
thinning viscosity It is therefore less go o d at describing shear ows of Boger
uids
FENE CR
In this mo del the extension b ehaviour remains of FENE type but the evo
lution of the quantity A is altered from the FENE P equation to give
a constant shear viscosity It b ecomes Chilcott Rallison
r
A I A
f R
Other rate equations
Several other ad hoc mo di cations of the Oldroyd B mo del have b een pro
p osed We add an extra term to equation to pro duce any or all of
CHAPTER INTRODUCTION
shear thinning nonzero N or a b ounded extensional viscosity The Oldroyd
B mo del may b e expressed as
r
p
p
E
Without violating any of the simple uid assumptions we may add an extra
term
r
p p
p
f E E
Some of the p opular choices for f are
Johnson Segalman
p p p
f E E E
Phan Thien Tanner PTT
p p p p p
f E fE E g Y tr
where
x
Y x e
Giesekus
This is a generalisation of the UCM uid using an anisotropic drag
force so we set and use
s
p p p
f E
where
The detailed expressions for N and N are given in
CONSTITUTIVE EQUATIONS THEORETICAL MODELS
Retarded motion expansions nth order uid
If the motion is weak and slow enough or corresp ondingly the longest
relaxation time is relatively short any simple uid may b e expanded as a
p erturbation to the Newtonian limit Rivlin Ericksen The small
quantity used in the expansion is either the Weissenberg number W often
W i or the Deborah number D e
W U L
D e T
These are b oth nondimensionalisations of the relaxation time W uses a
typical velocity gradient or shear rate while D e for unsteady ows uses
a typical timescale of the ow If these are b oth small then the retarded
motion expansion will b e valid and the form of the result is indep endent of
the sp eci c constitutive equation used
These equations are known as nth order uids according to how many
p owers of the small quantity are retained In particular the second order
uid is given Coleman Noll by
r
pI E E E E
where and are constants
These equations are useful where a phenomenon arises from the e ect of
very weak elasticity
3
In Larson incorrectly gives this as
r
pI E E E E
2 1
2
which leads to N in simple shear
1 1
CHAPTER INTRODUCTION
Doi Edwards mo del
The Doi Edwards mo del Doi Edwards is a sp eci c example
of a K BKZ uid Kaye Bernstein Kearsley Zapas for which
Z
t
X I I t t C t t X I I t t C t t dt
I I
C t t is the strain tensor accumulated b etween past time t and current
time t and I and I are its two invariants I tr C and I tr C
X is a general damping function which de nes a sp eci c mo del within the
class
The Doi Edwards mo del describ es melts in which individual molecules
move by reptating along their length This is given by
X I I s I I m s
with
X
exp s p m s G
p p
p
p
p o dd
and an approximate form for Currie by
ln J J I I
This mo del predicts a very high level of shear thinning even higher than
that observed in real p olymer melts
Mo dels with viscometric functions of
An ad hoc metho d of extending many mo dels to melt like b ehaviour to
include substantial shear thinning for example is to let some of the material
CONSTITUTIVE EQUATIONS THEORETICAL MODELS
p
E E j These functions may then parameters b ecome functions of j
b e determined by simple shear viscometry or assigned simple mathematical
forms
Convenient mathematical forms which go some way to approximating real
material quantities include the p ower law function
n
and the Carreau mo del
n
a
Generalised Newtonian uid
The Newtonian uid is given by equation To form the generalised
Newtonian uid the viscosity is allowed to dep end on The uid ex
s
hibits no normal stress e ects and is therefore not very useful for instability
calculations
CEF uid
The second order uid is given by
r
pI E E E E
If all three material prop erties and are allowed to b e functions of
this b ecomes the CEF equation Criminale Ericksen Filb ey This
is the simplest mo del to allow exact corresp ondence with all the observed
viscometric functions
CHAPTER INTRODUCTION
The Reiner Rivlin equation Reiner Rivlin is a sp eci c ex
ample of a CEF uid
pI E E E
White Metzner uid
If we p ermit the time constant in the Upp er Convected Maxwell uid
section to b e a function of we obtain the White Metzner mo del
White Metzner
r
p
p
I
This mo del can show normal stress e ects and shear thinning while retaining
the physical structure of a dumbbell mo del
Summary
The brief description ab ove indicates that a wide class of constitutive equa
tions is available and b etter descriptions generally necessitate more dimen
sionless groups The simplest equations that correctly describ e shear ows
of solutions and melts are the Oldroyd B and White Metzner mo dels and
we shall fo cus on these employing more sophisticated versions only when necessary
OBSERVATIONS OF INSTABILITIES
Observations of instabilities in ows of
non Newtonian uids
We o er here a brief and incomplete review of the range of instabilities seen in
the pro cessing of non Newtonian uids Fuller coverage of these phenomena
is given by Shaqfeh Larson and earlier by Tanner and Petrie
Denn
We b egin with interfacial instabilities section one of which is
also studied in chapter We then consider inertialess instabilities in ows
with curved streamlines section instabilities in extensional ows sec
tion and then more complex ows section Finally in sec
tion we discuss some instabilities observed in extrusion pro cesses
where there is neither an interface nor curvature of the streamlines
The channel ows studied in the bulk of this dissertation are relevant
to extrusion problems in that the ow inside an extrusion die has an e ect
on the ow downstream of the die exit and on the quality of the extruded
pro duct in addition since shear is a feature of almost all ows in con ned
geometries any instability of a shear ow has implications for p olymer pro
cessing in general
Interfacial instabilities in channel and pip e ows
Many practical pro cesses co extrusion multi layer coating lubricated pip e
lining involve multi layer ows of visco elastic liquids b ecause of the desir
able prop erties of many multi layer p olymeric solids A multi layer material
may have enhanced strength or a combination of the prop erties of its com
CHAPTER INTRODUCTION
Figure Schematic of three layer co extrusion with ow from left to right
p onents for example in fo o d wraps the coating may provide adhesion and
the core a moisture barrier Figure shows a schematic of three layer
extrusion in which the pro duct would b e co oled at the far right of the pic
ture to obtain a solid lm Usually a uniform interface is desired in the end
pro duct so instabilities are to b e avoided if p ossible
The detrimental e ect of interfacial instability is illustrated in gure
This shows how the optical quality of the nal pro duct is impaired by an
instability during pro cessing
Figure A photograph of the e ect of interfacial instability on the optical quality of a
three layer co extruded lm The lm on the left is the pro duct of a ow with an unstable
interface while that on the right was pro duced by a stable ow Mavridis Shro
OBSERVATIONS OF INSTABILITIES
Many exp eriments on the stability of multi layer ows in straight pip es
and channels have b een carried out by Yu Sparrow Lee White
and Khomami coworkers
Viscometric ows having curved streamlines
In Shaqfeh published a review of purely elastic instabilities Many
of the bulk instabilities i e not surface instabilities in this section are cov
ered by the review article and have a common mechanism the coupling of
a rst normal stress di erence N with curvature of the base ow stream
lines The mechanism therefore di ers from the co extrusion instability in
section
Taylor Couette
Inertial e ects at high Reynolds numbers cause the Taylor Couette instability
Taylor The uid forms toroidal roll cells with a regular spacing up
the column of uid shown in gure The mechanism for this instability
is centrifugal
First observed exp erimentally by Giesekus the purely elastic Taylor
Couette instability also app ears in the form of a cellular secondary ow It is
visually the same as the inertial instability but can o ccur at zero Reynolds
number The instability is driven by N which forms a ho op stress when
the streamlines are closed which has a destabilising e ect comparable to
but opp osite in sign to the centrifugal force in the inertial case It was pre
dicted theoretically for an Oldroyd B uid by Larson et al con rming
earlier Boger uid exp eriments by the same group in Muller et al It is
CHAPTER INTRODUCTION
interesting to note that Oldroyd B shows go o d quantitative agreement with
exp eriments on p olymer solutions for this shear dominated instability
Di erent mo des of elastic instability were
observed by Beris Avgousti and
the range of transitions was investigated by
Baumert Muller using Boger uids
Further theoretical work on the di erent pa
rameters a ecting the basic instability has
b een carried out by Shaqfeh et al and
by Larson et al In the fully nonlinear
exp erimental range a new phenomenon called
the Solitary Vortex Pair a pair of asymmetric
rolls isolated from any others was found and
a p ossible mechanism explained by Groisman
Steinberg in
If the cylinders are not quite concentric
the instability is somewhat mo di ed as ex
Figure Inertial Taylor rolls in
plained exp erimentally and theoretically by
a Couette device
Dris Shaqfeh and numerically using
a UCM uid by Chawda Avgousti
In Dean or Taylor Dean ow there is another di erent elastic instability
rst rep orted by Jo o Shaqfeh The same authors also studied the
e ect of inertia in and carried out a survey of exp erimental observations
coupled with calculations for an Oldroyd B uid in
OBSERVATIONS OF INSTABILITIES
Cone and plate Plate and plate
Both of these geometries consist of a rotating upp er surface cone or plate
and a xed lower surface plate b etween which the uid is sheared
Surface mo des
One elastically driven instability is distortion of the meniscus It takes one
of two forms either an axisymmetric smo oth distortion which indents the
middle of the meniscus known as edge fracture or an irregular distor
tion whose e ect lo oks like surface vortices referred to by Larson simply as
fracture
Figure The smo oth distortion known as edge fracture in a cone and plate geometry
Photograph taken from Hutton
The edge fracture e ect was observed rst by Hutton Tanner
Keentok analysed it in cone and plate ow and showed that it is crit
ically dep endent on the second normal stress di erence N This was con
rmed exp erimentally by Lee et al
The fracture e ect rst observed by Kulicke et al o ccurs for
entangled solutions and melts It is not known how deep into the uid the
CHAPTER INTRODUCTION
surface disturbance acts An exp erimental indicator here seems to b e the
b ehaviour of measured values of N when plotted against there is
a sudden change in slop e at the onset of fracture This is in common with
gross melt fracture an extrusion instability discussed in section
Bulk mo de
Using a Boger uid in a cone and plate device Jackson et al observed
a dramatic rise in b oth and N well after the b eginning of the motion
Following further exp eriments with Boger uids in b oth cone and plate and
plate and plate geometries a secondary ow was found by Magda Lar
son in indicating that Jackson s observations were caused by the
app earance of an instability in the ow This deduction was not immedi
ately accepted as recently as anomalous shear thickening e ects were
b eing rep orted from measurements made only in the cone and plate device
p ointed out by Tam et al
Exp eriments by McKinley et al thoroughly summarise the b ehaviour
Figure The irregular distortion known as fracture in a cone and plate geometry
Observations from Kulicke et al
OBSERVATIONS OF INSTABILITIES
of a p olyisobutylene polybutene solution in b oth these geometries nding
the conditions for onset of instability in terms of a critical Deb orah number
A more recent exp erimental discovery Byars et al is a second
critical radius which corresp onds to a higher critical Deb orah number in the
parallel plate geometry outside which the ow of p olyisobutylene restabilises
The earliest theoretical prediction of an elastic instability in this geometry
was by Phan Thien who predicted a spiral form for the secondary ow
However the exp erimentally observed secondary ow is not of this form but
consists rather of ring vortices at irregular radial spacings
The form of the instability is predicted theoretically by Oztekin et al
in the parallel plate geometry using the Oldroyd B uid and in the
cone and plate using a multi mode Giesekus mo del They compare
these predictions with exp eriments with p olyisobutylene p olybutene and
tetradecane uids The same group then extends this analysis to a systematic
study of the e ect of various parameters The cone and plate ow of
an Oldroyd B uid is analysed by Olagunju
Instabilities in extensional ows
The purest steady extensional ow in general use is that imp osed in a l
ament stretching rheometer Tirtaatmadja Sridhar In this device
two plates with uid b etween them are moved rapidly apart b oth the evo
lution of the radius of the resulting lament at its midp oint and the force
exerted on the endplates can b e measured However in industrial pro cessing
several other extensional ows are commonly used and it is here that the
rst instabilities were observed
CHAPTER INTRODUCTION
Instabilities in melt spinning
In a melt spinning or bre spinning pro cess a uid the melt is extruded
relatively slowly and then some distance down the pro cessing line taken up
rapidly onto a wheel Because the velocity at the wheel is much larger than
that at the exit of the extrusion die the bre undergo es extensional ow and
b ecomes much thinner during pro cessing A schematic representation of the
ow is given in gure the asp ect ratio of draw length to bre diameter
is many times larger in a real pro cess
Figure Schematic of a spin line ow
Draw resonance
Draw resonance is a p erio dic variation in bre diameter It o ccurs only for
constant take up sp eed at the sp o ol and not for constant force Pearson
Matovitch It is the result of a variation in cross sectional area of the
bre at the take up sp o ol causing a change in the spinline tension which in
turn enhances the upstream disturbance
For a Newtonian uid the instability was rst describ ed by Christensen
and Miller The rst classical one dimensional analysis using an
OBSERVATIONS OF INSTABILITIES
isothermal Newtonian uid and no gravity inertia surface tension or air
drag was p erformed indep endently by Kase et al in and Matovitch
Pearson in They b oth proved bre spinning to b e unstable for
draw ratios greater than
Using melts with constant shear viscosity the onset of draw resonance
o ccurs as for the Newtonian case at a draw ratio of just over indep en
dent of whether they thicken or thin under strain rst observed in by
Donnely Weinberger Cruz Saenz et al and Ishihara Kase
This last pap er along with Lamb also demonstrated an upp er
draw ratio ab ove which at least for short bres the ow restabilises
In general shear thinning enhances the instability lowering the critical
draw ratio by up to an order of magnitude Zeichner A linear stability
analysis was carried out by Fisher Denn for a shear thinning general
isation of the UCM uid and it is in qualitative agreement with isothermal
exp eriments
In recent years the eld of bre spinning has extended to include multi
layer lm casting in which layers are rst extruded and then stretched in a
broadly two dimensional geometry The application of linear stability anal
ysis to the corresp onding core annular ow has b een p erformed for di erent
uid combinations UCM skin with Newtonian core Lee and PTT
skin with Newtonian core Ji et al For b oth of these the visco elas
tic skin acts to stabilise the ow and delays the onset of draw resonance
Lee Park have p erformed one set of exp eriments in this eld using
linear low density p olyethylene in the core and low density p olyethylene for
the skin Their results are in qualitative agreement with isothermal linear
CHAPTER INTRODUCTION
stability calculations
Necking
Even for a constant force take up of the spun bre when draw resonance
do es not o ccur small indentations in the bre surface can grow as they are
convected down the bre This necking instability o ccurs for Newtonian u
ids but is usually inhibited in visco elastic uids b ecause of strain hardening
However if a melt is extension thinning as for example predicted by the
Doi Edwards mo del then samples in extension will neck and fail b efore
extensional viscosity measurements can b e made This makes veri cation
of mo dels for such uids very di cult The neck mechanism for failure of
spinnability has b een observed by Chen et al and Takaki Bogue
Cohesive failure
In melt spinning a bre may break with a mechanism quite distinct from
necking When the stresses in the uid exceed the cohesive strength of the
material it will fracture This mechanism was made clear by Ziabicki
Takserman Krozer using oils of dif
ferent molecular weights
The critical stresses observed for this phenomenon Vinogradov are
comparable to those observed at the onset of gross melt fracture in extrusion
section
Instabilities in lm blowing
The lm blowing pro cess consists of extrusion of an annulus of p olymeric
OBSERVATIONS OF INSTABILITIES
uid followed by its in ation an extensional ow by air injected along the
axis of the extrusion die The lm is then co oled and taken up onto a roll in
a manner similar to bre spinning
If insu cient air is injected the lm will not in ate and this pro cess
b ecomes little more than annular bre spinning As such it is of course
prone to draw resonance in exactly the same way as a normal bre
With su cient injected air to form a lm bubble three main types of
instability are observed all shown by Minoshima White The rst
called bubble instability by Larson is a p erio dic axisymmetric variation
in the bubble radius This was rst rep orted by Han Park There
is also a helical instability and nally a solidi cation instability in which
without changing the bubble shap e the solidi cation front moves back and
forth on the bubble surface All three of these phenomena are stabilised by
extension thickening
The rst linear stability analysis of this ow for a visco elastic uid UCM
was by Cain Denn who found instability to b oth blow up and col
lapse of the bubble More recently Andrianarahinjaka Micheau have
p erformed a numerical linear stability analysis and found a strong dep en
dence on the rheology of the uid and on the co oling pro cess
More complex ows
Most ow geometries involve b oth shear and extension As a consequence a
combination of mechanisms may arise to generate instabilities
CHAPTER INTRODUCTION
Instabilities in stagnation p oint ows
The archetypal stagnation p oint ow is a simple steady uniaxial straining
ow u x y z Exp erimentally this is approximated using an
opp osed jets device and its two dimensional equivalent using a cross slot
device illustrated in gure
Figure The cross slot device for a stagnation p oint ow
In exp eriments using concentrated solutions and melts a birefringent
strand of highly extended material forms along the extension axis At higher
ow rates this is replaced by a cylinder or pip e as observed by Odell et al
Both the birefringent line and the pip e have also b een predicted theoret
ically for a FENE mo del by Harlen et al
At still higher ow rates a phenomenon called are is observed in the
exp eriments in which the birefringent region is disturb ed and is spread
throughout the geometry Odell et al have suggested a mechanism involv
ing a molecular entanglement structure However Harris et al using
a linear lo cked dumbbell mo del nd a hydrodynamic instability to sinuous
waves on the birefringent strand A third p ossible explanation is given by the
OBSERVATIONS OF INSTABILITIES
presence of curved streamlines which can lead to the elastic shear instability
describ ed in section
Axisymmetric contraction ows
For b oth Newtonian and non Newtonian uids the basic stable steady ow
through a contraction where it exists consists of an extensional region near
the centre of the ow with a toroidal secondary vortex ow in the salient
corner Mo att as shown in gure
Figure Schematic of contraction ow with recirculating vortex
The ow is sub ject to several di erent mo des of instability rst observed
by Cable Boger and summarised by McKinley et al and
Ko elling Prud homme The choice of mo de is sensitive to contraction
ratio and uid rheology The centre region of the uid may swirl Bagley
Birks den Otter Ballenger White Oyanagi or pulse
and a lip vortex may form and b ecome quasip erio dic
CHAPTER INTRODUCTION
Numerically studies have b een made using the Giesekus mo del Hulsen
van der Zanden and the Oldroyd B mo del Yoo Na with
mo derate success at predicting the exp erimental observations of instability
This ow has curved streamlines as well as extension and so the mechanisms
of section may b e partly resp onsible for the observed instabilities
Planar contractions
In a planar geometry there are two corner vortices exp erimentally it is
observed that they may grow for a shear thinning solution Evans Walter
but not for Boger uids At high ow rates these vortices show
instability to three dimensional motions and the resultant ow is usually
time dep endent Giesekus
Wakes
The wake b ehind a falling sphere in a shear thinning uid may exhibit time
oscillatory ow Bisgaard in the region just b ehind the sphere where
high extensional stresses o ccur Chilcott Rallison and others How
ever for constant viscosity uids Boger UCM Chilcott Rallison PTT
there is no exp erimental or theoretical evidence of this instability Arigo
et al
McKinley et al have observed an instability which cannot b e an
e ect of shear thinning in the wake of a circular cylinder con ned in channel
ow of a visco elastic constant viscosity uid
More recently b oth Chmielewski Jayamaran and Khomami
Moreno have observed similar instabilities at Deb orah numbers of order
OBSERVATIONS OF INSTABILITIES
in exp eriments with p erio dic arrays of cylinders
Extrudate distortions and fracture
One of the most striking examples of ow instability is provided by evidence
of distortion of extrudates Many comp eting mechanisms are available many
complementary descriptions have b een provided
In Kolnaar Keller divide extrusion instabilities into three broad
classes surface distortions sharkskin p erio dic distortions with wavelengths
comparable with the extrudate diameter wavy instability and gross shap e
distortions gross melt fracture and show that each e ect has a di erent
stress criticality
The division into these three r egimes and particularly the separation of
wavy distortions from gross melt fracture has only taken place relatively
recently in the literature with these last two b eing group ed together with
misleading terminology as fracture
There are other unstable r egimes p ossible Vinogradov et al and
many more recent exp erimenters have found a global stick slip ow in which
the extrudate alternates b etween smo oth and distorted states while the up
stream pressure uctuates with the same frequency Compressibility of the
melt is a crucial ingredient here
It should b e noted also that all visco elastic uids exhibit the stable dis
tortion of die swell in extrusion in which the elastic recovery of the uid
after the straining and shearing ows inside the die causes the radius to swell
just after exit from the die This is not a true instability and will not b e
discussed further
CHAPTER INTRODUCTION
Sharkskin
The mildest instability that o ccurs in the extrusion of p olymer melts is a
surface distortion At its weakest it is little more than loss of gloss Rama
murthy and in its stronger form known as sharkskin it app ears as
a series of scratches across the extrudate mainly p erp endicular to the ow
direction Piau et al show clear photographs of the phenomenon an
example is shown in gure that distinguish it from other extrudate
distortions
Figure Exp erimental observation of the sharkskin instability reprinted from Piau et
al with p ermission from Elsevier Science The scale shown is mm long
OBSERVATIONS OF INSTABILITIES
It was rst rep orted by Clegg in and was shown by Benbow
Lamb and later by Moynihan et al to b e initiated at the exit of the
die This was con rmed more recently by Kolnaar Keller Vinogradov
et al demonstrated that sharkskin is asso ciated with high lo cal stresses
at the die exit and more recently Ramamurthy Moynihan et al
Hatzikiriakos Dealy Piau et al it has b een shown that in general
the die surface material or the presence of lubricants on the die wall has a
strong e ect on the app earance of sharkskin
Pomar et al considered solutions of linear low density p olyethylene
and o ctadecane they found that the onset of sharkskin was at constant ratio
of wall shear stress to plateau mo dulus However for general low density
p olyethylenes Venet Vergnes showed that wall shear stress is not the
unique determinant of sharkskin e ects and that chain branching and strain
hardening may inhibit the e ect
The current consensus is that sharkskin is caused by some kind of adhesive
fracture and or phase separation at the die exit due to the high lo cal stresses
there rather than any e ect o ccurring over the whole die length The current
literature prop oses several di erent mechanisms
Phase separation
Busse suggests that the p olymer will segregate under stress into
low and high molecular weight comp onents close to the die exit Chen
Joseph further showed how this could cause a short wave in
stability which could propagate out of the die and onto the extrudate
surface This p ossibility is discussed further in chapter Joseph
predicted the wave shap e of sharkskin using theory from the lubrication
CHAPTER INTRODUCTION
of heavy oils by water in pip elining
Wall slip
Hatzikiriakos simulates extrusion ow using slip conditions and
matches his own exp erimental data In the same vein Shore et al
use a Maxwell mo del with a rst order stick slip b oundary
condition to investigate stability On studying their system numerically
they nd that it to o can predict sharkskin Exp erimentally Tzoganakis
et al who quantify sharkskin by means of the fractal dimension
of the extrudate surface show that slip rst app ears in the die b efore
the app earance of sharkskin
For slipp ery dies Piau et al nd that there are two separate slip
regimes at low and high stresses with a transition zone in b etween In
b oth of these the extrudate shows a crack free surface This quanti es
the parameter r egimes where slip is exp ected for p olybutadiene and
may b e an indication that a lack of slip is needed for sharkskin
Molecular dynamics
Sharkskin is a small scale instability so it is p ossible that either of the
ab ove mechanisms will have its origin at the molecular scale However
there are some pap ers whose approach is purely molecular and whose
predictions cannot simply b e expressed in macroscopic terms
Stewart coworkers have prop osed a molecular based mo del
for wall slip which correctly predicts the wall b ehaviour at wall shear
rates b elow that critical for gross melt fracture
Wang et al consider sharkskin formation in linear low density
OBSERVATIONS OF INSTABILITIES
p olyethylene and using a mo del for molecular interactions with the
wall and with each other can predict sharkskin Their mechanism
is a combination of interfacial slip and cohesive failure due to chain
disentanglement
Cracking
Tremblay nds numerical evidence for a cracking mechanism for
sharkskin For linear low density p olyethylene El Kissi Piau
show that the exp eriments do not give conclusive evidence of a slip
mechanism but rather indicate a mechanism of cracking under high
tensile stresses In subsequent con rmation El Kissi et al they
watch the app earance of these cracks by using a p olymer with a long
relaxation time
Some degree of dep endence on die length has b een found by Moynihan
et al but this can b e explained by the assumption Kurtz that
the growth of the instability dep ends to some extent on the state of stress
of the material arriving at the die exit
Wavy distortions
The rst rep ort of any extrusion instability was by Nason in who
rep orted a helical form of extrudate at Reynolds numbers around
It has since b een observed at much lower Reynolds numbers of order
Tordella so inertia do es not seem to b e a critical ingredient An
example is shown in gure
CHAPTER INTRODUCTION
Helical extrudates can only form in cylindrical dies their slit die coun
terpart is regular large amplitude ribbing Atwoo d
This instability has b een largely neglected in the literature and either
dismissed as practically Newtonian or classed along with gross melt fracture
see b elow which often follows it However Kolnaar Keller have
shown that wavy motions originate in the interior of the die which indicates
a completely di erent mechanism from that for melt fracture Pomar et
al using solutions of linear low density p olyethylene and o ctadecane
Figure Helical instability observed by Piau et al reprinted with p ermission
from Elsevier Science The scale shown is mm long
OBSERVATIONS OF INSTABILITIES
found that the onset of a wavy instability where it existed at all was at
constant wall shear stress This supp orts the conclusion that the instability
takes place entirely inside the die
Sombatsompop Woo d using natural rubb er observed evidence
of steady slip at the walls and a helical disturbance within the capillary
Georgiou Cro chet suggest that visco elasticity is not vital for this
instability Simulating a compressible Newtonian uid with an arbitrary
nonlinear slip mo del for the wall b oundary condition they nd self excited
oscillations in the velocity eld and a wavy interface
However an alternative explanation is an elastically driven hydrodynamic
instability of the steady ow through the die This p ossibility is investigated
in depth in this dissertation
Gross melt fracture
Gross melt fracture which is a chaotic disturbance of the extrudate with
diameter variations of or more was rst observed and noted as such by
Benbow Lamb in However evidence of gross melt fracture is
available as early as Tordella where it o ccurred after the onset
of the wavy instability ab ove and was called spurt gure
Benbow Lamb also showed that there is unsteady ow inside the die
during the pro cess In fact Vinogradov et al the time dep endent ow
extends even into the inlet region of the die Becker et al in
rep ort high levels of noise in the pressure in the upstream reservoir during
pro cessing
Gross melt fracture cannot have an inertial mechanism b ecause it has
CHAPTER INTRODUCTION
b een observed for extremely low Reynolds numbers Tordella Shear
thinning is also not a critical factor even if it plays a part as illustrated by
Cogswell et al they observed melt fracture in a uid with constant shear
viscosity Similarly thermal heating is not critical Lupton Regester
showed that in exp erimental conditions shear heating will have an e ect
of at most degrees In addition Clegg had earlier failed to nd such an
instability for a Newtonian uid with viscosity very sensitive to temp erature
There is a strong di erence b etween the b ehaviour of broadly linear
and branched p olymers of which one manifestation is the frequency at on
set of the instability The linear p olymers show roughly the same frequency
throughout the ow development whereas the branched melts show a low
Figure Extrudates pro duced by Tordella at increasingly high ow rates the
rst shows a wavy instability the second gross melt fracture and the third is completely fragmented
OBSERVATIONS OF INSTABILITIES
frequency oscillation at onset which increases as the amplitude of the dis
tortion grows den Otter For branched p olymers most evidence is
consistent with a disturbance nucleated at the die entrance and gradually
decaying within the die Bagley et al give some early evidence
For linear p olymers on the other hand early evidence showed that the
distortion increased or at least did not decay with increasing die length
and thus the distortion could have b een nucleated inside the die den Otter
Ballenger et al However Kolnaar Keller show de nitively
that melt fracture originates at the entry of the die for the ow of linear
p olyethylene
The mechanism assumed to cause fracture from the die entry Benbow
Lamb is as follows an upstream oscillatory ow near the die en
trance could p ermit incorp oration of material with a di erent ow history
thus causing large scale extrudate defects b ecause of di erent elastic recovery
Such an upstream ow may o ccur as a result of the contraction ow insta
bilities of section This ts with observations for example Cogswell
Lamb that streamlining the die entrance can reduce the instability For
linear p olymers if this is the case the stability inside the die must still b e
marginal since the distortion do es not decrease with increasing die length
Marginally stable ow inside the die may o ccur for two distinct reasons
wall slip or constitutive instability
Wall slip seems to b e very dep endent on choice of uid For low density
p olyethylene it is generally agreed that slip do es not o ccur den Otter
whereas for high density p olyethylene the same pap ers rep ort some
observation of slip and stick slip is a p ossibility Particular evidence in
CHAPTER INTRODUCTION
favour of slip Vinogradov Ivanova is given by the dep endence of
the apparent wall shear rate on die radius
Wall slip is often mo delled by replacing the traditional no slip condi
tion at the wall with a slip law p ermitting a slip velocity which is some
functional of strain history at the wall see for example Pearson Petrie
Shore et al use a Maxwell mo del with a rst order stick slip
b oundary condition and nd with a numerical study that they can predict
melt fracture like structures Adewale et al makle a theoretical study
of the e ect of a stick slip mixed b oundary condition and nd a resulting
shear stress singularity They claim that this singularity is the cause of melt
fracture
The existence of slip is supp orted by exp erimental observations Benbow
Lamb Atwoo d Schowalter El Kissi Piau that during
gross melt fracture the ow inside the die lo oks like plug ow which may
b e intermittent On the other hand there is a general observation Rama
murthy Piau et al El Kissi Piau that the material used
for the die do es not have a strong e ect on gross melt fracture This suggests
that in fact the plug ow is caused not by slip at the wall but by e ective
slip within the material itself with the formation of a very thin region of
high shear next to the wall This could b e caused by a constitutive instability
describ ed in section
Ko opmans Molenaar raise the p ossibility of hydrodynamic insta
bility where a nonlinear constitutive equation is used with a prop osal to use
the energy balance equation to choose b etween di erent p ossible solutions of
the governing equations This metho d compromises b etween the alternative
OBSERVATIONS OF INSTABILITIES
mechanisms of slip at the wall and constitutive instability
Constitutive instability
If there is a non monotonicity in the stress shear rate curve for the uid
b eing extruded then a shear stress could b e attained that is a non unique
function of shear rate and the ow would undergo a sudden increase in ow
rate In practice this is very di cult to distinguish from slip This idea
seems to have b een put forward rst by Huseby
There are some uids which under simple shear show a non monotone
shear stress shear rate curve Examples include micellar solutions in which
surfactant molecules have aggregated to form wormlike structures Sp en
ley et al These materials can undergo a shear banding motion in
which two di erent shear rates are present at the same shear stress in a sim
ple shearing ow This was rst observed exp erimentally in and is
demonstrated in Decrupp e et al and Makhlou et al
Constitutive equations that show this non monotonicity include the ex
tended Doi Edwards mo del section which shear thins very rapidly at
high shear rates the Johnson Segalman mo del and the Giesekus mo del with
a solvent viscosity All have the generic form of stress curve in which the
shear stress passes through a maximum and a minimum as the shear rate
increases gure This b ehaviour can also b e derived from an extended
Doi Edwards mo del in which once the shear rate is larger than the slowest
Rouse relaxation rate the uid acts as if it is unentangled
McLeish Ball used this extended Doi Edwards mo del to predict
some critical features of spurt in p olybutadiene and p olyisoprene Vinogradov
CHAPTER INTRODUCTION
Shear rate
Figure Typical plot of shear stress against shear rate for a non monotonic constitutive
equation
et al
In simple shear exp eriments with shear banding uids two regions each
of one well de ned shear rate are typically observed with a steady interface
b etween them Espa nol et al have p erformed numerical simulations of
Couette ow of a Johnson Segalman uid and found the uid can mo del
the exp eriments including transient motion after start up with accuracy
showing a stable interface b etween the two regions
However Renardy uses the Johnson Segalman mo del and nds that
in Couette ow of a uid in which two regions use di erent shear rates at
the same stress level there is a short wave linear instability based around
the interface b etween the regions
Theoretical analyses of instability
Instabilities in inertialess parallel shear ows
There is a wealth of literature available on elastic p erturbations to highly
inertial ows Elasticity is here a small e ect and for the purp oses of this
THEORETICAL ANALYSES OF INSTABILITY
review we fo cus on ows having small inertia
For all such shear ows Preziosi Rionero claim to prove stability
by energy metho ds but there is a fundamental error in the pap er restrict
ing its class of p erturbations to a set which do not satisfy the momentum
equation the parent pap er by Dunwoo dy Joseph is correct but not
applicable in the limit of low Reynolds number
Squire s theorem states that in a two dimensional ow of a Newto
nian uid the most unstable disturbance is always two dimensional and so
three dimensional p erturbations need not b e considered section It also
holds for the Upp er Convected Maxwell uid and for the second order
uid if
Plane Poiseuille ow
The e ect of elasticity on the known inertial instability in plane Poiseuille
ow is destabilising However Ho Denn showed that Poiseuille ow of
an Upp er Convected Maxwell uid with no inertia is linearly stable to sinuous
or snakelike mo des They also explained the numerical accuracy problems
in previous pap ers which had led to claims that the ow was unstable The
corresp onding calculation for varicose sausage like mo des was p erformed by
Lee Finlayson and this study also found stability Finally Ghisellini
used an energy argument to prove the stability analytically
Similar results were found for the shear thinning Giesekus uid by Lim
Schowalter
However there is some hint of instability for a Giesekus uid Schleiniger
Weinacht studied Poiseuille ow and found multiple solutions along
CHAPTER INTRODUCTION
with some p ossible selection criteria These are prop osed as a mechanism
for spurt phenomena in extrusion Using the K BKZ mo del Aarts van de
Ven nd similar results with three steady states two of which are stable
It is debatable whether these two pap ers in each of which the base state is
not obvious from the start are showing ow or constitutive instabilities
Couette ow
Goro dtsov Leonov found all the eigenvalues of the stability equation
for inertialess Couette ow of the UCM uid and hence showed its stabil
ity Renardy Renardy extended this numerically to low Reynolds
numbers and maintained stability
Interfacial instabilities
The simplest multi layer shear ow and the rst to b e thoroughly anal
ysed is two layer strati ed Couette ow It has b een shown Yih that
viscosity strati cation of Newtonian liquids can cause a long wave inertial
instability for any nonzero Reynolds number However if the less viscous
layer is thin instability o ccurs at a nite Reynolds number Ho op er
Renardy In fact this instability if there is no surface tension may
exist for all wavenumbers Renardy
In lubricated pip elining a viscous core uid typically oil is lubricated by
a thin annulus of less viscous uid water Preziosi et al Strati cation
of viscosity can cause Yih s inertial instability and density strati cation also
leads to long wave instabilities The ow has b een investigated in detail by
Chen coworkers
THEORETICAL ANALYSES OF INSTABILITY
Moving away from Newtonian uids the e ects of shear thinning have
b een considered in di erent geometries by Waters Keely Kho
mami and Wong Jeng Yield uids have b een studied by
Pinarbasi Liakopoulos
Visco elastic uids can show a purely elastic instability at their interface
even in the limit of zero Reynolds number with matched viscosities The
linear stability of Oldroyd B and UCM uids has b een investigated by Li
Waters Keeley Renardy Chen coworkers
and contained in chapter of this thesis Wilson Rallison weakly
nonlinear analysis has b een undertaken by Renardy Li and Waters
Keeley used an incorrect stress b oundary condition at the interface Chen
and hence did not nd this instability
The elastic instability is driven by a jump in N at the interface and exists
in b oth the long and short wave limits In the long wave limit stability
dep ends on the volume fraction o ccupied by the more elastic comp onent
but the short wave limit dep ends only on the two materials and the shear
rate at the interface The mechanism of the long wave instability has b een
explained by Hinch et al and is summarised in chapter
Using more than two di erent uids but with the same physical mech
anisms causing the observed e ects theory has b een carried out by Su
Khomami for p ower law and Oldroyd uids and Le Meur
for PTT uids
Vertical ow in which the driving force is gravity rather than an applied
pressure gradient and will therefore di er from one uid to another where
there is a density di erence has b een investigated for visco elastic uids by
CHAPTER INTRODUCTION
Sang and Kazachenko
Free surface ows
As in the case of viscosity strati ed multi layer ows free surface ows are
susceptible to an inertial instability at any p ositive Reynolds number how
ever small Benjamin Yih Hsieh Because of the stabilising
in uence of surface tension long waves are the most unstable
Asymptotic long wave stability analysis has b een carried out on this ow
for the second order uid Gupta and the Oldroyd B uid linear anal
ysis Lai weakly nonlinear Kang Chen The linear analysis
for these two uids yields precisely the same result and shows that with any
nonzero elasticity the ow is unstable even at zero Reynolds number There
is therefore a purely elastic mechanism for instability Gupta
Using waves which are not asymptotically long or equivalently a lm
which is not thin Shaqfeh et al have investigated the Oldroyd B prob
lem numerically and showed that the growth rates are very small Because
this is a convective instability moving downstream with the uid it can b e
convected out of the ow b efore it is big enough to b e observed and indeed
it has never b een seen in exp eriment They also showed that elasticity has a
stabilising in uence on shorter waves
Flows with curved streamlines
A new dimensionless group to describ e the curved streamline instabilities
of section was introduced by Pakdel McKinley and explained
with more examples by McKinley et al In all cases the dominant
SCOPE OF THIS DISSERTATION
mechanism is a coupling of the curvature of the velocity eld with the rst
normal stress di erence N Some work sp eci c to the cone and plate and
parallel plate geometries has already b een discussed with the relevant exp er
imental observations in section
Flow in a curved channel also has curved streamlines and an elastic
instability was discovered for ow of an Oldroyd B uid by Jo o Shaqfeh
The unstable mo de is stationary whereas the Taylor Couette mo des are
overstable so this mechanism is distinct from that which causes instability
in the Taylor Couette ow Nonetheless the instability is fundamentally
dep endent on the coupling of streamline curvature with the normal stress
N
Stagnation p oint ow
The o ccurrence of a birefringent pip e in stagnation p oint ows and its pre
dicted instability has already b een discussed in section In a cross
slot ow Oztekin et al predict a purely elastic instability using the
Oldroyd B mo del for which the unstable mo de is fully three dimensional
unlike the pip es and are we have already seen The mechanism is due
to curved streamlines and normal stresses in common with many of the
instabilities of section
Scop e of this dissertation
As the brief account ab ove indicates the literature on instabilities in elastic
liquids ows is extensive and rapidly developing We have chosen to fo cus on
CHAPTER INTRODUCTION
exp erimentally observed instabilities in interfacial ows ows with curved
streamlines and extrusion in which inertial forces are small
In this dissertation we rst investigate theoretically an interfacial insta
bility which o ccurs in co extrusion chapter We then consider whether the
same mechanism can generate instability in straight streamline ows of a
single uid whose material prop erties are advected with the ow chapters
and We shall refer to this as a co extrusion instability even though only
one uid may b e present the fundamental mechanism is the same as that
for the co extrusion case This instability may provide a mechanism for the
helical distortions of extrudates as discussed in section
In chapter we consider the channel ow of a shear thinning uid and
nd evidence for another instability where the degree of thinning in the shear
viscosity is high The mechanism in this case is fundamentally di erent and
we refer to it as the thinning instability
By considering di erent constitutive mo dels with similar physical prop er
ties we determine a mathematical criterion for the co extrusion instability
dep ending on prop erties of the mo del itself not just on the velocity and
stress pro les of the base ow In chapter we give an example of two ows
with identical ow pro les and stresses but whose stability prop erties are
di erent as a result of di erences in formulation of the uid mo dels The
uids in question are generalisations of the two types of uid already used a
uid with advected prop erties like that of chapters and and a uid whose
viscometric prop erties are functions of the shear rate like that of chapter
The principal conclusions from the dissertation and suggestions for future
work that result from it are given in chapter
Chapter
Stability of Channel Flows
General Formulation
CHAPTER STABILITY OF CHANNEL FLOWS
Introduction
In the course of this dissertation we will use several di erent constitutive
equations all of rate type and all suitable for shear based ows
To avoid duplication a general constitutive equation is prop osed here
which contains all the subsequent equations as sp ecial cases The stability
problem is p osed in general terms when each individual uid is studied
later the sp eci c parameter values will b e substituted into the p erturbation
equations of section or section
Geometry
The basic geometry is a planar channel ow sketched in gure It is
convenient to scale lengths by the half width of the channel L velocities by
that on the centreline of the channel U and stresses by U L where
is a typical viscosity
Constitutive equations
We consider the following nondimensional constitutive equation
r u
r
C
pI E A W
CONSTITUTIVE EQUATIONS
y
U
y
y
x
y
Figure Flow geometry
r
A I A
p
E E
r
A here is the upp er convected derivative as de ned in section
There are four parameters W C and representing resp ectively a
relaxation time function relaxation time concentration of p olymer and the
solvent viscosity We then have several p ossible choices
a The Oldroyd B equation section is given by taking
constant W
b The UCM equation is C constant W
We may also de ne multicomponent Oldroyd B uids in which either
C or W is not a constant but a material function i e a function of the
Lagrangian variable x t that lab els each material p oint Because
is a material prop erty
D
D t
CHAPTER STABILITY OF CHANNEL FLOWS
In this case we will have the additional equation D C D t or
D W D t In investigating the stability of a steady planar channel
ow this description may b e simpli ed by identifying each material
p oint by its p osition y in the absence of any p erturbation
For either of these uids we x W indep endent of and except
for a multicomponent UCM uid
c The White Metzner uid section is C and
any function In order to b e de nite and to have an analytic base
state solution from which to start we limit ourselves initially to a
n
simple p ower law form of shear thinning W Note that
the limiting case n returns us to the UCM uid
Base solution
By symmetry we consider only the upp er half channel y in which
we exp ect U y to b e negative The nondimensionalisation xes the driving
pressure gradient P Denoting cross channel di erentiation that is di er
entiation with resp ect to y by a prime U we have the b oundary conditions
U U and U The base state quantities b ecome
U U y
jU j U
BASE SOLUTION
U U
A
A
U
P P C W P x
P P x C U W P y
A
P y P P x
and nally
C W U P y
For the given geometry and b oundary conditions we nd in particular
that if W then
P
R
C d
so for the Oldroyd B uid
P C and U y
and for the p ower law form of the White Metzner uid
n
n
n n
P and U y
n
The UCM uid is the combination of b oth these limits W n
and C We note that the limits commute to give P either
from equation or equation
The typical velocity pro le U y is shown on the diagram in g
ure
CHAPTER STABILITY OF CHANNEL FLOWS
Linear stability
We use the metho d of normal mo des to nd the temp oral stability of a given
ow There is a nonlinear system of equations of motion and b oundary
conditions for which one simple solution the given ow is known
N U
This simple solution has b een shown for the slit geometry and a selection of
constitutive equations in sections and
We imp ose an in nitesimal p erturbation having magnitude on this
base solution and linearise ab out the base state
N U N U L x y t O
U x y t
so that at leading order in
L x y t
U x y t
where L is a linear op erator on Because our basic solution U U y
U
dep ends only on one variable y we have
L x y t L y
U x y t U x y t
Boundary conditions p ermitting Fourier theory allows us to transform
all p ossible solutions in the x direction along the channel and in time
Z Z
ik x i t
u k y x y t e dxdt
with the inversion
Z Z
ik x i t
u k y e dk d x y t
LINEAR STABILITY
Now equation is equivalent to
Z Z
ik x i t
L y u k y e dk d
U x y t
in which we have used the linearity of L to change the orders of op erations
U
We deduce that
ik x i t
L y u k y e
U x y t
Now we simplify using
ik x i t ik x i t
u k y e ik ue
x
ik x i t ik x i t
u k y e i ue
t
and equation b ecomes an ordinary di erential equation ODE for u
as a function of y with k and as parameters
L ik d dy i y u y
U
In the channel geometry there are two zero b oundary conditions on U
at each wall slip velocity and p enetration velocity Now b ecause these are
b oth dep endent only on y they may b e expressed as and
for the two velocity comp onents of Referring back to equation
immediately gives u u for the velocity comp onents of u This
gives us a total of four zero b oundary conditions on u For a fourth order
di erential op erator L this system will b e overdetermined for sp eci ed k
U
and and therefore nontrivial solutions only exist for certain pairs k
CHAPTER STABILITY OF CHANNEL FLOWS
Temporal stability
There are two standard ways to pro ceed with this eigenvalue problem tem
p oral and spatial stability
Spatial stability analysis is the equivalent of imp osing a disturbance up
stream at a xed frequency and watching its evolution as it moves down
stream to see whether it grows This is achieved by xing a real value of
the frequency and solving for complex k If I m k is negative then the
exp onential term of equation will grow in space and the ow is unsta
ble In temp oral stability analysis the ow is given a disturbance of xed
wavenumber real k and its growth in time is found from the complex value
of
The spatial metho d dep ends on a disturbance which continues through
time whereas for temp oral stability an instantaneous disturbance is needed
but throughout the whole channel In reality the most likely disturbances
are lo calised in b oth space and time so the choice of metho d is somewhat
arbitrary
Here we use temp oral analysis the more common metho d in the existing
literature which is usually easier so the eigenvalue system is expressed as
k This expression is the dispersion relation and using it we sp ecify
real k to form an eigenvalue problem for complex If the imaginary part
of is p ositive for any k then the base ow is unstable
Weakly nonlinear stability
The next step in the pro cess of calculating stability would b e weakly nonlinear
PERTURBATION EQUATIONS FOR CHANNEL FLOW
theory in which the next term is included in the Taylor expansion given at
the b eginning of section
N U N U L N O
U U
Given a solution to the linear problem with eigenvalue information
ab out its nonlinear evolution may b e deduced
N U N O
U
where N is a quadratic op erator This can give us an evolution equation
U
for the amplitude of and near a bifurcation p oint marginally stable can
yield information ab out stable solutions close to our unstable base state
However a weakly nonlinear study is di cult unless the linear problem
is soluble analytically In this dissertation we limit ourselves to the linear
stability problem This will give conditions for instability but we will not b e
able to predict the form of disturbance seen in exp eriments
Perturbation equations for channel ow
We denote p erturbation quantities by either lower case letters A a
W etc or subscripts
Conservation of mass r u b ecomes
ik u v
and we use a streamfunction y such that
u and v ik
CHAPTER STABILITY OF CHANNEL FLOWS
to satisfy this automatically
The p erturbation momentum equation b ecomes
ik
ik
The p erturbation evolution equations give
C c C
p ik a A A
W W W
c C C
a A A k
W W W
c C C
a p ik
W W W
i ik U a
A ik A A A ik U a
i ik U a ik A A k U a A
i ik U a ik A k
In addition b ecause C and W are material quantities
D C c D t D W D t
PERTURBATION EQUATIONS FOR CHANNEL FLOW
It is convenient to de ne the streamline displacement so that a streamline
moves to
y exp ik x i t
where is the Lagrangian co ordinate describing the streamline Because the
streamline is a material surface D D t and to linear order
y y exp ik x i t
which gives
j W j
W W
0 0
k
c C
W
i ik U ik
The system is a fourth order di erential equation for
and so we need four b oundary conditions
The no slip no p enetration b oundary condition still holds for the p ertur
bation ow as it did for the base ow and gives two b oundary conditions
at y
CHAPTER STABILITY OF CHANNEL FLOWS
Figure Sinuous p erturbation left and varicose right
As we are considering only the upp er half channel the remaining b ound
ary conditions are to b e placed on the centreline y If the p erturbation
is sinuous the x velocity at the centre is unchanged under the transforma
tion y y and is therefore zero so must b e an even function of y
Alternatively for a varicose p erturbation there must b e no y velocity on the
centreline of the channel so is o dd Figure shows the nature of sinuous
and varicose p erturbations We have therefore
Sinuous s
Varicose v
Because all these b oundary conditions are zero the condition that we are
lo oking for a nontrivial solution provides the over determination mentioned
in section The eigenvalue problem k is then de ned in general by
equations
Alternative form for the equations
A mathematically unfortunate and as we shall see later physically inappro
priate feature of the equations in section arises in considering cases where
ALTERNATIVE FORM FOR THE EQUATIONS
C or W is a material function so that C or W is nonzero For a natural
and imp ortant limiting case in which C or W is actually a discontinuous
function the equations ab ove give rise to pro ducts of generalised functions
which are awkward to deal with
Motivated by the physical understanding of the way in which the insta
bility op erates chapter we are led to rewrite the equations
adv
We divide the stresses stretches and pressure into two parts
dyn adv dyn adv dyn adv
a a a p p p where x is the part of quantity X
owing to advection of C or W by the streamline displacement alone
adv
x c X C X W
The advection contributions are then given as
adv
p c W C W
adv
a
adv
a c WU C U
adv
a c W U C W U
adv adv
adv
c W U C C C U
CHAPTER STABILITY OF CHANNEL FLOWS
The remaining dynamic quantities are then driven by these advected p er
turbations for which the governing equations are known The momentum
dyn
equations now b ecome omitting the sup erscript
ik ik P y fC W C g
ik
Note the app earance of a b o dy force on the right hand side of equation
arising from the advection of the base state solutions The constitutive equa
tions are now
C
p ik a
W
C
a f P y C g k
W
C
p ik a
W
W P y
i ik U a ik
W C
A f P y C g A ik U a
WP
a ik i ik U
W C
f P y C g A k U a
a ik A k i ik U
W
EXISTENCE OF ROOTS
and the kinematic condition is
i ik U ik
Equations prove to b e easier to manipulate than those of
section b ecause of the limited number of o ccurrences of derivatives of C
and W Taken together with the unchanged b oundary conditions
they de ne k
We have not made any alteration to the equations in the case in which
b oth C and W are constants but is a function of This is b ecause the
p erturbation to
k j
0
do es not dep end on the streamline displacement but rather on the p er
turb ed ow itself Our simpli cation was based on the observation that part
of the p erturbation was due to the advection of base state quantities by the
streamline displacement Where this advection do es not o ccur there is no
such simpli cation
Existence of ro ots
The linear equations we have de ned for the p erturbation may b e combined
to give a single fourth order di erential equation for Where the equations
are regular a small change in k will cause only a small change in and in
Therefore for most of parameter space a ro ot which has b een found for
some real k may b e analytically continued to higher and lower values of k
CHAPTER STABILITY OF CHANNEL FLOWS
The exceptions to this principle arise at singularities of the system When
the co e cient of the highest derivative is zero ro ots may app ear or disapp ear
For instance in the Oldroyd B uid as describ ed in section the highest
derivative is multiplied by the term i ik U W so there is a continuous
sp ectrum a line of stable ro ots at k U i W setting this term equal
to zero The eigenmo des corresp onding to this sp ectrum are a set of delta
functions across the channel
f k U y i W y y g
We are not interested here in the continuous sp ectrum as such but in its
e ect on the discrete sp ectrum Discrete ro ots with eigenfunctions which
are regular functions may disapp ear while moving in k space by merging
into the continuous sp ectrum
Other instances of singular equations may o ccur if k U or if n for
a White Metzner uid
Numerical metho d
It is generally necessary to solve these equations numerically The metho d
used follows Ho Denn
For each prescrib ed k an estimated value of is chosen based on asymp
totic limits and parameter continuation Using this guess we have a fourth
order di erential equation for with four b oundary conditions
Let us supp ose we are lo oking for a sinuous eigenmo de Then the b ound
NUMERICAL METHOD
ary conditions are
Since the system of equations is linear any solution is sp eci ed by the values
of and its derivatives at the centre line
Therefore any solution to condition must b e a linear combination of
the two functions and whose ngerprints are
and
If is the true value then there exists some linear combination of and
for which the two wall b oundary conditions are satis ed
where j j denotes the determinant
The numerical metho d consists of integrating the two basis functions
and numerically by a fourth order adaptive Runge Kutta Merson
metho d provided by the Numerical Algorithms Group NAG from to
CHAPTER STABILITY OF CHANNEL FLOWS
across the channel This nds the determinant ab ove and then the Newton
Raphson metho d is used to nd a zero of this determinant as a function of
The integration routine also involves renormalisation of the two basis
vectors at several stations across the channel If there is one vector which is
growing exp onentially as it moves across the channel and the other decays
or grows more slowly then small numerical errors will cause b oth vectors
to tilt in their vector space towards the growing mo de and to grow with
it For large values of the exp onent which o ccur for high k or large W this
parallelisation of the eigenfunctions can cause the app earance of spurious
ro ots Therefore p erio dically the vectors are orthonormalised by a Gram
Schmidt pro cedure
a a jaj
new
b b a b a
int new new
b b jb j
new int int
P
p
where a b a b and jaj a a Even with this pro cedure in place
i i
there are some regions of parameter space where numerical solution is not an
option For example very close to the singular limit of the White Metzner
equations n the problem is extremely sti and tiny errors grow out of
control as the integration progresses In regions such as this precautions have
to b e taken against spurious results caused by numerical errors It is sensible
to change numerical parameters such as the number of renormalisations the
size of the maximum error allowed from the integration routine or the small
value at which the determinant is deemed to b e zero and check that the
ro ots do not change Ro ots of the analytic problem should b e insensitive to
NUMERICAL METHOD
these numerical parameters which distinguishes them from solutions to the
numerical problem which are not true ro ots of the original analytic problem
The co de is checked against published results in sp eci c cases Oldroyd
B and UCM and against asymptotic limits To nd a single value of
where the parameters are all of mo derate size takes three to four seconds on
a machine with Sp ecfp However if the parameter values are extreme
and the problem is nearly singular for example if n in the system of
chapter a single function evaluation can take around ve minutes
CHAPTER STABILITY OF CHANNEL FLOWS
Chapter
Co extrusion Instability of
Fluids having Matched
Viscosities but Discontinuous
Normal Stresses
CHAPTER COEXTRUSION INSTABILITY
Introduction
We consider in this chapter the co extrusion of two elastic liquids in a channel
Our principal aim is to identify the physical pro cesses which generate and
inhibit instability of the interface b etween the uids The bulk of the work
in this chapter has b een published in the Journal of Non Newtonian Fluid
Mechanics
When there is a jump in viscosity and the Reynolds number is nonzero
an instability is present even for Newtonian liquids Yih We therefore
isolate the central non Newtonian asp ect by considering a case where viscos
ity varies continuously across the interface and only normal stress jumps
This leads us to use the constant viscosity Oldroyd B mo del Additionally
we restrict attention here to the simplest case where inertia is negligible
We also neglect surface tension at the interface b etween the two uids but
section contains a discussion of the imp ortance of surface tension
For long wavelength disturbances Hinch et al provide a mechanism
for and calculation of the growth of disturbances resulting from normal
stress This mechanism op erates only when the coating is more elastic than
the core and has zero growth rate as the wavelength tends to in nity
In the short wave limit Chen Joseph found a new instability for
UCM uids whose growth rate is nite in contrast to the long wave mo de
In order to understand the mechanisms of instability we extend these
analyses to nd the in uence of solvent viscosity absent for the UCM uid
and of wavelength where this is mo derate In later chapters we will consider
these mechanisms as p ossible destabilisers of the ow of a single continuously
strati ed uid
INTRODUCTION
We shall nd in the absence of surface tension that
interfaces b etween di erent Oldroyd B uids are unstable to short wave
disturbances except at high concentration and high elasticity
the growth rate b ecomes indep endent of wavelength in this limit
the symmetry of the disturbance whether varicose or sinuous whether
planar or axisymmetric is irrelevant to the mechanism at work
The central conclusion from our analysis is therefore that surface tension
must b e su ciently large if elastic interface disturbances are to b e damp ed
We examine the imp ortance of surface tension e ects in section
Related theoretical work on this phenomenon was p erformed by Su and
Khomami in who studied analytically and numerically the stability of
two layer die ow Renardy in who tackled the short wave
b ehaviour of two layer Couette ow and Chen also in who
found a short wave instability in core annular ow More recently Laure et
al have studied two and three layer Poiseuille ow of Oldroyd B uids
with long to mo derate wavelength p erturbations incorp orating the e ects of
surface tension viscosity strati cation density strati cation and inertia
We formulate the problem for arbitrary interface disturbances in sec
tion which provides a numerical problem against which our asymptotics
can later b e tested The asymptotic limit of long waves is explained in
section Within the short wave limit asymptotic limits of dilute and
concentrated solutions and high and low relative elasticity are identi ed in
section Conclusions are given in section
CHAPTER COEXTRUSION INSTABILITY
Statement of the problem
The basic geometry is a planar channel ow sketched in gure with
di erent Oldroyd B uids o ccupying the regions jy j jy j resp ec
tively
y
y
y
x
y
y
Figure Flow geometry
We supp ose that b oth uids have the same constant shear viscosity but
di erent normal stress co e cients
Inertia is neglected as is surface tension at the interface b etween the two
uids The nondimensional groups that remain are then
the relative width of the inner uid
C the ratio of the elastic zero shear viscosity contribution to the solvent
contribution the concentration of p olymer
W the Weissenberg number the ratio of uid relaxation time to a typical
ow time
We require that the concentration parameter C should b e matched in the
two uids so that each has the same shear viscosity C The two uids
STATEMENT OF THE PROBLEM
may have di erent Weissenberg numbers We call these W and W If
W W there is e ectively only one uid present
Base state
We use a standard Oldroyd B equation within each uid which in the no
tation of chapter is given by W C W Thus the base
state b ecomes Poiseuille ow
U y
W y W y
A
A
W y
P P C W C x
P C x C W y C y
A
C y P C x
Because we are using the same value of C within each uid but di erent
values of W has a discontinuity of magnitude cy W at each inter
face resulting from the quadratic growth in normal stress with shear rate for
Oldroyd B uids This jump is p ermissible only b ecause the interfaces are
parallel to the x axis
The fundamental mechanism of the elastic instability as shown in
is that the tilting of this interface exp oses the jump in and this drives a
p erturbation ow which may in its turn cause the p erturbation to grow
CHAPTER COEXTRUSION INSTABILITY
Perturbation equations
The p erturbation equations within each uid are
ik
ik
C
p ik a
W
C
a k
W
C
p ik a
W
a ik A A A ik y a i ik U
W
a ik A A k y a i ik U
W
i ik U a ik A k
W
We have b oundary conditions at the wall and centreline of the channel
Sinuous s
Varicose v
STATEMENT OF THE PROBLEM
At the interface y exp ik x i t we require continuity of U u
and of the traction n These conditions b ecome continuity of
ik and
Substituting the base state quantities into the equations of section
and denoting di erentiation with resp ect to y by D we obtain
C C
D k D k a ik D a a
W W
a ik W y ik W y D i ik y
W
W y D y a
i ik y a ik W k W y D y a
W
i ik y a k W y ik D
W
Equation here is the vorticity equation and arise
from the evolution of A
The b oundary conditions are
Sinuous D D s
Varicose D v
D
C
D k a ik C W
W
CHAPTER COEXTRUSION INSTABILITY
C C
D D k D a ik a a
W W
where the square brackets denote a jump across y
We have nally a kinematic b oundary condition stating that the inter
face is a material surface
i ik ik
As it stands the problem ab ove is not amenable to analytic solution
b ecause of the explicit y dep endence of the co e cients in equations
to So in order to investigate the b ehaviour of k in the various
regions of parameter space we rst lo ok at the extreme values and identify
physical mechanisms
The long wave limit k !
The long wave limit k was rst studied in this geometry by Chen
Hinch et al found the leading order b ehaviour for general constant
viscosity non Newtonian uids For the Oldroyd B uid in particular the
problem can b e p osed as a regular asymptotic expansion in k We anticipate
that the frequency of the disturbance will have the form
k u i k
where u is the velocity of the interface and thus the Deb orah num
b er asso ciated with the p erturbation is prop ortional to k and so is small
In other words the p erturbation elastic stresses resp ond quasi statically to
THE LONG WAVE LIMIT k
the ow and give rise to a Newtonian stress having shear viscosity C
within each uid
Because the disturbance wavelength is much longer that the channel
width the leading order p erturbation ow is in the x direction the pres
sure across the channel is uniform and so the ow pro le is linear for a
sinuous mo de and quadratic if the mo de is varicose
The velocity must b e continuous across y and the tangential stress
balance across the interface gives
+
C u ik C W
For a sinuous mo de sketches of sinuous and varicose p erturbations are
shown in gure on page the pressure gradient must b e an o dd function
and is thus zero across the channel We then obtain
y y
u y
C
y y
and mass conservation gives the streamfunction at the interface as
Z
f
s
u dy
C
where
f
s
For a varicose mo de the p erturbation pressure gradient along the channel
must b e such as to make the p erturbation volume ux along the channel
CHAPTER COEXTRUSION INSTABILITY
vanish After some algebra we thus obtain
y y
u y
C
y y y
and
f
v
C
where
f
v
Since our eigenvalue condition is
k k
we deduce
C
k i W W f k O k
C
The leading order term is the convection rate at the undisturb ed interface
which indicates that the dynamics is based at the interface The leading order
growth rate of the instability is
C
W W f k
C
Since f for all this gives instability to sinuous mo des if
s
and only if W W i e if the outer uid is the more elastic relaxes more
slowly Varicose mo des are unstable where W W f i e where
v
p p
or W W and either W W and
THE LONG WAVE LIMIT k
Figure Representation of the mechanism of the long wave sinuous interfacial instabil
ity The diagram is xed in the frame of reference for which the disturbance is stationary
so the walls are moving from right to left
Figure Representation of the mechanism of the long wave varicose interfacial instabil
ity Stable mo des are on the left unstable on the right The diagram is xed in the frame
of reference for which the disturbance is stationary so the walls are moving from right to
left
The mechanism of the instability is as follows illustrated in gures
and
Let us supp ose that W W i e the more elastic uid is on the outside
The tilt in the interface caused by the imp osed p erturbation exp oses the
jump in In order to balance this stress jump the uid resp onds by
owing in the directions shown by the arrows
CHAPTER COEXTRUSION INSTABILITY
Incompressibility then requires that the ow conserve mass which it do es
by recirculating as shown in dotted lines In the sinuous case gure
there is one eddy across the whole channel which enhances the streamline
p erturbation In the varicose case gure the symmetry leads to one
eddy in each half of the channel If the inner uid o ccupies a large enough
p
fraction of the cross section this eddy will o ccur within the
inner uid thus destabilising the interface otherwise the eddy is close to the
wall and the interface is stable
If W W then all these p erturbation ows and therefore the stability
of the base state are reversed
Note that the growth rate of the instability tends to zero like k so the
very long wave limit k is neutrally stable for all Weissenberg numbers
The short wave limit k ! 1
This limit was rst considered by Chen Joseph
The Oldroyd B uid has no intrinsic lengthscale Thus in the short wave
limit the wavelength k b ecomes the only lengthscale de ning the ow At
leading order as k the p erturbation is con ned within O k of the
interface So the walls of the channel the p erturbation of the other interface
and the variation in base shear rate with y have no e ect The results there
fore apply to any interface without surface tension there is no distinction
here b etween sinuous or varicose planar or axisymmetric disturbances all of
which have the same growth rate
If we move to a frame of reference travelling with the interface i e at
THE SHORT WAVE LIMIT k
sp eed move the level y to the interface and rescale all lengths
with k we obtain at leading order the equations for a p erturbation to
unbounded simple shear ow of two uids having a constant shear rate
gure
C C
D D a i D a a
W W
a i W D W D a i i y
W
i i y a W D a
W
i i y a W iD
W
The following functions must b e continuous at the interface y
D
C
D a iC W
W
C C
and D D D a i a a
W W
and the kinematic condition reduces to
while the p erturbation decays away from the interface
y as y
CHAPTER COEXTRUSION INSTABILITY
We note that k has disapp eared from the problem and thus this set of
equations gives instability at a growth rate that is constant as k The
symmetry of the unbounded system means that exchanging W and W is
equivalent to reversing the direction of shear which reverses the convective
part of but do es not change the growth rate
LL
L
L
L
L
L
L
L
L
L
Figure Unbounded Couette ow of two uids
Note that neither nor any of the dimensionless uid parameters C W
and W has b een rescaled in this reformulation here is the same as in
section except for the subtraction of a term k inherent in the
change of frame of reference
The e ect of layer thickness on short waves
Thin inner uid layer
The driving term for the instability derives from the jump in the base state
normal stress i e the nal term in equation ik C W or equiv
alently ik C W where is the shear rate at the interface For a
very narrow inner uid layer this shear rate is very small so any destabilising
e ect is small also
THE SHORT WAVE LIMIT k
We take k to b e an order quantity i e we are able to consider waves
having length k One consequence of this is that we may not neglect
the e ect of the presence of the centreline so we consider a ow in which
only the outer uid is unbounded
The eigenfunction is then an even function of y within the inner uid i e
cosh y y sinh y and a decaying exp onential in the outer uid
The eigenvalue at leading order convecting with the interface is
k
C
i W W k e O
C
The growth rate of the instability is given by the imaginary part of the
eigenvalue
k
C
W W k e
C
which is p ositive indicating instability if the inner uid is more elastic
W W Where is negative the ow is stable to p erturbations of this
kind in this limit
In the unstable case has a maximum in k at k gure Note
that very short waves k do not have the fastest growth rate rather it
is at k
Thin outer uid
In this limit the shear rate remains O and so the equations within the
thin uid are no simpler than those in the bulk of the ow The leading order
system of equations still cannot b e solved analytically
CHAPTER COEXTRUSION INSTABILITY
I m
Wavenumber k
Figure Growth rate k exp k plotted against wavenumber k for xed small
1
The function has a maximum at k
Representative case
For the rest of the study of short waves since we are interested in unbounded
Couette ow for convenience we put the interface at the representative p oint
i e the shear rate is at the upp er interface The geometry is
simple two dimensional unbounded shear ow of sup erp osed uids with the
interface parallel to the direction of ow as shown in gure
Even this reduced problem requires numerical solution b ecause of the
advective terms prop ortional to y in equations but is a
function only of C W and W and the central features may b e identi ed
by examining extreme cases
THE SHORT WAVE LIMIT k
The e ect of p olymer concentration
on short waves
Concentrated solution C but arbitrary ow strength W
As the concentration parameter C tends to or in the notation of chapter
the Oldroyd B uid e ectively loses its solvent contribution giving the
Upp er Convected Maxwell uid UCM This uid was studied in Couette
ow by Renardy and the stability problem solved analytically The
neutral stability curves in the W W plane are shown in gure We note
that the interface is stable if either W W or W and W are widely
di erent
Dilute solution C
In the dilute limit b oth uids are nearly Newtonian All the p erturbation
quantities may b e expanded as p owers of C and the eigenvalue problem solved
iteratively using Maple The eigenvalue is
C W W W W
iC W W W W
W W i W i E i W W i E i W
O C
R
xt
e
dt is the exp onential integral function where E x
t
We note that the e ect of elasticity app ears only at order C At leading
order in C this shows neutral stability as exp ected when W W but
CHAPTER COEXTRUSION INSTABILITY
10
9 S 8
7 U 6
5
4
U W 3
2
1 S 0
0 1 2 3 4 5 6 7 8 9 10
W
1
Figure Neutral stability curves for C k U unstable S stable
2
There is stability if W W or if W and W are very di erent from each other
1 2 1 2
Elsewhere the interface is unstable
instability for al l other W W This is qualitatively di erent from Renardy s
result for UCM ab ove
Intermediate concentrations
Because the concentrated limit is stable for widely separated values of W and
W but the dilute limit is not there must exist some critical concentration
C ab ove which restabilisation o ccurs We de ne this concentration by
cr it
lo oking at the b ehaviour of as W increases with W For C C
cr it
the imaginary part of is always p ositive whereas for C C the growth
cr it
rate is negative as W Numerical results show C Ab ove this
cr it
THE SHORT WAVE LIMIT k
value we can nd pairs of Weissenberg numbers for which the ow is stable to
p erturbations of short wavelength Indeed for these values the ow is found
using the metho ds of section to b e stable to all wavelengths provided
the long wave mo des are also stable for example at C
W the ow is stable for all W
The e ect of ow strength on short waves
The weak ow limit W W
If b oth Weissenberg numbers tend to zero keeping their ratio constant we
can expand the streamfunction stretches and eigenvalue as Poincar e series
for arbitrary C and obtain
2
C
W W W W
2
C
3
C
W W W W W W C W W W W
4
C
3
i C
W W W W
5
C
C C W W W W W W
O W
i
We note for these slow weak ows that the growth of the disturbance
is inhibited by the total zero shear viscosity C not just the solvent
viscosity Furthermore b ecause in the limit W there is no elasticity
i
the growth rate must vanish as W Our result for is consistent with
i
this requirement i e this is not a spurious instability having large growth
rate that would violate a slow ow expansion section
CHAPTER COEXTRUSION INSTABILITY
In a slow ow expansion as explained in chapter any simple uid may
n
b e regarded as an nth order uid with an error of O W Surprisingly p er
haps the rst term giving a nonzero growth rate is O W thus we anticipate
i
that a fourth order uid would not b e able to capture this b ehaviour We
exp ect that a sixth order uid would b e needed but although such a result
would b e universal the result is unlikely to b e useful in view of the large
number of undetermined co e cients in the slow ow expansion
Strong ow limit high elasticity W
In the large Weissenberg number limit at leading order the stretches a
and a in equations scale as W for y O so that
the vorticity equation loses its highest derivative Thus the limit is
singular suggesting the app earance of b oundary layers In order to simplify
the problem we consider the case where a Newtonian uid o ccupies one
half plane y which is equivalent to setting W with strong elastic
forces only in y
Detailed analysis reveals that two lengthscales arise in the elastic uid
near the interface First there is a b oundary layer of width W across
which the pressure is approximately constant but the velocities D and
the shear stress change by a factor of order W Within this layer
there is a balance b etween the growth or decay of the p erturbation and
the elastic relaxation terms in the evolution of the stretches Second on
a lengthscale of order unity remembering that all lengthscales have b een
rescaled by k here there arises an incompressible elastic deformation in
y
which elastic forces dominate viscous forces so that y e and r p
THE SHORT WAVE LIMIT k
5
0 y
-5
-0.5 0 0.5
x
Figure Streamlines of the p erturbation to Couette ow unstable case C
W The Weissenberg number W is large enough for an elastic b oundary layer to b e
2 2
apparent at the interface
CHAPTER COEXTRUSION INSTABILITY
3
2
1
0 y -1
-2
-3
-0.5 0 0.5
x
Figure Streamlines of the p erturbation to Couette ow stable case C UCM
uid W As in gure the value of W is su ciently high for the elastic b oundary
2 2
layer to b e seen
NUMERICAL RESULTS
The advection due to shear remains imp ortant
Typical streamlines of the p erturbation ow calculated numerically are
shown in gures and showing the elastic b oundary layer
The eigenvalue is of order W but its sign is not obvious and needs
to b e determined numerically as a function of C As noted ab ove we nd
that the growth rate changes sign at C
This limit is di cult to analyse and the result may b e surprising b ecause
although the jump in is large W W is large the growth rate is nev
xx
ertheless small O W The reason is that two comparably large e ects
are in comp etition the highly elastic uid strongly resists the p erturbation
with the b oundary layer acting rather like an extra surface tension This
greatly reduces the growth rate and can for su ciently high concentrations
C C prevent the instability
cr it
Arbitrary ow strength
We conclude from the asymptotic cases examined ab ove that for xed C
C the growth rate of disturbances will b e small and p ositive if W W is
cr it
small since this reduces the driving force and small in magnitude if W W
is large since the elastic resistance then rises Thus there is an intermediate
value of W W at which growth is largest as shown in gure
Numerical results
Away from the limits k section and k section a
numerical study was used
CHAPTER COEXTRUSION INSTABILITY
0.04
0.035
0.03
0.025
0.02
0.015
I m
0.01
0.005
0
-0.005
0 5 10 15 20
W
Figure Growth rate against W when C W the dotted curve is the
2 1
asymptote found for small W As W the growth rate is prop ortional to W but
2 2
the co e cient is not known analytically
NUMERICAL RESULTS
The system is a fourth order di erential equation for y
with three p oint b oundary conditions Because of the interfacial jump condi
tions the simple numerical metho d describ ed in chapter needs extending
Instead of integrating from the centreline to the wall the basis stream
functions are integrated to the interface from b oth the wall and the centreline
The four jump conditions at the interface then b ecome de ning through
the kinematic b oundary condition a matrix equation of the form
M
where is the vector of co e cients of each of the four linearly indep endent
basis functions Thus the determinant of M needs to b e made zero A
Newton Raphson pro cedure is used as in the case with no interface
Figures to show representative cases of the b ehaviour of the
growth rate with wavenumber k for the most unstable disturbances
In this illustration we have chosen the value for which sinuous and
varicose long wave mo des are b oth unstable if the outer uid is more elastic
and b oth stable otherwise The sinuous mo de is the more unstable for long
waves but can b e overtaken by the varicose mo de at some wavenumber
of order unity The mo des necessarily have the same b ehaviour for very short
waves
If the outer uid is less elastic then long waves are stable and the b e
haviour is more complicated The least stable mo de at long waves is again
that identi ed by Chen but continuation of this mo de to high wavenum
b ers do es not necessarily give the least stable short wave mo de Similarly
continuation of sinuous and varicose mo des may also lead to di erent ro ots
of the short wave problem A selection of the typical growth rate curves is
CHAPTER COEXTRUSION INSTABILITY
0.045
0.04
0.035
0.03
0.025
0.02
I m
0.015
0.01
0.005
0
0 1 2 3 4 5 6 7 8
Wavenumber k
Figure Growth rates plotted against k for a case where b oth long and short waves
are unstable C W W The outer uid is the more elastic and
1 2
the fastest growing mo de is sinuous with k The solid curve is the sinuous mo de the
p oints show the varicose mo de
NUMERICAL RESULTS
0.016
0.014
0.012
0.01
0.008
0.006
I m 0.004
0.002
0
-0.002
-0.004
0 1 2 3 4 5 6 7 8
Wavenumber k
Figure Growth rates plotted against k for a case where long waves are unstable and
short stable C UCM W W The outer uid is the more
1 2
elastic and the fastest growing mo de is sinuous with k The rst p eak is the sinuous
mo de and the p oints show the varicose mo de
CHAPTER COEXTRUSION INSTABILITY
0.05
0
-0.05
-0.1
I m
-0.15
-0.2
-0.25
0 1 2 3 4 5 6 7 8
Wavenumber k
Figure Growth rates plotted against k for a case where long waves are stable and
short unstable C W W The inner uid is the more elastic
1 2
and the fastest growing mo de has k The varicose mo de p oints almost always has
the higher growth rate two di erent sinuous mo des are shown solid lines
NUMERICAL RESULTS
0
-0.2
-0.4
-0.6
-0.8
-1
I m
-1.2
-1.4
-1.6
-1.8
0 1 2 3 4 5 6 7 8
Wavenumber k
Figure Growth rates plotted against k for a case where b oth long and short waves
are stable C UCM W W The inner uid is the more elastic
1 2
all mo des decay The sinuous mo des are solid lines the p oints are the varicose mo des
In the short wave limit the disp ersion relation is a p olynomial which has ve ro ots
four of which are marked here with small squares The fth has a decay rate of and
is not on the scale of this graph
CHAPTER COEXTRUSION INSTABILITY
plotted in gures and Note that when one of the Weissenberg
numbers is zero there is a mo de for which long waves decay at rate W i e
the relaxation rate of the elastic uid This corresp onds to an elastic wave
con ned in the elastic uid close to the interface analysis of equations
reveals a b oundary layer of width k which relaxes at the molecular
relaxation rate
For dilute solutions with C small gure the most unstable distur
bances are either short wave or have k of order unity i e the disturbance
wavelength is comparable with the channel width Thus the long wave anal
ysis though su cient to demonstrate instability may b e a p o or predictor
of the type wavelength and growth rate of instability seen exp erimentally
For large C gure Renardy s analysis for UCM as k
gives a quintic p olynomial for having ve distinct ro ots We are able to
nd the least stable branch for b oth sinuous and varicose mo des and all the
mo des we nd match up with the short wave limit
Conclusions
We have investigated the stability to linear p erturbations of a two uid planar
Poiseuille ow of Oldroyd B uids matching all the uid prop erties except
elasticity while neglecting inertia and surface tension
For dilute Oldroyd B uids short waves are unstable over the full range
of Weissenberg numbers W W Near the UCM limit however there is
1
We are grateful to the referee of for p ointing this out
CONCLUSIONS
instability for only a nite range of W W The instability o ccurs for small
W at O W so the phenomenon should not b e limited to Oldroyd B uids
As the wavelength tends to zero the growth rate remains nite
Long waves are unstable to sinuous mo des if the coating is more elastic
than the core and to varicose mo des if the more elastic uid o ccupies a small
enough fraction of the channel The short wave b ehaviour of the system is
indep endent of the type of p erturbation sinuous or varicose and of which
uid forms the core and which the coating
Since the long wave stability characteristics are reversed by interchang
ing the core and coating the short and long wave limits are fundamentally
di erent from one another In most cases the short wave mo de is unstable
and having nite growth rate is predicted to grow faster than the long wave
mo de whose growth rate is small The largest growth rate usually o ccurs
for some mo derate wavelength comparable with the channel width
We have not b een able to verify this wavelength prediction with reference
to existing exp eriments Mavridis Shro have investigated this ge
ometry but their photographs are taken after pro cessing and the instability
has developed well into the nonlinear r egime The distortions of the unstable
interface are irregular in shap e and length but a dominant lengthscale in
the single available photograph of the interface app ears to b e an order of
magnitude smaller than the channel width i e within our short wave r egime
An ingredient neglected in our short wave analysis is surface tension A
dimensionless group that measures the relative imp ortance of normal stress
CHAPTER COEXTRUSION INSTABILITY
and surface tension e ects is an elastic capillary number
N
T k T k
Here T is the surface tension co e cient k the wavenumber of the p erturba
tion at the interface and N the jump in the rst normal stress di erence
Starred quantities are in their dimensional form is the uid relaxation
time and the shear rate at the interface so that is the Weissenberg
number is the shear viscosity
For viscous p olymeric liquids with P and surface tension co e cient
T dyne cm at a shear rate of s and relaxation time of
order s this imp oses the strong requirement that the wavelengths considered
should exceed cm Since however the elastic capillary number scales with
at shear rates of s the neglect of surface tension is justi ed provided
that the wavelength is in excess of cm In a channel of width cm this
value of k lies on the short wave plateau of gures
The mechanism of all the instabilities here is the jump in N This varia
tion in N is present also in elastic liquid ows where N varies continuously
across the channel and for which surface tension vanishes It may therefore
b e exp ected that continuous strati cation may also generate instability This
observation is the motivation for the work in chapter
Chapter
Instability of Fluid having
Continuously Strati ed Normal
Stress
CHAPTER CONTINUOUSLY STRATIFIED FLUID
Introduction
Purely elastic instabilities of homogeneous Oldroyd B uids are known to
o ccur in ows having curved streamlines section and in plane parallel
ows of co extruded uids where the viscosity is constant but where there is
a discontinuity in normal stress chapter We do not know of any case in
which a plane parallel ow of a uid with constant viscosity but whose elastic
prop erties are continuously strati ed across the channel has b een shown to
b e unstable
A strong suggestion that such an instability could arise is given by the
physical explanation of long wave instabilities for co extruded elastic liquids
provided by Hinch et al The idea discussed in chapter is that a
p erturbation of the interface exp oses the jump in normal stress at the in
terface which drives a ow One might guess then that if a variation in
normal stress o ccurs over a thin enough region the same mechanism will
generate an instability in a continuously strati ed uid Indeed at rst
sight this long wave mechanism suggests that even a Poiseuille ow of an
Oldroyd B uid with uniform elastic prop erties will b e unstable b ecause the
cross stream variation of normal stress due to the variation of the shear rate
across the channel app ears rapid to a long wave disturbance However such
a ow is stable section The long wave argument ab ove is in general
erroneous for a uid with continuously varying prop erties b ecause it fails to
take prop er account of convective e ects As shown in chapter in a ow
with a uid uid interface there is a long wave instability having wavenum
b er k The streamfunction is convected with velocity u the streamwise
velocity at the interface and the growth is a second order e ect Thus the
INTRODUCTION
complex frequency is
k u i k O k
where is a constant having magnitude of order unity Now if instead
the uid prop erties vary over a distance d that is small compared with the
width of the channel di erent uid layers within this region are convected
at velocities which di er by a quantity of order d Thus di erent layers
generate p erturbation elastic stresses that are out of phase with each other
by an amount of order k d In order for the phase di erence not to swamp
the e ect of the jump which is of order k we need k d k and therefore
d k Very long waves with k d are exp ected to b e stabilised
We show in this chapter that channel ows of elastic liquids having
constant viscosity and continuously varying elasticity can b e sub ject to in
stabilities For the somewhat arti cial mo del problem we have chosen b oth
long wave asymptotic section and numerical section solutions are
available
Since however the calculation shows that the mechanism is robust we
anticipate that it will app ear to o in other channel ows that involve rapid
variations in material prop erties leading to rapid changes in N Another
example of such a ow is investigated in chapter
CHAPTER CONTINUOUSLY STRATIFIED FLUID
Geometry and constitutive mo del
We consider an Oldroyd B uid whose constitutive equation is given in di
mensional form by
P I rU r U G A
r
A I A
where U is the velocity eld the stress tensor and A the microstructural
con guration tensor The elastic mo dulus G and time constant are each
functions of a Lagrangian variable x t that lab els each material p oint
Because is a material prop erty
D
D t
In investigating the stability of a steady planar channel ow sketched
in gure this description may b e simpli ed by identifying each material
p oint by its p osition y in the absence of any p erturbation In e ect we
regard the continuously strati ed Oldroyd B uid as a multi layer co extru
sion ow in which an in nite number of nozzles feed uid into the channel
each at its own level
In a steady parallel ow the uid has viscosity G at level y
In order to isolate the instability mechanism asso ciated with normal stress
and not viscosity variation we choose the mo dulus so that G C
where C is a constant The viscosity is then C indep endent of as
1
The deviatoric stress may b e shown using equation to satisfy the alternative form
r
r
G( )
1 1
of the Oldroyd B equation E E
( ) ( )
GEOMETRY AND CONSTITUTIVE MODEL
y
y
x
y
y
Figure Flow geometry
for a homogeneous uid The rst normal stress co e cient on the other
hand is C which we may vary arbitrarily by suitable choices of
We make the equations dimensionless using the solvent viscosity the
half channel width L and the centreline uid velocity U De ning W
U L the equations b ecome
C
P I rU r U A
W
r
A A I
W
It should b e remembered that W is convected with material particles thus
W W and x y t where D D t
This is the equation given in chapter by the conditions n
C constant The base state is therefore
U y
W y W y
A
A
W y
CHAPTER CONTINUOUSLY STRATIFIED FLUID
P C W C W y y C
A
y C P C W
P P C W C x
It remains to choose the pro le W We consider a symmetric pro le
so that W W A convenient choice for is then
W
arctan q W W
in which W changes by an amount W across a layer centred at of
dimensionless width q d L In the limit q
W W W
W
W W W
so we should recover the two uid co extrusion problem of chapter
Perturbation ow
Referring back to chapter and using the transformed equations of sec
tion we nd that the p erturbation equations are
ik C y W ik
ik
and the p erturbation stresses are
C
p ik a
W
PERTURBATION FLOW
C
a k
W
C
a p ik
W
in which
i ik y a ik W y W y ik
W
W y y a
i ik y a ik W W y k y a
W
a ik W y k i ik y
W
i ik y ik
with b oundary conditions
Sinuous s
Varicose v
The advantage of the transformation in section is now apparent the
form of these equations is identical to those for which W is a constant except
for the app earance of W y in equation
CHAPTER CONTINUOUSLY STRATIFIED FLUID
In the familiar co extrusion case chapter involving two distinct uids
W is zero within either uid and integration of equation across the
interface yields the continuity condition
ik C W
where is the p osition of the interface and brackets denote a jump across it
This limiting case would b e imp ossible in the formulation of section
Long wave asymptotics
The considerations discussed in the introduction suggest that the most likely
candidate for a long wave k instability is a layer having width of
order k In consequence we consider the distinguished limit in which q q k
and k with q xed In that case except in a layer of size k around
y W takes constant values W W for y and W W for
y As a result the ow has the dynamics of co extruded Oldroyd B
uids with matched viscosities C So far as the outer ow is concerned
we may imp ose by equation a discontinuity in tangential stress
at y The magnitude of this discontinuity which may b e complex is
as yet unknown and must b e determined by analysis of the thin layer itself
We anticipate that the frequency of the disturbance will continue to have the
form of equation
k u i k
and thus the Deb orah number asso ciated with the p erturbation is small
The p erturbation elastic stresses resp ond quasi statically to the ow so giv
LONG WAVE ASYMPTOTICS
ing rise to a Newtonian stress having shear viscosity C
Outer ow
As in chapter the outer solution may b e expressed entirely in terms of
We obtain
f
C
where f f for a sinuous mo de f f for a varicose mo de and
s v
f s
s
v f
v
Inner solution
Within the thin layer the relevant lengthscale for y is k Because k is
small the pressure within the layer is constant and the x momentum equa
tion reduces to
C iC k y W
where equation
ik i ik y
Since the layer is thin we may expand ab out y writing
y with O k
CHAPTER CONTINUOUSLY STRATIFIED FLUID
Then from the expression for we have
k i k
where might itself b e complex so that
d dW
C iC k
d i d
It follows as exp ected that as k and are continuous across the
layer and that has the value in equation or Further
more to b e self consistent we must have
C C k d d
Z
d
d C k
d
Z
dW d
d iC
i
Substituting for gives the disp ersion relation for
Z
dW d C
d
i iC f
where f f for a sinuous mo de and f f for a varicose
s v
mo de
For co extrusion of two distinct uids dW d W and thus
C f
W
C
If on the other hand we consider a continuous variation for W equa
tion then
dW W q
d q
LONG WAVE ASYMPTOTICS
and thus
Z
q d
q i i
This integral is easily evaluated by the calculus of residues giving
i i q R e
i
i i q R e
We deduce that if j j q there are no acceptable mo des of the kind
sought but that if j j q then
j j j j q
All these estimates are con rmed by a full long wave matched asymptotic
expansion included in section
We conclude that in the long wave limit
a Disturbances are propagated with the velocity U at the zero of the
arctan pro le
b The growth rate of disturbances k is p ositive if both the analogous
two uid situation is unstable and the interface is sharp enough i e
the thickness d of the layer is such that d j jk L
c In this case the eigenvalue still in its nondimensional form to the full
problem b ecomes
k ik q k O k
d Increasing by increasing W results in an increase in the largest
value of q for which the eigenvalue exists This pro cess cannot b e
extended b eyond the condition k
CHAPTER CONTINUOUSLY STRATIFIED FLUID
e For any nonzero width d of the layer no asymptotically long wave
disturbance is available
f Because f f where b oth sinuous and varicose disturbances
s v
are unstable sinuous mo des are always more unstable than varicose
ones as in the long wave limit of the co extrusion instability
We cannot tell from a long wave analysis the fate of waves which have wave
lengths comparable with the channel width For this purp ose we need a full
numerical solution describ ed in the next section
Numerical results
Equations de ne a fourth order ordinary di erential equa
tion for the unknown eigenvalue We solve this problem as describ ed in
chapter using orthogonalisation sho oting from the centreline b oundary
conditions to the wall
For very small values of q the eigenvalues were found as exp ected to b e
similar to those for the two uid problem As q increased however and the
width of blurring increased the eigenvalues quickly b ecame neutral For
q in most parameter ranges no interfacial mo des stable or unstable
could b e found An example set of results is shown in gure In ows for
which the maximum growth rate is larger the instability p ersists for higher
values of q
In the long wave limit the growth or decay is lost earlier than for mo der
ate length waves The reason for this is the di ering convection on di erent
sides of the interface as explained in the introduction and is shown clearly
APPENDIX LONG WAVE ASYMPTOTICS
0.045
0.04 q
q
0.035
0.03
q
0.025
I m
0.02
0.015 q
0.01
0.005 q
0
0 0.5 1 1.5 2 2.5 3 3.5 4
Wavenumber k
Figure Decay of instability for increased blurring Sinuous mo des C W
W
in the analysis of section The numerical and asymptotic results are in
agreement
Analytically the disapp earance or app earance of a mo de must b e asso
ciated with a singularity of the governing equations in this case the term
i ik U b ecomes zero at the p oint where the mo de app ears This term
comes from the material derivative D D t and is present only b ecause of the
advection of W as a material quantity We exp ect therefore that this ro ot
should exist only where there is advection of material prop erties
App endix Long wave asymptotics
In this app endix we show the detailed asymptotic expansion used in sec
tion We use k as our small parameter The set of equations we are
CHAPTER CONTINUOUSLY STRATIFIED FLUID
solving is
i ik y a ik W y W y ik
W
W y y a
i ik y a ik W W y k y a
W
i ik y a ik W y k
W
C
a p ik
W
C
a k
W
C
a p ik
W
ik C y W ik
ik
ik i ik y
W
W arctan y q W
APPENDIX LONG WAVE ASYMPTOTICS
W W
q
W imp ortant
W W
Figure Three regions for asymptotic analysis of a uid in which W varies sharply with
with b oundary conditions
Sinuous s
Varicose v
We set q q k and assume q to b e p ositive
There are clearly three regions gure two outer regions consisting of
the main b o dy of each uid where the Weissenberg number is not noticeably
varying and the equations are those of a normal Oldroyd B uid and the
interior of the varying layer in which the W term plays a large part
Outer region
We consider the two outer regions rst Here at leading order in k the
equation b ecomes
CHAPTER CONTINUOUSLY STRATIFIED FLUID
Our two outer solutions are in the uid nearer the wall
E y E y
and in the centre uid
Sinuous F F y s
Varicose F F y v
Inner region
Within the inner region we exp ect all the p erturbations to vary rapidly on
a lengthscale k so we rescale the cross channel co ordinate y k and
work in terms of We let d The resultant equations have an
obvious scaling so we let k a W k a W a W We
also rescale the stresses s k s k and s As in the
co extrusion case we set
k ik
De ning z k the new equations are
k W ik W k ik W
ik W k W k id
W k d W k W
k W ik W k ik W
W k k d k W k
APPENDIX LONG WAVE ASYMPTOTICS
k W ik W k id W k W k
s k p ik d C
s d k C
s k p id C
iC k k z dW ik s ds
is ds
i k z
Equation tells us that we need to work to order k in order to see
the e ect of the strati cation We set k k O k
Order Calculation
Solving at leading order we obtain
A B C
We know that we want equivalence with the two uid problem as the layer
thickness go es to zero so we imp ose B C at this p oint
Order k Calculation
The next order of solution gives
A B C
Again we use our prior knowledge to deduce that C
CHAPTER CONTINUOUSLY STRATIFIED FLUID
Order k Calculation
For matching onto the outer solutions we do not need to nd exactly we
simply need to nd the change in and across the layer Let us lo ok at
rst
Third derivative
The third derivative k d Hence we lo ok again at equation
substituting in the de nitions of s and We may assume that dW as
j j leaving it smaller than any relevant p ower of k Hence
C iC k W i iC k W d
iC k W d k d d ik p O k
where indicates the change in value over the layer We can use the fact
that d k to show that the change in is of order k and is therefore
zero to leading order
Second derivative
d across the layer We have solved two Finally we need to nd
orders in k at the next order using all the solution values from the rst two
orders we nd
C
d i A dW i
C
W W
arctan q we have dW q q This W Since W
leaves us with an integral to nd d which may b e easily solved by contour
APPENDIX LONG WAVE ASYMPTOTICS
metho ds to give
Z
C W
d i A q d
C q i
q R e
C
A W
C
q R e
Matching
We have six constants to determine E E and either F and F or F and
F from the outer and A and B from the inner A do es not app ear at
leading order when we rescale to the outer solution
In order to nish the calculation we must sp ecify either varicose or
sinuous type mo des by way of example we show the varicose case
The six matching equations are in order down to
A
C B C B
C B C B
C B C B
B G
C B C B
C B C B
C B C B
E
C B C B
C B C B
C B C B
E
C B C B
C B C B
C B C B
F
A A
F
where
q R e
C
G W
C
q R e
The solvability condition for nontrivial solutions of this matrix system
b ecomes
G Gf
v
CHAPTER CONTINUOUSLY STRATIFIED FLUID
The equivalent equation for a sinuous mo de is
Gf G
s
In either case
C
G W
C
where is the corresp onding eigenvalue for the two uid case Applying
equation we obtain
q
if R e and
q
otherwise These equations can only b e self consistent if j j q other
wise the eigenvalue do es not exist We reached the same conclusion in a less
formal way in equation of section
Chapter
Channel Flows having
Variations in Polymer
Concentration
CHAPTER CONCENTRATION STRATIFICATION
Introduction
In chapter we generalised the instability caused by a jump in W to that
arising from a steep gradient in W As we will prove explicitly in chapter
this instability dep ends up on the fact that the relaxation time W is taken
to b e a material prop erty and therefore to b e carried with each uid parcel
For a real uid this assumption may not b e correct such small scale
advection cannot b e veri ed exp erimentally and it is not clear from physical
arguments that relaxation time is necessarily a prop erty of a uid and not
in any way a prop erty of its strain rate environment
The other dimensionless parameter in the Oldroyd B mo del is concen
tration of p olymer which corresp onds to a real physical quantity that is
easily understo o d The concentration in any material parcel changes only
through slow cross streamline di usion of the p olymer molecules which may
b e assumed to b e negligible if the p olymers are long and the concentration
gradients not to o great We shall return to this condition in section
In this chapter therefore we consider a generalised Oldroyd B uid
for which the concentration is a continuously strati ed material prop erty
C C which varies steeply across the channel We b egin with the Heav
iside limit for C equivalent to the ow of two distinct uids
The new complication in this chapter is that since the base state viscosity
is discontinuous so is the velocity gradient at the interface Thus although
this problem is in some ways more natural than that studied in chapter it
is also more di cult We shall assume that the relaxation time do es not vary
across the channel for this calculation The extension to the case in which
b oth C and W are strati ed is straightforward
BASE STATE
Base state
As shown in chapter the unidirectional channel ow pro le is given as
U P y C
WU WU
A
A
WU
P P C W P x
P P x CW U P y
A
P y P P x
In these expressions the concentration C may dep end on y In the case
where
C jy j
C
C jy j
the undetermined quantities ab ove b ecome
C C
P
C C
jy j C y
P
U
C C
C y C jy j
This is the Newtonian pro le for two uids of viscosity C and C
resp ectively sketched in gure for a case where C C Note that it
involves a discontinuity in U at y There is additionally a discontinuity
in N of magnitude WP C C at the interface
CHAPTER CONCENTRATION STRATIFICATION
U
Figure Flow pro le U in the channel for the case of two separate uids in which the
higher viscosity uid is on the outside C C
1 2
Stability problem for two uids
Governing equations
Each uid is an Oldroyd B uid and so the p erturbation equations for each
uid are as in chapter
ik
ik
C
p ik a
W
C
a k
W
C
p ik a
W
a ik A A A ik y a i ik U
W
STABILITY PROBLEM FOR TWO FLUIDS
i ik U a ik A A k y a
W
a ik A k i ik U
W
with the streamline p erturbation de ned as
i ik U ik
We have b oundary conditions at the wall and centreline of the channel
Sinuous s
Varicose v
At the interface y exp ik x i t we require continuity of U u
and of the traction n These conditions b ecome continuity at y
of U ik and It is the app earance of the U term
here involving the discontinuity of U that di ers from chapter
Long wave limit
In the limit of small k we p ose the series
k ik O k
and expand all quantities in k The details of this analysis are in section
CHAPTER CONCENTRATION STRATIFICATION
Sinuous mo de
Using for convenience the notation
g
we obtain at leading order for a sinuous mo de
C C C
g C C g C C
and thus
C C C C
g C C g C C g C C
We note that the disturbance do es not convect with the interface as it do es
for the matched viscosity case of chapter At this leading order the uids
are Newtonian so we may check this result by comparing with the ow in
the same geometry of two Newtonian uids with di erent viscosities
1
Consider the channel ow of two Newtonian uids with viscosities The p er
1 2
turbation to the interface exp oses the jump in U due to the viscosity di erence causing
a discontinuity in velocity at the interface a p erturbation to the velocity integral Q
Z Z
+ 1
Q U dy U dy U
2 1
0 +
R
1
The extra part of Q U must then b e balanced by the p erturbation Q u dy and
0
since the p erturbation ow for a sinuous mo de is linear and the shear stress continuous
the velocity pro le is
y y
U
2
u y
1 2
y y 1
STABILITY PROBLEM FOR TWO FLUIDS
The growth rate which o ccurs at order k is
M U
M U
g C C
M U C
g C C
where
WP C C C
M
g C C C C
WP g C C
WP
g C C C C
WP C C
M
g C C C
U P
C C
C C
WP
C C
and
C C
P
g C C
Mass conservation for the p erturbation ow then gives
U
2
2
1
2
g
1 2
and hence as in equation
2
2 1 2
k U k
2
g g
1 2 1 2
CHAPTER CONCENTRATION STRATIFICATION
The growth rate is prop ortional to the Weissenberg number W indicating
once again that it is normal stress that drives the instability and as we
would exp ect the case C C is neutrally stable However in other cases
the sign of this growth rate is not obvious The disturbance is stable for some
we have triples C C and unstable for others For instance at
C C
W
W
W
W
in which we observe that instability may o ccur when either C C or
C C dep ending on the sp eci c parameter values
In the limit the growth rate b ecomes
W C
C
C
W C C
C C
so the mo de is unstable if C C i e if the large outer uid has the higher
viscosity
As we have
C C C
W
C C C C
C
W C C
C
which again indicates weak instability at order if C C
STABILITY PROBLEM FOR TWO FLUIDS
Varicose mo de
For a varicose mo de the scalings of the equations are similar The leading
order growth rate has b een calculated using Maple
We lo ok at the limits of small and large For a slender inner uid
we have
C C C
W C C
C C
whose sign is not obvious
We exp ect that in the wall limit the two interfaces are to o far
apart to feel each other s e ect The growth rates should b e the same for
sinuous and varicose mo des We do indeed obtain
C
W C C
C
Figure shows a plot of growth rate W against interfacial p osition
for the illustrative case C C In this gure we see that the
sinuous mo des generally have the higher long wave growth rate However it
is p ossible for varicose mo des alone to b e unstable for small if C
then the growth rate is WC C for the varicose mo de while
the sinuous mo de is stable
Numerical results
In this section we are studying the interfacial stability of two co extruded u
ids with di erent p olymer concentrations The purp ose of this is to generate
an understanding of the instability In the next section we will generalise to
CHAPTER CONCENTRATION STRATIFICATION
0.35
0.3
0.25
0.2
0.15
W 0.1
0.05
0
-0.05
0 0.2 0.4 0.6 0.8 1
Figure Plot of the growth rate of long waves against interface p osition for the
representative case C C The solid line is for sinuous mo des and the p oints
1 2
for varicose mo des The sinuous mo des usually have higher growth rate and are always
unstable whereas the varicose mo des are stable for
STABILITY PROBLEM FOR TWO FLUIDS
the case of a steep concentration gradient within a single uid i e a blurred
interface
The interfacial mo de is exp ected to have a short wave asymptote of nite
growth rate as we saw for the case of uids with di ering elasticities How
ever this short wave analysis is not relevant to a blurred interface which
has a short lengthscale of its own so we do not investigate short waves but
rather move on to a numerical study of waves of mo derate wavelengths
0.025
0.02
0.015
I m 0.01
0.005
0
0 0.5 1 1.5 2 2.5 3 3.5 4
Wavenumber k
Figure Plot of growth rate against wavenumber for a representative case in which
b oth sinuous and varicose mo des are unstable C C and W The
1 2
solid line shows the b ehaviour of sinuous mo des and the p oints the b ehaviour of varicose
mo des
The equations are solved using the numerical metho d describ ed in chap
ter integrating from the wall and centre line to the interface and applying
jump conditions there Some typical plots of growth rate against wavenum
b er are shown in gures and The long wave asymptote I m k
CHAPTER CONCENTRATION STRATIFICATION
0
-0.005
-0.01
-0.015
-0.02
-0.025
I m
-0.03
-0.035
-0.04
-0.045
0 0.5 1 1.5 2 2.5 3 3.5 4
Wavenumber k
Figure Plot of growth rate against wavenumber for a representative case in which
b oth sinuous and varicose mo des are stable C C and W The
1 2
solid line shows the b ehaviour of sinuous mo des and the p oints the b ehaviour of varicose mo des
STABILITY PROBLEM FOR TWO FLUIDS
is evident in these curves and where long waves are unstable there is a wave
length on the same scale as the channel width which is most unstable It is
this wavelength which we might exp ect to survive longest during blurring
of the interface
1
0.5
0 y
-0.5
-1
-0.5 0 0.5
x
Figure Streamlines of the p erturbation ow for co extruded uids at two di erent
concentrations The most unstable wavenumber k is chosen for a sinuous mo de
with C C and W
1 2
Figure shows the streamlines of the p erturbation ow for the most
unstable p oint on the sinuous plot of gure These are of similar form to
the schematic streamlines shown in gure on page b ecause the mecha
nism for long wave instability here is the same as that in chapter However
the discontinuity in viscosity pro duces streamlines with discontinuous slop e
at the interface
CHAPTER CONCENTRATION STRATIFICATION
Single uid with varying concentration
Referring back to chapter and substituting the advected stresses we nd
that the p erturbation equations for a continuously strati ed uid are
ik ik W P y C C
ik
C
p ik a
W
C
f P y C g k a
W
C
a p ik
W
W P y
a ik A f P y C g i ik U
W C
ik A a U
WP
i ik U a ik P y C g f
W C
A k U a
i ik U a ik A k
W
i ik U ik
SINGLE FLUID WITH VARYING CONCENTRATION
with b oundary conditions
Sinuous s
Varicose v
As for the case of varying W chapter these equations are the same
as for a single unstrati ed Oldroyd B uid except for the terms involving
in equations
The connection b etween these extra terms and their delta function coun
terparts for the case of two separate uids discussed ab ove is clear in equa
tion the term ik W P y C C b ecomes ik i e it
is a jump in N Similarly in equations and the term
P y C b ecomes U i e a jump in velocity gradient caused by
the jump in viscosity
Sp eci c C pro le
As an illustration we take C as
C
C C arctan q
so that the limit q is a Heaviside function equivalent to the two uid
C C and C C C case with C
As for the W case discussed for long waves in section we can
divide the uid into two regions an outer in which the uid has the same
CHAPTER CONCENTRATION STRATIFICATION
resp onse as for the interfacial problem and an inner in which the variation
of C is imp ortant As noted ab ove the real part of the eigenvalue for the two
uid case here do es not corresp ond to the interfacial velocity Nevertheless
the p erturbation ow is still driven entirely by forcings at the interface and
so the interface region is central to the dynamics The scalings are therefore
the same as for the W problem As the blurring is increased di erential
advection of di erent layers within the interface causes the forcing due to N
from these di erent layers to drift out of phase It follows that no mo de can
b e found for very long waves with q k and that the form of b ecomes
Aq k
for some constant A Unfortunately in this case the eigenvalue for the two
uid problem is not simply expressible so a full analytic solution of the
general problem is not practicable
A numerical study of the same form as that in chapter reveals that the
resp onse to blurring of the interface is indeed as exp ected gure shows
an example Ro ots are shown for q and The mo de
disapp ears when I m As for the W case higher values of q can b e
attained that still show instability by increasing the two uid growth rate
in this case by increasing W However the condition in the long wave
limit that k means that this pro cess cannot continue inde nitely
thus no ro ots could b e found for q
CONCLUSIONS
0.025
0.02 q
0.015
q
I m 0.01
q
0.005
q
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Wavenumber k
Figure Growth rate plotted against wavenumber for an increasingly blurred C pro le
W C C The long wave asymptotic growth rate quadratic in k
for q from section table is shown in p oints At q the eigenvalue
has ceased to exist
Conclusions
We have studied a uid which mo dels a susp ension of microscopic dumbbells
whose concentration C varies This concentration is advected with the uid
i e cross streamline di usion is neglected If the di usion were to b e taken
into account the increased blurring of the interface could provide a stabilis
ing mechanism However a dumbbell di uses through its own diameter on a
timescale determined by Brownian motion which like the relaxation time
is order Thus the cross channel di usion rate scales as the square of the
ratio of dumbbell size to channel width We have already assumed that this
ratio is small in order to use continuum mechanics for a channel of cm and
dumbbells of size m the di usion rate is of order We thus conclude
CHAPTER CONCENTRATION STRATIFICATION
that di usion is unimp ortant here
For this material we nd similar results to those for the less physical
but analytically more tractable uid of chapter As the unstable interface
b etween two uids is blurred the growth rate of the instability decreases
and nally the mo de ceases to exist Waves having wavelengths comparable
to the channel width survive the longest in this pro cess
The imp ortant result of this chapter is evidence of an elastic instability in
a uid with no sharp interface and no curvature of the base state streamlines
and for which there is a sound physical basis
We are not aware of any exp eriment in which the concentration of p olymer
across a channel has b een delib erately varied Such exp eriments would b e of
interest Our calculation suggests that steep concentration gradients must b e
included if our instability is to arise concentration gradients due to imp erfect
upstream mixing are unlikely to b e su cient
The analysis of this chapter has a b earing on a prop osed mechanism
for the sharkskin instability describ ed in section Chen Joseph
suggest that the high stresses near the wall can cause the p olymer molecules
to migrate inwards creating a depleted wall region They then prop ose that
the interface dividing this region from the bulk of the uid would b e sub ject to
an interfacial instability of the type discussed in chapter and section
Our analysis shows that a thin outer layer with lower viscosity than the
bulk will b e stable to long waves in the absence of inertia and that a small
amount of blurring of the interface will stabilise short waves An interface
created by migration will never b e truly sharp and therefore the purely
elastic instability mechanism they prop ose could not o ccur in practice at
APPENDIX DETAILS OF THE LONG WAVE EXPANSION
least on a continuum scale
App endix Details of the expansion for
long waves
In this section we show the full details of the asymptotic expansion used in
section
Leading order calculation
In the base state
C C
P
g C C
where g
Then at leading order for a sinuous mo de
C y y
U
g C C
C C y y
where
C C
U P
C g C C
Hence
C C C
g C C g C C
CHAPTER CONCENTRATION STRATIFICATION
and
C C C C
g C C g C C g C C
For reference we note that at the interface
U
C
g C C
C y
U
g C C
C y
C y
U
g C C
C y
Order k calculation
The quantities involved in the jump conditions at order k are
iW P U C
C C
g C C C
and
iW P C U C C
C
g C C C C
iW P U g C C
i iW P U
g C C C C
APPENDIX DETAILS OF THE LONG WAVE EXPANSION
where
WP C C
W C C C C
g C C
We denote
C i M U i
C i M U C
to obtain
y
i M U
g C C
y
y y
M i U
C y y
M i U
g C C
C C y y
and therefore
M U
i M U
g C C
M U C
g C C
CHAPTER CONCENTRATION STRATIFICATION
Growth rate
The growth rate which o ccurs at order k is
M U
k M U
g C C
M U C
g C C
where
WP C C C
M
g C C C C
WP g C C
WP
g C C C C
C C WP
M
g C C C
U P
C C
C C
WP
C C
and
C C
P
g C C
as stated in section
Chapter
Channel Flows of a
White Metzner Fluid
CHAPTER CHANNEL FLOWS OF A WHITE METZNER FLUID
Introduction
In chapters and we demonstrated the instability of channel ows for
two forms of generalised Oldroyd B uid with a mechanism dep ending on
gradients or jumps in the rst normal stress di erence N We referred to
this as a co extrusion instability and noted that it will only o ccur if the
gradient in N is large
We now consider a more realistic constitutive equation for a p olymer melt
one which will show a shear thinning viscosity while also having gradients in
N The White Metzner uid has these prop erties and in addition sp ecialises
to a uid we have already studied which therefore provides a helpful reference
p oint for numerical studies We de ne the uid by setting C and
c in the formulation of chapter In other words there is no
solvent viscosity or variation in the elasticity parameter W There remains
at our disp osal the choice of the shear dep endent viscosity
The simplest mathematical form of stress as a function of shear rate which
shows shear thinning is a p ower law
n
W
where n Note that although is well de ned everywhere its
n
derivative may not exist at and furthermore W has a sin
gularity at for n This technical di culty is discussed further in
section It could b e remedied by using say a Carreau viscosity func
tion de ned in section but only at the cost of introducing at least one
more dimensionless group and this is a course we prefer not to take here
In the sp ecial case n we recover the UCM uid
GOVERNING EQUATIONS FOR A CHANNEL FLOW
In a simple shear ow where u y the White Metzner mo del has
the stress
n n
p W W
A
n
p W
Governing equations for a channel ow
Statement of the equations
The base ow is given by chapter as
n n
U y y
The remaining base state variables are then
n
n
y jU j U
n
n
n n
n
y W
n
n n
n n
W y W y
n n
A
A
n
n
W y
n
P P W P x
P P x WP y P y
A
P y P P x
n
n
P
n
CHAPTER CHANNEL FLOWS OF A WHITE METZNER FLUID
Figure Base ow velocity pro le for two values of n On the left is the parab olic
pro le of n a UCM uid while a shear thinning uid with n is on the right
The velocity pro le U is sketched in gure for two values of n for n
it is parab olic b ecause the uid has a constant viscosity and for small n it
is close to plug ow through most of the channel with b oundary layers near
y
The linearised momentum equations for a p erturbation to this ow are
as in section
ik
ik
a p
W
a
W
a p
W
The pressure p may b e eliminated from equations to give a
vorticity equation
a k a ik a a
GOVERNING EQUATIONS FOR A CHANNEL FLOW
The evolution equations for the p erturbations a are
a i ik U
A ik A A A ik U a
i ik U a ik A A k U a A
i ik U a ik A k
where
n
n n
n
y W
n
is the relaxation time asso ciated with the base state and is its p ertur
bation value given as
n
n n
n
y k W n
n
Finally the b oundary conditions b ecome
Sinuous s
Varicose v
Our aim as usual is to determine the eigenvalue for a nontrivial solution
to this set of equations
CHAPTER CHANNEL FLOWS OF A WHITE METZNER FLUID
It is notoriously easy to generate spurious numerical instabilities from
equations of this kind We therefore start by constructing asymptotic ap
proximations against which our numerics can b e checked In addition the
availability of well known solutions for n provides further checks to our
results
A mathematical di culty
In forming a single equation for from the governing equations
we would have to take derivatives of the term i ik U
0
n n
which will involve derivatives of y This will give terms prop ortional
n n n n
to y and to y which can cause a problem near the centre line
y if n
Physically this di culty is caused by the form of the relaxation function
which also acts as shear viscosity
n
W
which is non smo oth at When we imp ose a base state shear rate
and add a p erturbation the resultant relaxation time is
n n n
W W n W
expanding as a Taylor series providing is nonzero Near the centre line
may b e smaller than and the principle of linearisation breaks down
We note however that the p erturbation to the shear rate is prop or
tional to k For a varicose mo de is an o dd function of y and so
at the channel centre line Thus the linearisation is rescued and
LONG WAVES
the system has a removable singularity at y Numerically we may start
the integration a small distance from the channel centre and b ecause the
singularity is removable the magnitude of this distance do es not a ect the
results
For a sinuous mo de however the singularity is not removable and if a
numerical patch is introduced or equivalently a Carreau viscosity function
is used the size of the patch has a marked e ect on the results particularly
as n Therefore for the remainder of this chapter we will consider
varicose mo des only satisfying equation v
Long waves
In the long wave limit as k we expand in p owers of k It is convenient
to rescale the Weissenberg number W by the wall shear rate n n
w
n
to give W W n n note that this gives us W W when
wall
n The leading and rst order calculations are summarised here
Order calculation
From integration of equation and substitution of equation we
obtain
n n
y
A B y i
W
where A and B are constants of integration For varicose mo des must
b e an o dd function of y so the condition at y gives A Because
the equation is linear we may set B without loss of generality We are
CHAPTER CHANNEL FLOWS OF A WHITE METZNER FLUID
lo oking for a solution of equation satisfying and
Integrating twice and applying the condition at y and at y
gives
n n
n ny n y
i y y
W n n
Finally the condition on at y yields
n i
W n
We have shown that in nitely long waves k are stable for all n and W
This eigenvalue corresp onds to the pure relaxation mo de i W in the case
n
Order k calculation
Setting i ik U we write
k O k
k O k
k O k
a a k a O k
ij
ij ij
It may seem counter intuitive at rst sight that the growth rate should
have a term of order k Let us consider the transformation k k b ecause
the p erturbation is of the form exp ik x i t this corresp onds to a physical
p erturbation of wavelength k in a system in which the x direction is re
versed Now the real part of represents convection of the disturbance and
LONG WAVES
the imaginary part represents growth so we would exp ect to b e reversed
r
by this transformation but to b e unchanged Thus
i
r i
k k i k
Any mo de as de ned ab ove
i r i
i k i
i
do es not app ear to satisfy this condition if is nonzero However the
presence of a corresp onding mo de with complex conjugate term of order k
can rescue the situation If we lab el such a complex conjugate pair of ro ots
by A and B then
i r i
k i k i
A
i r i
i k k i
B
i r i
k i k i
A
i r i
k i k i
B
The growth rate of mo de A for negative k is that of mo de B for p ositive
k and vice versa Figure shows a sketch of the b ehaviour of the real
and imaginary parts of these ro ots with k We conclude that an O k term
may legitimately app ear in and indeed this is the b ehaviour for the UCM
i
mo del n
When the leading order quantities are substituted into the rst order ex
pansion in small k the result is a third order ODE for If we de ne
W W a a W and a a W so that all quantities
x are indep endent of W we obtain an equation whose co e cients are not
functions of W We deduce that the form of is a function of n only
n
n n
y
n
CHAPTER CHANNEL FLOWS OF A WHITE METZNER FLUID
r
k
B
A
a
a
a
a
a
a
i
a
a
a
a
a
a
B
A
Figure Schematic of the b ehaviour of two complex conjugate ro ots of the long wave
system as k moves through p ositive and negative values
n n
i i y
n n
n
y y n a n
n n
n n
n
y i i n a n
n n
a ia C
In general we cannot nd an analytic solution of these equations but in
the Maxwell limit n they are soluble Solution and application of the
b oundary conditions yields
G
p
p
arctan
LONG WAVES
Ro ots of this equation app ear in two complex conjugate pairs
i
i
We can demonstrate that no further ro ots exist by using winding number
techniques G is analytic in away from branch cuts at and
on the real axis These branch cuts are caused by the logarithm
p
inherent in b oth and arctan for complex
It is straightforward to check that G do es not have a zero for either j j
or j j It follows that the number of ro ots of G N G is given as
I
G
d N G
i G
where the integral is p erformed around the contour shown in gure
t t
Figure The plane showing a schematic of the contour of integration used for
1
determination of the winding number N G
CHAPTER CHANNEL FLOWS OF A WHITE METZNER FLUID
The winding number N G is the number of times that the function G
go es round the origin as traverses the contour of the integral Plotting
G in the complex plane for values on the contour and counting the times
this curve encircles the origin gives the number of ro ots of G In this case
N G
Long waves and low n
For a highly shear thinning uid with n we consider the joint limit for
long waves in which k and n with the ratio k W n constant
We assume that will b e an order quantity and check this a posteriori
The full equations are
n n
a a a i a
W W
n
i i a W y n in W y U
W
n n
W y U a n y i
W W
n
i i a n in U
W
n
U a n W y
W
n n n
i i a i U y
W W W
n n
W ny
LONG WAVES
Neglecting all but leading order terms we obtain
a a Ay A
i a in n
n n
W ny
This leaves us with
n
Ay
i iA y
W n
which is satis ed to leading order in small n by
n
Any A
y B sinh i y
W
in which we have applied the condition that
The wall b oundary conditions and give
A A
B i cosh i
W
A
B sinh i O n
and so
W i coth i
In the limit or k for small nite n this b ecomes
i
W
CHAPTER CHANNEL FLOWS OF A WHITE METZNER FLUID
-1.48
-1.5
-1.52
-1.54
-1.56
-1.58
I m -1.6
-1.62
-1.64
-1.66
-1.68
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
R e
Figure Behaviour of as increases in the long wave shear thinning limit of a White
Metzner uid The Weissenberg number is W All values of give stable
which is indeed the n limit of the leading order expression for k
given by equation with W W n
For other values of is shown in the complex plane in gure the
ow is stable for all
A sample plot of the eigenfunction for the case n and
W for which i is given in gure and the resulting
streamlines of the p erturbation ow in gure
LONG WAVES
R e
0 0.2 0.4 0.6 0.8 1
y
Figure Streamfunction plotted across the channel for the long wave shear thinning
limit of a White Metzner uid The Weissenberg number is W and n k
so The b oundary conditions at y are not satis ed exactly but the error is of
order n compared with the magnitude of in the bulk of the channel
CHAPTER CHANNEL FLOWS OF A WHITE METZNER FLUID
1
0.5
y 0
-0.5
-1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
Figure Streamlines of the p erturbation ow for the asymptotic limit of low n and low
k The streamfunction is shown in gure and the parameter values are n k
W
SHORT WAVES
Short waves
Analytic scalings for short waves
The White Metzner uid has no intrinsic lengthscale like the Maxwell uid
to which it sp ecialises The only lengthscales in the problem then are the
channel width normalised to in dimensionless form the wavelength k
and any scale o ccurring in the base ow
If we take a very highly shear thinning uid n then the base state
velocity has steep gradients in a thin layer near the wall This layer gives a
third lengthscale to the ow given in dimensionless form as n
For very short waves shorter than n if n is small we exp ect all the
lengths in the problem to scale with the wavelength k and the disturbance
to b e lo calised at some p oint in the channel
While it is p ossible that a mo de may b e lo calised anywhere the distin
guished p oints are near the centre of the channel and the wall y We
concentrate on wall mo des here
We let z k y d denote z and let k We rescale a
ij
by a factor of k and as we would exp ect the relevant shear rate is the
rate at the wall n n We de ne the relaxation time at the
w
n
wall W W n n as we did for the long wave limit of
w
section to obtain
i i z a i W d W nd
w w
w
W
a W n
w w
CHAPTER CHANNEL FLOWS OF A WHITE METZNER FLUID
i i z a W nd a n
w w
w
W
i i z a W id
w w
W
d a id a a
with b oundary conditions
as z
Note the similarity of these equations to equations for the
Oldroyd B uid of chapter with shear rate This similarity is
w
the reason for using W Although di erent b oundary conditions are to b e
applied the only di erence b etween these equations and those for a Maxwell
uid is the co e cient of the highest derivative and the nal term in equa
tions and Both these changes come from the p erturbation
caused by the in nitesimal change in e ective shear rate The equations are
of course the same as for the Maxwell uid when n
For the case n using the eigenfunctions of Goro dtsov Leonov
the leading order eigenvalue as k is
w
i W
w
where W W W and This ro ot is stable
w w
for all values of W
SHORT WAVES
For other values of n these equations are not soluble analytically but
as for the case of two Oldroyd B uids in chapter we can obtain this
short wave limit numerically
Numerical results for short waves
In the short wave limit a numerical study reveals only one ro ot with the
wall boundary layer scaling outlined ab ove This ro ot is known in the limit
n and may b e continued to lower n
We note that as n if we keep the original Weissenberg number W
xed b oth W and b ecome O n Thus if is to have any e ect in
w
equations we exp ect n as n
Because W b ecomes small as n we revert here to the original Weis
senberg number W For any xed value of W the ro ot remains stable as
n decreases until n b ecomes small n when instability may o ccur
dep ending on the value of W Thus su ciently short waves grow though at
nite rate for any xed n
This is a new instability It di ers from the co extrusion instability of
chapters and Because it o ccurs only for small but nite n we refer to
it as a shear thinning instability Two examples W and W are
shown in gures and and sample streamfunctions from each curve are
shown in gure Figure shows the p erturbation streamlines for the
unstable mo de at W n Numerical di culties o ccur for smaller
n b ecause of the large values of and the high shear rates involved
CHAPTER CHANNEL FLOWS OF A WHITE METZNER FLUID
0.2
0
-0.2
-0.4
-0.6
I m -0.8
-1
-1.2
-1.4
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Power law index n
Figure Imaginary part of the complex growth rate of short wave disturbances plotted
against p ower law index n for White Metzner uids with W solid line and W
p oints The mo de remains stable as long as n is mo derate but instability is observed
for n The values at n corresp ond with equation
2.2
2
1.8
1.6
1.4
1.2
R e 1
0.8
0.6
0.4
0.2
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Power law index n
Figure Real part of the eigenvalues shown in gure
SHORT WAVES
1.8
1.6
1.4
1.2
1
0.8
j j
0.6
0.4
0.2
0
-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0
Channel co ordinate
Figure The mo dulus of the complex streamfunction j j plotted across the b oundary
layer Both functions are for a value of n The solid lines are for a stable case
W and the p oints for an unstable case W
CHAPTER CHANNEL FLOWS OF A WHITE METZNER FLUID
0
-0.5
-1 z
-1.5
-2
-0.5 0 0.5
x
Figure Streamlines of the more unstable streamfunction from gure The distur
bance decays away from the wall at the top of the picture W n
INTERMEDIATE WAVELENGTHS NUMERICAL RESULTS
Intermediate wavelengths numerical
results
Following the b ehaviour of ro ots of in the complex plane is di cult and we
pro ceed by parameter continuation As a rst step therefore we consider
the mo des of the UCM uid n as k varies
In this limit long waves show four stable ro ots all coalescing at k
These four ro ots can all b e found numerically and may b e analytically con
tinued into the r egime of nite k As gure shows one of these ro ots
ceases to exist at k This is b ecause the problem has a continuous sp ec
trum which is stable a branch cut in the plane This is a non analytic
region and so analytic continuation is invalid The disapp earing ro ot we
shall refer to it as ro ot A enters the continuous sp ectrum at the branch
p oint where k
Of the three ro ots which exist for high k only one ro ot B matches the
short wave asymptotics of section This is b ecause the others do not
ob ey our assumed scaling but rather have k O as k In
other words they b ecome concentrated around the centre line rather than
the wall This limit has not b een studied b ecause of the singularity in the
White Metzner uid at zero shear rate However the Maxwell uid is well
p osed for the entire channel and for that case these two decaying ro ots are
acceptable physical mo des In the very short wave limit they are con ned
within a central region of plug ow and represent the stable resp onse of the
uid to small p erturbations ab out a state of rest
Turning now to the more general problem in which n there is
CHAPTER CHANNEL FLOWS OF A WHITE METZNER FLUID
still a continuous sp ectrum of mo des lo cated at y y where the factor
n n
i ik y is zero
n
n i
n n n
y y y k y
W n
Figure shows this sp ectrum in the plane for W k for a few
illustrative values of n We note that the extreme values of the continuous
sp ectrum are given by y for which
n
i n
W n
and y for which unless n
k
-0.3 B
-0.35 C
-0.4
-0.45
I m
-0.5 A
-0.55 D
-0.6
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Wavenumber k
Figure Growth rates plotted against wavenumber k for an Upp er Convected Maxwell
uid with W The long wave k asymptotes shown are calculated in section
and the short wave limit in section
INTERMEDIATE WAVELENGTHS NUMERICAL RESULTS
0.5
0
-0.5 n
-1
n
-1.5
-2
I m -2.5
n
-3
-3.5
-4
-4.5
0 0.2 0.4 0.6 0.8 1
R e
Figure The continuous sp ectrum in the plane for a White Metzner uid W
and k three di erent values of n are shown
CHAPTER CHANNEL FLOWS OF A WHITE METZNER FLUID
Because the continuous sp ectrum reaches the line of neutral stability
whenever n analytic continuation of ro ots will only b e of limited value
Ro ots may cease to exist and this metho d cannot guarantee to nd all ac
ceptable ro ots of the disp ersion relation
As an illustration gures and show all the ro ots we have found
for as n varies when W k There are four ro ots which are highly
unstable for small n B E F and G only one of which ro ot B continues
in n up to n Another ro ot H is weakly unstable and also disapp ears
at a critical value of n We know the four ro ots A B C and D as de ned
in gure at n only two of them A and B continue to exist for
small n Ro ot A was rst continued downward in n at lower k b efore moving
to k and continues to exist and is stable b eyond the scale of gure
when n is small
1
0.8
0.6 B
0.4
0.2 G
0
H
I m -0.2
F C
-0.4
-0.6
D E
-0.8 A
-1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Power law index n
Figure Plot of the growth rate against p ower law index of ro ots found numerically
for the White Metzner uid W k
INTERMEDIATE WAVELENGTHS NUMERICAL RESULTS
2 C
H
G
D F
1.5 E
1
R e B
0.5 A
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Power law index n
Figure Plot of the real part of the eigenvalues shown in gure
CHAPTER CHANNEL FLOWS OF A WHITE METZNER FLUID
We are unable to continue our calculation to very small n The di culty
is numerical As n the equations b ecome singular and the streamfunc
tion derivatives increasingly large Small errors in the function values are
ampli ed by the integration until they swamp the true value and no ro ot
can b e found
It is interesting to consider the asymptotic b ehaviour of these four ro ots
as n At rst glance the growth rate I m seems to b e growing
without b ound at least for the most unstable ro ot B However plotting
this ro ot logarithmically b oth as log I m against n and as log I m
against log n it b ecomes clear that the ro ot is in fact exp onential in n and
is therefore b ounded though large as n Since we exp ect very short
waves to grow at rate I m n as n this b ounded growth rate can
only apply to any xed nite value of the wavenumber k Figure shows
the b est p ower law t and the b est exp onential t to the imaginary part of
ro ot B Its real part shown in gure starts to change rapidly shortly
b efore the calculation fails The region of this variation is to o short to show
conclusively whether this change is of p ower law or exp onential form Thus
we cannot tell whether or not the real part of stays b ounded as n
The other three strongly unstable ro ots E F and G are similar to one
another Figure shows that they have b ounded real parts as n and
their imaginary parts are also b ounded consistent with exp onential growth
rates as for ro ot B
In gure we show the shap e of typical unstable eigenfunctions The
1
Spurious numerical ro ots of the type outlined in section are sometimes found
but these are not ro ots of the disp ersion relation
INTERMEDIATE WAVELENGTHS NUMERICAL RESULTS
3
2.5
2
1.5
I m
1
0.5
0
0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Power law index n
Figure Plot of the low n b ehaviour of the growth rate of ro ot B for a White Metzner
uid with W k The p oints show the numerical data while the two solid curves are
( 22 5n)
the b est p ower law and exp onential ts The exp onential curve I m e
gives the b etter t
CHAPTER CHANNEL FLOWS OF A WHITE METZNER FLUID
two curves shown are for n and n allowing some insight into
the dep endence of the streamfunction shap e on n The scalings of this func
tion will b e discussed in section Figure shows the p erturbation
streamlines asso ciated with the n curve of gure
j j
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Channel co ordinate y
Figure Two typical eigenfunctions for small values of n The mo dulus of the stream
function is plotted for ro ot B at W k The solid line is for n and the
p oints for n
To complete the picture of this instability a study of its wavenumber
dep endence is needed For no values of k and W did we nd instability for
n We choose two representative sets of parameter values n
W for which all ve of the unstable mo des we have found are present
and short waves are also unstable and n W for which short waves
are known to b e stable and only three mo des have b een found at k
Figures and show the imaginary and real parts of the ve ro ots
at n The most unstable of these corresp onds to the known short wave
INTERMEDIATE WAVELENGTHS NUMERICAL RESULTS
1
0.5
0 y
-0.5
-1
-0.5 0 0.5
x
Figure Streamlines of the p erturbation ow for an unstable White Metzner uid
W k and n The mo de chosen is ro ot B Much of the variation in the ow
takes place close to the wall
CHAPTER CHANNEL FLOWS OF A WHITE METZNER FLUID
ro ot as k and to the known long wave ro ot as k The other four
ro ots cease to exist when the waves b ecome long k small by merging into
the continuous sp ectrum For this case the most unstable disturbance is a
short wave
Three of the ve ro ots in gures and ro ots B E and F are
su ciently widely spaced to b e clearly distinguishable on the plot but the
other two ro ots G and H are very close together In fact they are close to
a mo de crossing p oint as their b ehaviour demonstrates shown on a larger
scale in gures and
Finally the three ro ots at n are shown in gures and
In this case the most unstable disturbance has k of order unity The corre
sp onding eigenfunction is shown in gure
Highly shear thinning limit n !
From the numerical studies of sections and we observe that very
highly shear thinning uids with n can show instability We would
therefore like to understand the scalings appropriate for the case in which
n
The highest derivative in the p erturbation equations which comes
from a has a co e cient prop ortional to n This is b ecause the p erturbation
shear stress is of order n The equations are not regular in the limit n
and therefore we exp ect some kind of b oundary layer structure to emerge in
this limit Indeed as n most of the variation in base state velocity
takes place very close to the wall so we exp ect to see b oundary layers in this
HIGHLY SHEAR THINNING LIMIT n
2 B
1.5
1
0.5
E F
0
I m
G H
-0.5
-1
-1.5
0 1 2 3 4 5 6 7 8
Wavenumber k
Figure The wavenumber dep endence of the growth rate of the ve unstable ro ots
for a White Metzner uid with W and n The numerical results are all shown
as p oints and the solid line is an extrap olation of the b ehaviour of the most unstable ro ot
1
B as k It is given by I m I m k where is the short wave ro ot
found by the program of section
CHAPTER CHANNEL FLOWS OF A WHITE METZNER FLUID
3.5 F
3
G H
2.5 E
2 B
1.5
R e
1
0.5
0
0 1 2 3 4 5 6 7 8
Wavenumber k
Figure The real parts of the eigenvalues shown in gure The solid line is an
extrap olation of the b ehaviour of ro ot B the most unstable ro ot as k matching up
with the numerical results for short waves from section W and n
HIGHLY SHEAR THINNING LIMIT n
0.025
0.02
0.015
0.01
I m
0.005
0
-0.005
1 1.2 1.4 1.6 1.8 2
Wavenumber k
Figure Close up of part of gure The two ro ots shown ro ots G and H are
close in parameter space to a p oint where they cross
2.2
2
1.8
1.6
R e
1.4
1.2
1
1 1.2 1.4 1.6 1.8 2
Wavenumber k
Figure The b ehaviour of the real parts of ro ots G and H close to their mo de crossing
This plot shows the real part of the curves shown in gure
CHAPTER CHANNEL FLOWS OF A WHITE METZNER FLUID
0.1
0 B
-0.1
-0.2
-0.3 E
-0.4
I m
-0.5
-0.6
-0.7 H
-0.8
0 0.5 1 1.5 2 2.5 3 3.5 4
Wavenumber k
Figure The wavenumber dep endence of the growth rate of the three ro ots for a
White Metzner uid with W and n
3 H 2.5
2 E
1.5
R e
1 B
0.5
0
0 0.5 1 1.5 2 2.5 3 3.5 4
Wavenumber k
Figure The wavenumber dep endence of the convective part of the eigenvalue of the
three ro ots for a White Metzner uid with W and n This plot shows the real
part of the curves whose imaginary parts are in gure
HIGHLY SHEAR THINNING LIMIT n
0.6
0.5
0.4
0.3
j j
0.2
0.1
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Channel co ordinate y
Figure The eigenfunction of the most unstable mo de ro ot B for a White Metzner
uid with n and W The wavenumber is k and the mo dulus of the
streamfunction is shown
CHAPTER CHANNEL FLOWS OF A WHITE METZNER FLUID
region
We consider the simpli cation which may b e made to the equations in
the limit of small n while still retaining all the terms which are physically
and mathematically imp ortant
Governing equation
We will consider only the leading order b ehaviour as n Throughout
this section there will b e an implicit O n on the right hand side of
each equation We neglect terms which are clearly not of leading order from
the full equations for general n to obtain
a k a ik a a
n
y
i ik
W n
n
y
a W y n W y ik ik W y k W y a
n
n
y
a n W y k ik W a
n
a ik W y k
with b oundary conditions
HIGHLY SHEAR THINNING LIMIT n
Let us rst consider the region far from the wall Here is approximately
constant and so b ecause each of the comp onents of a is prop ortional to
it plays no part in the dynamics of equation In order to maintain the
shear stress n at leading order the ow must vary on a lengthscale of n
across the whole channel This scaling is not obvious in the eigenfunctions
of gure This is b ecause the values of n there are only n and
n i e n and n neither of which is very small The
physical origin of the n lengthscale is that rapid y variations are required
to generate a shear stress of order unity Since this is likely to b e the region
of slowest variation of which we exp ect to have large derivatives close to
the wall we deduce that a is always large compared to a This allows us
to p erform the rst integral of the vorticity equation equivalent to stating
that the p erturbation pressure is constant across the channel
ik a a A a
We now need keep only those terms of a which may somewhere b e as
large as a The neglect of many terms is obvious but we lo ok in detail at
n
the term a y n This needs to b e as large as a to have an e ect which
n n n
can only happ en if y n is large Where y n is large y W n
and the term is only as large as a Therefore we neglect it everywhere
and equation b ecomes
a W y ik
CHAPTER CHANNEL FLOWS OF A WHITE METZNER FLUID
We also observe that
a ik
Substituting in we obtain
ik a a A a
n
y
i ik
W n
a a W y ik
n
y
W y k ik W a n ik
n
The term ik a a in the vorticity equation scales as For it to
have any in uence we need to b e of the same order as n in other
words the scale of variation of needs to b e no faster than n This can
only o ccur far from the wall where is constant and since derivatives of
are large we have at leading order
ik a a ik W y ik
k W k y
and the vorticity equation b ecomes
a k W k y Ay B
We de ne A A i ik and W W i ik to obtain
n
ik W y
n y A y Ay
n
W n
nW y
HIGHLY SHEAR THINNING LIMIT n
where
y k ik W W k y
with R e and we have applied the b oundary conditions at y to
obtain B This equation contains all the information ab out the leading
order dynamics as n Note that we have not yet made any assumption
ab out the size of except that j k j is not small
The b oundary conditions are
We have therefore shown that the n dynamics are controlled by
a second order ODE rather than the full fourth order ODE equa
tions
Physical meaning of the reduced equation
In order to b egin to understand the mechanism of instability we need to
know the physical origin of the four di erent terms in equation It
may b e re expressed as
ik W
n y Ay
where
y k ik W k N
CHAPTER CHANNEL FLOWS OF A WHITE METZNER FLUID
and
n
i ik y W n
i ik
We lo ok in detail at equation
The rst term n comes from the shear stress part of the vorticity
n
equation The shear stress is W so the p erturbation to it is
n
nW n The p erturbation to the shear rate is k
so the p erturbation shear stress is approximately n
The second term comes from the coupling of the evolution of a to
a Physically this coupling o ccurs even in the Maxwell limit n
and corresp onds to the e ect of the base state shear rate on the
p erturbation vorticity
The third term y is more complicated It contains contribu
tions from the inelastic part of the stress the W limit from the
material convection of A by the p erturbation velocity and from the
base state rst normal stress di erence N
Finally the constant A which came from integrating the vorticity equa
tion is the p erturbation pressure across the channel
The form of shows that there is a wall layer In the bulk of the uid
n
j i ik j y W n and convection is more imp ortant than relaxation
n
Close to the wall however the relaxation term y W n is of order n
and therefore imp ortant The balance of these two terms if is order unity
o ccurs at y n log n so there is a wall layer of width n log n
HIGHLY SHEAR THINNING LIMIT n
Numerical results for small n
We consider here the case in which k W i e the parameter values
for which we saw the most interesting b ehaviour in section We exp ect a
numerical solution of this mo del problem to show qualitatively similar results
to the full problem but we would only exp ect quantitative agreement in the
limit n Because the di erential equation is singular as n neither
problem is numerically soluble very close to this value so the quantitative
agreement cannot easily b e checked
Figure shows the b ehaviour of all the ro ots we have obtained for this
mo del problem as n varies There is an unstable mo de whose growth rate
increases as n decreases as we found for the full problem In this case it is
not clear whether the growth rate scales as a p ower of n or an exp onential
However this reduced problem is capable of pro ducing growth rates which
are clearly exp onential in n as shown in gure for a lower value of k
The convection b ehaviour of this ro ot is shown in gure the evidence
suggests that it is also nite as n
We also consider the dep endence of the growth rates on wavenumber as
we did in section Figures and show the growth and convection
rates of the two ro ots we have found plotted against the wavenumber for
the two cases n and n It should b e noted that the reduced
equation also has a continuous sp ectrum in this case at
n
i y
k y y
W n
CHAPTER CHANNEL FLOWS OF A WHITE METZNER FLUID
0.8
0.6
0.4 U
0.2
0
-0.2
I m -0.4
-0.6
-0.8 S -1
-1.2
0.1 0.15 0.2 0.25 0.3
Power law index n
Figure Plot of the growth rate against p ower law index of ro ots found numerically
for the reduced White Metzner equation W k as for the full case in gure
The ro ots are lab elled U and S for convenience only in gure we will see that ro ot S
can b e unstable
3.2
3
2.8
2.6
R e S
2.4
2.2 U
2
0.1 0.15 0.2 0.25 0.3
Power law index n
Figure Plot of the convective part of the eigenvalues shown in gure
HIGHLY SHEAR THINNING LIMIT n
2
1.5
1
I m 0.5
0
-0.5
0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
Power law index n
Figure Plot of the growth rate against p ower law index of ro ot S of the reduced
White Metzner equation where the mo de is unstable Here W and k The
p oints show the numerical results and the solid line is an exp onential t
2
1.8
1.6
1.4
R e
1.2
1
0.8
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
Power law index n
Figure Plot of the convection of the eigenvalue from gure There is no sign of
unbounded growth as n
CHAPTER CHANNEL FLOWS OF A WHITE METZNER FLUID
which is equivalent to
i
n k
W
This is the reason for the disapp earance of the more unstable ro ot as k
decreases
Because of the thin layers in the base state of this ow regions of size n
close to the wall the second order equation we have derived is not appro
priate for short waves with k n When the wavelength is short there
are two comp eting short lengthscales and we have not investigated this sit
uation Therefore the increase in growth rate in gure as k increases
should not b e a cause for concern the results are only valid for wavelengths
of the same order as the channel width i e k O
Conclusions
We have considered the stability of channel ow of a White Metzner uid
We nd numerical evidence of an instability when n This instability
is purely elastic in origin as there is no inertia in our calculation
The detailed mechanism for the instability remains to b e elucidated
There is no interface in the ow and the base state streamlines are not
curved so this is neither an interfacial nor a Taylor Couette instability
Nonetheless N is a crucial ingredient
For the smallest values of n investigated short waves are the most un
stable for larger n waves with length comparable with the channel width
grow fastest They convect with an intermediate region of the uid neither
CONCLUSIONS
0.8
0.6
0.4
0.2 U
0
-0.2
I m -0.4
-0.6 S -0.8
-1
-1.2
0 0.5 1 1.5 2 2.5 3
Wavenumber k
Figure Plot of the growth rate against wavenumber for two ro ots of the reduced
White Metzner equation The Weissenberg number is W and the solid lines show the
case n the p oints n
3.5 S 3
2.5
2
1.5
R e
1 U
0.5
0
0 0.5 1 1.5 2 2.5 3
Wavenumber k
Figure Plot of the convection rate of eigenvalues shown in gure
CHAPTER CHANNEL FLOWS OF A WHITE METZNER FLUID
the wall which is stationary nor the central region which has velocity in
dimensionless co ordinates and corresp onds to k
APPENDIX PRELIMINARY ASYMPTOTICS FOR n
App endix Preliminary asymptotics for
n !
In this app endix we present some preliminary working towards an asymp
totic solution of equation of section as n
In order to solve the equation we will have to assume a scaling for and
check our solution for self consistency at the end Motivated by the results
of section we assume that is O
Outer
n
Far from the wall where y n equation b ecomes
n y Ay
with b oundary condition
This is solved by a multiple scales metho d using fast scale y n and slow
frequency mo dulation scale y The leading order solution is
M y Ay
sinh y
y y n
for some constant M In order to determine the form of the co e cient of
sinh y y n we have worked to order n and applied a secularity
condition
The structure of this solution is exp onential and has parts which grow
as either y or y This is one reason why the numerical solution has
CHAPTER CHANNEL FLOWS OF A WHITE METZNER FLUID
problems for small values of n Whichever direction is chosen for the inte
gration an error can increase by a factor of exp n as the integration
takes place where R e k
Wall layer
n
The outer solution is valid for y n Therefore for the regions close to
n
the wall we will need y n so we put y n log x Then the region
n
y n The derivatives are n b ecomes x
d x
x
d
x
dy dx n
and
x x d
x
d d x d x d
x x x
x
dy dx n n n
Equation b ecomes
ik W xd A
x
x d xd n An
x
x
W x
nW x
where
k ik W W k
with R e The b oundary conditions are
We nd that the region in which this scaling applies divides into two
further regions
APPENDIX PRELIMINARY ASYMPTOTICS FOR n
Inner
First we solve the inner equation in the region closest to the wall x
Equation b ecomes
A
x d xd ik W xd
x x
x
W x
and we solve at leading order
A
ik W
x x d
x
W ik W
and hence
ik W
x A
x
W ik W ik W
This has no undetermined constant b ecause we have applied b oth the wall
b oundary conditions
Middle layer
The biggest term we neglected in the wall layer was of order nx so the next
scale is nx Equation b ecomes
d d ik W d d
d d
W
and therefore
ik W
G d
d
W
We cannot integrate this directly but rather give the form of as
Z
ik W
dt t
G G
t
W t
n
We have now solved the equation in three asymptotic regions the remain
ing task is to match these solutions together to determine
CHAPTER CHANNEL FLOWS OF A WHITE METZNER FLUID
Matching
Inner to middle layer
We match the wall region to the middle layer rst The co ordinates of the
two regions are x near the wall and in the middle region where nx
We match with an intermediate variable n x n where
The two functions to match b ecome
ik W ik W
A n
n
inner
W ik W ik W
and in the middle
Z
1
ik W
n
dt t
G G
middle
W i ik t t
n
Expanding we have
A A
ik W ik W
O n n
inner
ik W ik W ik W
and
ik W ik W
G n G n
ik W ik W
n G O n
middle
ik W ik W
We deduce that
ik W
An
G
W ik W
and
A
G
W ik W
APPENDIX PRELIMINARY ASYMPTOTICS FOR n
Outer to middle layer
In order to match the outer layer to the middle we need a variable which
will move b etween the scales of n and n log n This is complicated in itself
another complication is the fact that we do not know the full solution in the
middle region except as an integral
If we use instead the na ve metho d and take the limit as of the
middle layer we obtain
ik W
G W log G
middle
Z
ik W
t dt
ik W
G W log n
t
t W
n
which is to match against
M y Ay
sinh y
outer
y y n
Substituting the de nition y n log n we obtain
y
ik W
G W G
middle
n
Z
ik W
dt t
ik W
G W
t
t W
n
In theory we can extend this matching until is determined The di culty
is the app earance of logarithms here as noted by Van Dyke all the
logarithmic terms may have to b e included in the matching pro cess and we
have not satisfactorily resolved this di culty
CHAPTER CHANNEL FLOWS OF A WHITE METZNER FLUID
Chapter
Criterion for Co extrusion
Instability
CHAPTER CRITERION FOR COEXTRUSION INSTABILITY
Introduction
We have indicated in chapters and that even for a continuously strati
ed Oldroyd B uid the Lagrangian convection of material prop erties either
a time constant or a concentration is an essential ingredient for the devel
opment of the co extrusion instability In other words we claim that a
di erent uid in which material prop erties are not convected may display
the same base state velocity pro le and stress distribution and yet the un
stable co extrusion mo de may b e absent
We establish this p ossibility by reference to a sp eci c case In this chap
ter we use b oth the White Metzner formalism in which material prop erties
dep end on shear rate and the Oldroyd B formalism of chapters and in
which they are Lagrangian prop erties We de ne two uids with almost iden
tical pro les of velocity and stress in a channel ow and show the e ect of
the mo del type on the stability sp ectrum in section Section discusses
this e ect and its implications for constitutive mo delling
A mo del problem
We consider the channel ows of two di erent uids The rst is a generali
sation of the White Metzner uid of chapter and the second a combination
of the two Oldroyd B uids of chapters and
1
Mathematically the co extrusion mo de comes into existence at a singular p oint of the
governing equations The existence of this p oint dep ends on the presence of a material
derivative which is precisely a derivative with resp ect to a Lagrangian co ordinate This
derivative is only relevant where some material prop erty undergo es Lagrangian convection
A MODEL PROBLEM
White Metzner uid
The White Metzner uid as it is de ned in chapter applied to a two
dimensional channel ow exhibits a rst normal stress di erence N which
is quadratic in distance across the channel This prop erty is indep endent of
the relaxation function
In order to compare with a uid which shows the co extrusion instability
we need a uid which will show steep gradients in N across the channel
By adding a solvent viscosity a physically sensible mo di cation we can
reduce the e ect of on the shear stress and thus increase its e ect on
the normal stress The case constant b ecomes the simple Oldroyd B
uid whereas in chapter it gave us the UCM uid
In the notation of chapter this generalisation of the White Metzner
uid is given by C C W W may b e any monotonic
function
Oldroyd B uid
We want to de ne a uid whose material prop erties are convected but whose
base state velocity and stress pro le are identical to the generalised White
Metzner uid already sp eci ed ab ove Neither the mo del of chapter in
which the Weissenberg number is convected with the uid nor that of chap
2
In the p erturbation equations of section we will see that W only ever app ears
0
as part of C W and so we may without loss of generality set W Physically this
0 0 0
is simply a matter of taking the elastic timescale into the de nition of or equivalently
using a di erent nondimensionalisation for However for easy sp ecialisation to other
uids it is more convenient to leave b oth C and W to vary
0 0
CHAPTER CRITERION FOR COEXTRUSION INSTABILITY
ter in which the convected prop erty is p olymer concentration gives su
cient freedom
Therefore we de ne a generalised Oldroyd B mo del in which b oth C and
W are p ermitted to vary with Lagrangian distance across the channel C
W Both C and W will undergo Lagrangian convection In the formula
tion of chapter we set and W
Base state
White Metzner uid
The base state for channel ow of our generalised White Metzner uid is
jU j U
U U
A
A
U
P P C W P x
P P x C U W P y
A
P y P P x
and nally
C W U P y
Equation de nes which may only b e computed numerically for most
functions
CHOICE OF PARAMETERS
Oldroyd B uid
The base state for the generalised Oldroyd B uid b ecomes
WU WU
A
A
WU
P P C W P x
P P x CW U P y
A
P y P P x
C U P y
Choice of parameters
White Metzner uid
We use an arctan pro le for
arctan q
q and For small This gives us a mo del with parameters C W
values of C and q it is capable of giving high gradients of N
U P y
y jP y j W N C P
We therefore select the sp eci c mo del parameters used for demonstration
to b e C q W and
CHAPTER CRITERION FOR COEXTRUSION INSTABILITY
Oldroyd B uid
We choose the functions C and W for the Oldroyd type uid so that
its base state velocity pro le pressure and stresses are the same as those of
the White Metzner type uid
For the White Metzner uid the relevant quantities are
C W U P y
N C U W
P P C W P x
and for the Oldroyd uid
C U
N CW U
P P C W P x
Therefore for equivalence b etween the two uids in the base state we
need U and the stresses to b e the same in each and so
W
and
C C W W
We integrate the White Metzner form numerically to nd as a function
of y and hence nd y which should b e equal to We obtain a go o d
approximate t shown in gure with
y tanh y
PERTURBATION EQUATIONS
Perturbation equations
White Metzner uid
Referring back to equations of chapter the p erturbation
equations for the generalised White Metzner uid b ecome
ik
ik
C
0
p ik a
W
0
C
0
a k
W 0
4
3.8
3.6
3.4
3.2
3
0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7
y
Figure Plot of across the channel p oints with the analytic function y solid line
CHAPTER CRITERION FOR COEXTRUSION INSTABILITY
C
0
a p ik
W
0
a i ik U
ik A A A ik U a A
a ik A A k U a i ik U A
i ik U a ik A k
k
These may b e solved numerically in the same way as those in chapters
and
Oldroyd B uid
We use the advected form section of the p erturbation equations and
note that C W C W to obtain
ik ik P y fCW C g
ik
NUMERICAL RESULTS
C
p ik a
W
C
a f P y C g k
W
C
p ik a
W
W P y
a ik i ik U
W C
A f P y C g A ik U a
WP
a ik i ik U
W C
f P y C g A k U a
a ik A k i ik U
W
i ik U ik
Numerical results
The two uids we have chosen app ear identical in terms of their base ow
stresses Figures and show part of their stability sp ectra for varicose
mo des In each case the only mo des not shown here are more stable than
the ones shown and may safely b e ignored It may b e seen that the Oldroyd
uid has an unstable mo de for mo derate values of k as discussed in chapters
CHAPTER CRITERION FOR COEXTRUSION INSTABILITY
0.05
0
-0.05
-0.1
I m -0.15
-0.2
-0.25
-0.3
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Wavenumber k
Figure The least stable part of the sp ectra of the generalised uids The solid lines are
for the Oldroyd type uid the horizontal line is part of its continuous sp ectrum and the
p oints are for the White Metzner type uid Only the Oldroyd uid shows the unstable mo de
-0.24
-0.245
-0.25
-0.255
-0.26
-0.265
I m -0.27
-0.275
-0.28
-0.285
-0.29
-0.295
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Wavenumber k
Figure A close up of the lower region of gure demonstrating the close corresp on
dence b etween the two sp ectra
CONCLUSIONS
and No such mo de exists for the White Metzner uid which is stable to
p erturbations of all wavenumbers
The stable parts of the sp ectra are very similar as shown in the close up
in gure This indicates that if a more blurred pro le were chosen for
which the Oldroyd co extrusion mo de ceases to exist as shown in chapters
and to all extents and purp oses the two mo dels would b e equivalent
Conclusions
The extra eigenvalue in the Oldroyd sp ectrum compared with the White
Metzner sp ectrum indicates a fundamental di erence b etween the two uids
In a simple Poiseuille ow the former is unstable and the latter stable
The presence of the convective derivative i ik U in equation
of the Oldroyd mo del p ermits a singularity in the equations at which a
neutral mo de may app ear It is this mo de discussed in chapter which
b ecomes unstable The White Metzner mo del has no such term and hence
no such singularity and no such neutral mo de As a result it cannot exhibit
the co extrusion form of instability
These two uids suggest that two criteria must b e met for this form of
instability First there must b e su ciently steep gradients in some physical
quantity in a thin region of base ow This region might b e a vortex sheet
or a layer of steep gradients of stress For illustration we treat it as a region
of large stress gradients as in the uid of chapter
In general the states of stress will b e di erent on the two sides of this
small region Let us assume that these two states of stress are such that the
CHAPTER CRITERION FOR COEXTRUSION INSTABILITY
interface b etween two uids owing with these stresses would b e unstable
For the ows of chapter this corresp onds to choosing W and W so that
the corresp onding interfacial ow of chapter would b e unstable
Second we require that the material prop erties b e advected with the ow
It follows that the existence of an instability is not dictated by the base state
alone but dep ends also on the constitutive b ehaviour Supp ose for example
that the uid p ossesses a second timescale T over which the uid relaxes
some material prop erty
The Oldroyd limit
In this limit a material parcel retains its value of W throughout its
life Therefore the timescale on which W changes for any uid parcel
is e ectively in nite T
The White Metzner limit
In this case on the other hand W W i e a parcel s Weissenberg
number is an instantaneous function of the applied rate of strain W has
no memory of past values but rather reacts to the ow on a timescale
T
The White Metzner uid for which T is stable where the Oldroyd
uid for which T is not This may suggest a critical value of T b elow
which the co extrusion instability do es not exist This second timescale T
is an imp ortant prop erty for comparison of di erent constitutive mo dels and
one which has not b een considered b efore
Chapter
Conclusions
CHAPTER CONCLUSIONS
The fo cus of this dissertation is to examine the stability of planar channel
ows of visco elastic liquids
We have shown theoretically the existence of two purely elastic instabil
ities whose mechanisms dep end neither on the presence of an interface nor
on streamline curvature We do not b elieve that these instabilities have b een
identi ed b efore
The rst instability dep ends on advection of material prop erties and is
an extension of and shares a mechanism with the interfacial instability that
arises in co extrusion chapter
The second instability introduced in chapter is predicted to o ccur in
a highly shear thinning White Metzner uid and is discussed b elow along
with suggestions for future work to elucidate the mechanism
Co extrusion instability
For co extrusion of two uids our theoretical understanding of the linear
instability problem is now more or less complete Earlier work had
identi ed a long wave instability mechanism which we have now extended
to short waves and numerically to waves of all wavelengths
We have p osed the question as to whether the same mechanism can op er
ate when uid prop erties are continuously strati ed It can provided there
remain su ciently sharp gradients in normal stress and provided material
prop erties are advected by the ow
The way in which blurring of an interface is able to stabilise a ow has
b een identi ed For a two uid case exp eriments have shown the instability
that we have discussed Further exp eriments are needed for comparison with
CHAPTER CONCLUSIONS
the continuously strati ed case that we have identi ed for example with
p olymer solutions having cross channel variations in concentration
Shear thinning instability
In chapter we considered a highly shear thinning White Metzner uid
and found that for su ciently large degree of shear thinning p ower law
co e cient n there is an instability provided that the normal stress
N do es not shear thin more strongly than the shear viscosity
This instability is not driven by steep gradients in normal stress b ecause
the pro le of N across a slit is quadratic for such a uid However the
presence of normal stress is essential to the instability mechanism
In the long wave limit we have shown that the mo de restabilises even
for small values of n comparable with the wavenumber Thus the mechanism
dep ends on having waves with wavelengths comparable with the channel
width and shorter
Several extensions to this theory would b e desirable First the White
Metzner mo del though commonly used for p olymer melt ows particularly
shear ows has a weak physical foundation and the extent to which the
conclusions are functions of the mo del chosen needs to b e explored Second
the p ower law mo del leads to a spurious non smo othness in uid prop erties
at the channel centre We have therefore explored only varicose instability
mo des for which the singularity disapp ears Sinuous mo des may well turn
out to b e less stable Third the technical task of extending the analysis
to axisymmetric ows of greater practical imp ortance should b e undertaken
Channel ows are directly relevant to cylindrical extrusion which is used
CHAPTER CONCLUSIONS
in the creation of p olymeric bres Fourth and most imp ortant exp eri
ments are needed to examine the stability of channel ows for uids whose
viscometric prop erties thin but whose normal stresses remain signi cant at
high shear rates Our analysis suggests that the class of extrusion instabil
ity called wavy instability may originate dynamically in the extruder rather
than b eing a consequence of stick slip as is often supp osed
Nomenclature
Roman characters
A a Polymer stretch
C c Polymer concentration
D Derivative with resp ect to y
d Width of thin layer in a blurred pro le
E Rate of strain tensor
E Exp onential integral
F Spring force in a dumbbell
F Bo dy force
G Elastic mo dulus
H Heaviside function
I Identity matrix
k k Dimensional and dimensionless wavenumber
L Channel half width
N Winding number
N N First and second normal stress di erences
n Power law co e cient
NOMENCLATURE
P p Pressure
P Dimensionless pressure gradient
Q Velocity integral
q Dimensionless width of thin layer in blurred pro le
q Rescaling of q by k where k is small
R Vector representation of dumbbell spring
t Time
T Secondary timescale
T T Dimensional and dimensionless surface tension co e cients
Transpose of a matrix
U Base x velocity at channel centreline
U u v Velocities
W Weissenberg number
W Wall Weissenberg number
x y Eulerian co ordinates
Greek characters
Factor for evolution of a advection and relaxation
Dimensional and dimensionless shear rate
Small quantity
Perturbation to interface or streamline Stokes drag co e cient
Lagrangian co ordinate Shear viscosity
Position of uid uid interface
Fluid relaxation time
NOMENCLATURE
Parameter k W n in the small k small n expansion of chapter
Indicator function for presence of solvent viscosity
Typical shear viscosity
Fluid density
Stress tensor
Dimensionless relaxation time
Streamfunction
Complex frequency
NOMENCLATURE
Bibliography
A C T Aarts and A A F van de Ven Transient b ehaviour and stability
p oints of the Poiseuille ow of a KBKZ uid Journal of Engineering
Mathematics
K E P Adewale O Olabisi A A Oyediran and V O S Olunloyo A stick
slip problem related to melt fracture in p olymer pro cessing op erations
International Polymer Processing
H Andrianarahinjaka and P Micheau Instability phenomena in lm
blowing pro cess a linear stability analysis Comptes Rendus de
l Acad emie des Sciences S erie II Fascicule B M ecanique Physique
Chimie Astronomie
M T Arigo D Ra jagopalan N Shapley and G H McKinley The sedi
mentation of a sphere through an elastic uid Steady motion Jour
nal of Non Newtonian Fluid Mechanics
B T Atwoo d Wal l slip and extrudate distortion of high density
polyethylene PhD thesis Department of Chemical Engineering Prince
ton University Princeton
BIBLIOGRAPHY
E B Bagley and A M Birks Flow of p olyethylene into a capillary
Journal of Applied Physics
E B Bagley S H Storey and D C West Post extrusion swelling of
p olyethylene Journal of Applied Polymer Science
R Y Bai K P Chen and D D Joseph Lubricated pip elining stabil
ity of core annular ow Exp eriments and comparison with theory
Journal of Fluid Mechanics
T F Ballenger I J Chen J W Crowder G E Hagler D C Bogue and
J L White Polymer melt ow instabilities in extrusion investigation
of the mechanism and material and geometric variables Transactions
of the Society of Rheology
T F Ballenger and J L White The development of the velocity eld
in p olymer melts in a reservoir approaching a capillary die Journal of
Applied Polymer Science
B M Baumert and S J Muller Flow visualisation of the elastic Taylor
Couette instability in Boger uids Rheologica Acta
J Becker P Bengtsson C Klason J Kubat and P Saha Pressure
oscillations during capillary extrusion of high density p olyethylene In
ternational Polymer Processing
J J Benbow and P Lamb New asp ects of melt fracture Society of
Petroleum Engineers Transactions
BIBLIOGRAPHY
T B Benjamin Wave formation in laminar ow down an inclined plane
Journal of Fluid Mechanics
A N Beris and M Avgousti Visco elastic ow instabilities inception
and non linear evolution In P Moldenaers and R Keunings editors
Theoretical and Applied Rheology Proceedings of the X International
Congress on Rheology volume pages Belgium Elsevier
B Bernstein E A Kearsley and L J Zapas A study of stress relaxation
with nite strain Transactions of the Society of Rheology
R B Bird R C Armstrong and O Hassager Dynamics of Polymeric
Liquids volume Fluid Mechanics John Wiley and Sons Inc
R B Bird O Hassager R C Armstrong and C F Curtiss Dynamics of
Polymeric Liquids volume Kinetic Theory John Wiley and Sons
Inc
R B Bird and J M Wiest Constitutive equations for p olymeric liquids
Annual Review of Fluid Mechanics
C Bisgaard Velocity elds around spheres and bubbles investigated by
laser doppler anemometry Journal of Non Newtonian Fluid Mechan
ics
D V Boger A highly elastic constant viscosity uid Journal of Non
Newtonian Fluid Mechanics
BIBLIOGRAPHY
W F Busse Two decades of high p olymer physics a survey and
forecast Physics Today September
J A B Byars A Oztekin R A Brown and G H McKinley Spiral
instabilities in the ow of highly elastic uids b etween rotating parallel
disks Journal of Fluid Mechanics
P J Cable and D V Boger A comprehensive exp erimental investigation
of tubular entry ow of visco elastic uids I Vortex characteristics
in stable ow American Institute of Chemical Engineering Journal
P J Cable and D V Boger A comprehensive exp erimental investigation
of tubular entry ow of visco elastic uids I I The velocity eld in stable
ow American Institute of Chemical Engineering Journal
P J Cable and D V Boger A comprehensive exp erimental investigation
of tubular entry ow of visco elastic uids I I I Unstable ow American
Institute of Chemical Engineering Journal
J J Cain and M M Denn Multiplicities and instabilities in lm blowing
Polymer Engineering and Science
P J Carreau Rheological equations from molecular network theories
PhD thesis University of Wisconsin Madison
B W Char K O Geddes G H Gonnet B L Leong M B Monagan and
S M Watt Maple V Library Language Reference Manual Springer
Verlag and Waterloo Maple Publishing
BIBLIOGRAPHY
A Chawda and M Avgousti Stability of visco elastic ow b etween ec
centric rotating cylinders Journal of Non Newtonian Fluid Mechanics
I J Chen G E Hagler L E Abb ot D C Bogue and J L White In
terpretation of tensile and melt spinning exp eriments on low density
and high density p olyethylene Transactions of the Society of Rheology
K P Chen Elastic instability of the interface in Couette ow of
visco elastic liquids Journal of Non Newtonian Fluid Mechanics
K P Chen Interfacial instability due to elastic strati cation in concen
tric co extrusion of two visco elastic uids Journal of Non Newtonian
Fluid Mechanics
K P Chen Short wave instability of core annular ow Physics of
Fluids A
K P Chen Hyp erb olicity and change of type in steady co extrusion
ow of Upp er Convected Maxwell uids Journal of Non Newtonian
Fluid Mechanics
K P Chen and D D Joseph Long wave and lubrication theories for
core annular ow Physics of Fluids A
K P Chen and D D Joseph Elastic short wave instability in extrusion
ows of visco elastic liquids Journal of Non Newtonian Fluid Mechan
ics
BIBLIOGRAPHY
K P Chen and Y Zhang Stability of the interface in co extrusion ow
of two visco elastic uids through a pip e Journal of Fluid Mechanics
M D Chilcott and J M Rallison Creeping ow of dilute p olymer so
lutions past cylinders and spheres Journal of Non Newtonian Fluid
Mechanics
C Chmielewski and K Jayamaran Elastic instability in cross ow of
p olymer solutions through p erio dic arrays of cylinders Journal of Non
Newtonian Fluid Mechanics
R E Christensen Extrusion coating of p olypropylene Society of
Petroleum Engineers Journal July
P L Clegg Elastic e ects in the extrusion of p olythene In P Mason
and N Wookey editors The Rheology of Elastomers pages
Pergamon Press New York
F N Cogswell J G H Gray and D A Hubbard Rheology of a uid
whose b ehavior approximates to that of a Maxwell b o dy Bul letin of
the British Society of Rheology
F N Cogswell and P Lamb The mechanism of melt distortion Plastics
and Polymers
B D Coleman and W Noll An approximation theorem for function
als with applications to continuum mechanics Archive for Rational
Mechanics and Analysis
BIBLIOGRAPHY
W O Criminale Jr J L Ericksen and G L Filb ey Jr Steady shear ow
of non Newtonian uids Archive for Rational Mechanics and Analysis
P K Currie Calculations on the Doi Edwards mo del for concentrated
p olymer systems In G Astarita G Marrucci and L Nicolais editors
Rheology Volume Principles Pro ceedings of the Eighth Interna
tional Congress on Rheology pages Naples Italy
J P Decrupp e R Cressely R Makhlou and E Capp elaere Flow bire
fringence exp eriments showing a shear banding structure in a CTAB
solution Col loid and Polymer Science
J L den Otter Mechanisms of melt fracture Plastics and Polymers
J L den Otter Some investigations of melt fracture Rheologica Acta
M Doi and S F Edwards Dynamics of concentrated p olymer systems
Brownian motion in equilibrium state Journal of the Chemical Society
Faraday Transactions II P
M Doi and S F Edwards Dynamics of concentrated p olymer systems
Molecular motion under ow Journal of the Chemical Society Faraday
Transactions II P
M Doi and S F Edwards Dynamics of concentrated p olymer systems
Constitutive equation Journal of the Chemical Society Faraday
Transactions II P
BIBLIOGRAPHY
M Doi and S F Edwards Dynamics of concentrated p olymer systems
Rheological prop erties Journal of the Chemical Society Faraday
Transactions II P
G J Donnelly and C B Weinberger Stability of isothermal b er spin
ning of a Newtonian uid Industrial and Engineering Chemistry Fun
damentals
I M Dris and E S G Shaqfeh On purely elastic instabilities in eccentric
cylinder ows Journal of Non Newtonian Fluid Mechanics
J Dunwoo dy and D D Joseph Systematic linearisation for stability of
shear ows of visco elastic uids Archive for Rational Mechanics and
Analysis
N El Kissi and J M Piau The di erent capillary ow r egimes of en
tangled p olydimethylsiloxane p olymers macroscopic slip at the wall
hysteresis and cork ow Journal of Non Newtonian Fluid Mechanics
N El Kissi and J M Piau Adhesion of linear low density p olyethylene
for ow regimes with sharkskin Journal of Rheology
N El Kissi J M Piau and F Toussaint Sharkskin and cracking of
p olymer melt extrudates Journal of Non Newtonian Fluid Mechanics
BIBLIOGRAPHY
P Espa nol X F Yuan and R C Ball Shear banding ow in the Johnson
Segalman uid Journal of Non Newtonian Fluid Mechanics
R E Evans and K Walters Flow characteristics asso ciated with abrupt
changes in geometry in the case of highly elastic liquids Journal of
Non Newtonian Fluid Mechanics
R E Evans and K Walters Further remarks on the lip vortex mech
anism of vortex enhancement in planar contraction ows Journal of
Non Newtonian Fluid Mechanics
J D Ferry Viscoelastic properties of polymers John Wiley Sons
Inc second edition
R J Fisher and M M Denn Draw resonance in melt spinning Applied
Polymer Symposium
G C Georgiou and M J Cro chet Time dep endent compressible ex
trudate swell problem with slip at the wall Journal of Rheology
R Ghisellini Elastic free energy of an Upp er Convected Maxwell uid
undergoing a fully developed planar Poiseuille ow a variational re
sult Journal of Non Newtonian Fluid Mechanics
H Giesekus Die Elastizit at von Fl ussigkeiten Rheologica Acta
BIBLIOGRAPHY
H Giesekus On the stability of ow of visco elastic uids Rheologica
Acta
H Giesekus Nonlinear e ects in the ow of visco elastic uids in slits
and holes Rheologica Acta
V A Goro dtsov and A I Leonov On a linear instability of a plane par
allel Couette ow of visco elastic uid Journal of Applied Mathematics
and Mechanics
A Groisman and V Steinberg Solitary vortex pairs in visco elastic
Couette ow Physical Review Letters
A S Gupta Stability of a visco elastic liquid lm owing down an
inclined plane Journal of Fluid Mechanics
C D Han and J Y Park Studies on blown lm extrusion Bubble
instability Journal of Applied Polymer Science
O G Harlen E J Hinch and J M Rallison Birefringent pip es the
steady ow of a dilute p olymer solution near a stagnation p oint Jour
nal of Non Newtonian Fluid Mechanics
O J Harris and J M Rallison Instabilities of stagnation p oint ow of a
dilute p olymer solution Journal of Non Newtonian Fluid Mechanics
S G Hatzikiriakos The onset of wall slip and sharkskin melt fracture
in capillary ow Polymer Engineering and Science
BIBLIOGRAPHY
S G Hatzikiriakos and J M Dealy Wall slip of molten high density
p olyethylene Sliding plate rheometer studies Journal of Rheology
E J Hinch O J Harris and J M Rallison The instability mechanism
for two elastic liquids b eing co extruded Journal of Non Newtonian
Fluid Mechanics
T C Ho and M M Denn Stability of plane Poiseuille ow of a highly
elastic liquid Journal of Non Newtonian Fluid Mechanics
A P Ho op er Long wave instability at the interface b etween two viscous
uids thin layer e ects Physics of Fluids
D Y Hsieh Mechanism for instability of uid ow down an inclined
plane Physics of Fluids A Fluid Dynamics
M A Hulsen and J P P M van der Zanden Numerical simulation of
contraction ows using a multimode Giesekus mo del Journal of Non
Newtonian Fluid Mechanics
T W Huseby Hyp othesis on a certain ow instability in p olymer melts
Transactions of the Society of Rheology
J F Hutton Fracture and secondary ow of elastic liquids Rheologica
Acta
BIBLIOGRAPHY
H Ishihara and S Kase Studies on melt spinning Simulation of
draw resonance using Newtonian and p ower law viscosities Journal of
Applied Polymer Science
K P Jackson K Walters and R W Williams Rheometrical study of
Boger uids Journal of Non Newtonian Fluid Mechanics
C C Ji J C Yang and W S Lee Stability of Newtonian PTT co ex
trusion b er spinning Polymer Engineering and Science
M W Johnson Jr and D Segalman Mo del for visco elastic uid b e
havior which allows non a ne deformation Journal of Non Newtonian
Fluid Mechanics
Y L Jo o and E S G Shaqfeh Visco elastic Poiseuille ow through a
curved channel a new elastic instability Physics of Fluids A Fluid
Dynamics
Y L Jo o and E S G Shaqfeh The e ects of inertia on the visco elastic
Dean and Taylor Couette ow instabilities with application to coating
ows Physics of Fluids A Fluid Dynamics
Y L Jo o and E S G Shaqfeh A purely elastic instability in Dean and
Taylor Dean ow Physics of Fluids A Fluid Dynamics
BIBLIOGRAPHY
Y L Jo o and E S G Shaqfeh Observations of purely elastic instabilities
in the Taylor Dean ow of a Boger uid Journal of Fluid Mechanics
D D Joseph Steep wave fronts on extrudates of p olymer melts and
solutions Lubrication layers and b oundary lubrication Journal of
Non Newtonian Fluid Mechanics
F Kang and K P Chen Nonlinear elastic instability of gravity driven
ow of a thin visco elastic lm down an inclined plane Journal of
Non Newtonian Fluid Mechanics
S Kase T Matsuo and Y Yoshimoto Theoretical analysis of melt
spinning Part Surging phenomena in extrusion casting of plastic
lms Seni Kikai Gakkai Shi Japanese Journal of Textile Machinery
A Kaye Non Newtonian ow in incompressible uids Part A
general rheological equation of state Note College of Aeronautics
Cran eld UK
T A Kazachenko Instability of the interface b etween two highly elastic
liquids in a vertical cylindrical channel Zhurnal Fizicheskoi Khimii
M Keentok A G Georgescu A A Sherwoo d and R I Tanner The
measurement of the second normal stress di erence for some p olymer
solutions Journal of Non Newtonian Fluid Mechanics
BIBLIOGRAPHY
B Khomami Interfacial stability and deformation of two strati ed
p ower law uids in plane Poiseuille ow Stability analysis Journal
of Non Newtonian Fluid Mechanics
B Khomami Interfacial stability and deformation of two strati ed
p ower law uids in plane Poiseuille ow Interface deformation Jour
nal of Non Newtonian Fluid Mechanics
B Khomami and L D Moreno Stability of visco elastic ow around
p erio dic arrays of cylinders Rheologica Acta
B Khomami and M M Ranjbaran Exp erimental studies of interfa
cial instabilities in multilayer pressure driven ow of p olymeric melts
Rheologica Acta
K W Ko elling and R K Prud homme Instabilities in multi hole con
verging ow of visco elastic uids Rheologica Acta
J W H Kolnaar and A Keller A temp erature window of reduced ow
resistance in p olyethylene with implications for melt ow rheology
Implications for ow instabilities and extrudate distortion Polymer
R J Ko opmans and J Molenaar The sharkskin e ect in p olymer
extrusion Polymer Engineering and Science
W M Kulicke H E Jeb erien H Kiss and R S Porter Visual obser
vation of ow irregularities in p olymer solutions at theta conditions
Rheologica Acta
BIBLIOGRAPHY
W M Kulicke and R S Porter Irregularities in steady ow for non
Newtonian uids b etween cone and plate Journal of Applied Polymer
Science
S J Kurtz Die geometry solutions to sharkskin melt fracture In
B Mena A Garcia Rejon and C Rangel Nafaile editors Advances in
Rheology volume Polymers pages Universidad Nacional
Aut onoma de M exico Mexico City
W Lai Stability of an elastico viscous liquid lm owing down an
inclined plane Physics of Fluids
P Lamb Analysis of fabrication pro cesses In Advances in Polymer
Science and Technology number in So ciety of Chemical Industry
monograph series pages So ciety of Chemical Industry Lon
don England
R G Larson Constitutive equations for polymer melts and solutions
Butterworths Boston Massachusetts
R G Larson Instabilities in visco elastic ows Rheologica Acta
R G Larson S J Muller and E S G Shaqfeh The e ect of uid rheology
on the elastic Taylor Couette instability Journal of Non Newtonian
Fluid Mechanics
R G Larson E S G Shaqfeh and S J Muller A purely elastic instability
in Taylor Couette ow Journal of Fluid Mechanics
BIBLIOGRAPHY
H M Laun Description of the nonlinear shear b ehavior of low density
p olyethylene melt by means of an exp erimentally determined strain
dep endent memory function Rheologica Acta
P Laure H Le Meur Y Demay J C Saut and S Scotto Linear stability
of multilayer plane Poiseuille ows of Oldroyd B uids Journal of Non
Newtonian Fluid Mechanics
H Le Meur Non uniqueness and linear stability of the one dimensional
ow of multiple visco elastic uids RAIRO Mathematical Mod
el ling and Numerical Analysis Modelisation Math ematique et Analyse
Num erique
B Lee and J L White An exp erimental study of rheological prop erties
of p olymer melts in laminar shear ow and of interface deformation
and its mechanisms in two phase strati ed ow Transactions of the
Society of Rheology
C S Lee B C Tripp and J J Magda Do es N or N control the onset
of edge fracture Rheologica Acta
K C Lee and B A Finlayson Stability of plane Poiseuille and Couette
ow of a Maxwell uid Journal of Non Newtonian Fluid Mechanics
W S Lee Linear stability of a concentric two phase b er spinning
Journal of the Chinese Institute of Chemical Engineers
BIBLIOGRAPHY
W S Lee and C W Park An exp erimental study of a bicomp onent b er
spinning International Polymer Processing
C H Li Stability of two sup erp osed elasticoviscous liquids in plane
Couette ow Physics of Fluids
F J Lim and W R Schowalter Pseudo sp ectral analysis of the stabil
ity of pressure driven ow of a Giesekus uid b etween parallel planes
Journal of Non Newtonian Fluid Mechanics
F J Lo ckett On Squire s theorem for visco elastic uids International
Journal of Engineering Science
J M Lupton and R W Regester Melt ow of p olyethylene at high rates
Polymer Engineering and Science
M J MacDonald and S J Muller Shear rheology of p olymer solu
tions near the critical condition for elastic instability Rheologica Acta
J J Magda and S G Baek Concentrated entangled and semidilute
entangled p olystyrene solutions and the second normal stress di erence
Polymer
J J Magda and R G Larson A transition o ccurring in ideal elastic
liquids during shear ow Journal of Non Newtonian Fluid Mechanics
J J Magda J Lou S G Baek and K L de Vries Second normal stress
di erence of a Boger uid Polymer
BIBLIOGRAPHY
R Makhlou J P Decrupp e A Aitali and R Cressely Rheo optical
study of worm like micelles undergoing a shear banding ow Euro
physics Letters
M A Matovich and J R A Pearson Spinning a molten thread
line steady state isothermal viscous ows Industrial and Engineering
Chemistry Fundamentals
H Mavridis and R N Shro Multilayer extrusion exp eriments and
computer simulation Polymer Engineering and Science
G H McKinley R C Armstrong and R A Brown The wake instabil
ity in visco elastic ow past con ned circular cylinders Philosophical
Transactions of the Royal Society of London Series A Physical Sci
ences and Engineering
G H McKinley J A Byars R A Brown and R C Armstrong Observa
tions on the elastic instability in cone and plate and parallel plate ows
of a p olyisobutylene Boger uid Journal of Non Newtonian Fluid Me
chanics
G H McKinley A Oztekin J A Byars and R A Brown Self similar
spiral instabilities in elastic ows b etween a cone and a plate Journal
of Fluid Mechanics
G H McKinley P Pakdel and A Oztekin Rheological and geometric
scaling of purely elastic ow instabilities Journal of Non Newtonian
Fluid Mechanics
BIBLIOGRAPHY
G H McKinley W P Raiford R A Brown and R C Armstrong Non
linear dynamics of visco elastic ow in axisymmetrical abrupt contrac
tions Journal of Fluid Mechanics
T C B McLeish and R C Ball A molecular approach to the spurt e ect
in p olymer melt ow Journal of Polymer Science part B Polymer
Physics
J C Miller Swelling b ehaviour in extrusion Society of Petroleum
Engineers Transactions
W Minoshima and J L White Instability phenomena in tubular
lm and melt spinning of rheologically characterized high density
low density and linear low density p olyethylenes Journal of Non
Newtonian Fluid Mechanics
H K Mo att Viscous and resistive eddies near a sharp corner Journal
of Fluid Mechanics
R H Moynihan D G Baird and R Ramanathan Additional ob
servations on the surface melt fracture b ehavior of linear low density
p olyethylene Journal of Non Newtonian Fluid Mechanics
S J Muller R G Larson and E S G Shaqfeh A purely elastic transition
in Taylor Couette ow Rheologica Acta
H K Nason A high temp erature high pressure rheometer for plastics
Journal of Applied Physics
BIBLIOGRAPHY
J A Odell A J M uller and A Keller Non Newtonian b ehaviour of
hydrolysed p olyacrylamide in strong elongational ows a transient
network approach Polymer
D O Olagunju Elastic instabilities in cone and plate ow small gap
theory Zeitschrift f ur Angewandte Mathematik und Physik
J G Oldroyd On the formulation of rheological equations of state
Proceedings of the Royal Society A
Y Oyanagi A study of irregular ow b ehaviour of high density
p olyethylene Applied Polymer Symposium
A Oztekin B Alakus and G H McKinley Stability of planar stagnation
ow of a highly visco elastic uid Journal of Non Newtonian Fluid
Mechanics
A Oztekin and R A Brown Instability of a visco elastic uid b etween
rotating parallel disks analysis for the Oldroyd B uid Journal of
Fluid Mechanics
A Oztekin R A Brown and G H McKinley Quantitative prediction
of the visco elastic instability in cone and plate ow of a Boger uid
using a multimode Giesekus mo del Journal of Non Newtonian Fluid
Mechanics
P Pakdel and G H McKinley Elastic instability and curved streamlines
Physical Review Letters
BIBLIOGRAPHY
J R A Pearson and M A Matovich Spinning a molten threadline sta
bility Industrial and Engineering Chemistry Fundamentals
J R A Pearson and C J S Petrie On melt ow instability of extruded
p olymers In R E Wetton and R W Whorlow editors Polymer Systems
Deformation and Flow pages Macmillan London England
A Peterlin Ein u der endlichen Molek ull ange auf die Gradientenab
h angigkeit des Staudinger Index Makromolekulare Chimie
C J S Petrie and M M Denn Instabilities in p olymer pro cessing Amer
ican Institute of Chemical Engineering Journal
N Phan Thien Coaxial disk ow of an Oldroyd B uid exact solution
and stability Journal of Non Newtonian Fluid Mechanics
N Phan Thien and R I Tanner A new constitutive equation derived
from network theory Journal of Non Newtonian Fluid Mechanics
J M Piau N El Kissi and A Mezghani Slip ow of p olybutadiene
through uorinated dies Journal of Non Newtonian Fluid Mechanics
BIBLIOGRAPHY
J M Piau N El Kissi F Toussaint and A Mezghani Distortions of
p olymer melt extrudates and their elimination using slipp ery surfaces
Rheologica Acta
J M Piau N El Kissi and B Tremblay In uence of upstream insta
bilities and wall slip on melt fracture and sharkskin phenomena dur
ing silicones extrusion through ori ce dies Journal of Non Newtonian
Fluid Mechanics
A Pinarbasi and A Liakopoulos The e ect of variable viscosity on
the interfacial stability of two layer Poiseuille ow Physics of Fluids
A Pinarbasi and A Liakopoulos Stability of two layer Poiseuille ow of
Carreau Yasuda and Bingham like uids Journal of Non Newtonian
Fluid Mechanics
G Pomar S J Muller and M M Denn Extrudate distortions in
linear low density p olyethylene solutions and melt Journal of Non
Newtonian Fluid Mechanics
L Preziosi K P Chen and D D Joseph Lubricated pip elining stability
of core annular ow Journal of Fluid Mechanics
L Preziosi and S Rionero Energy stability of steady shear ows
of a visco elastic uid International Journal of Engineering Science
A V Ramamurthy Wall slip in viscous uids and in uence of materials
of construction Journal of Rheology
BIBLIOGRAPHY
M Reiner A mathematical theory of dilatancy American Journal of
Mathematics
M Renardy and Y Y Renardy Linear stability of plane Couette ow of
an Upp er Convected Maxwell uid Journal of Non Newtonian Fluid
Mechanics
Y Y Renardy Instability at the interface b etween two shearing uids
in a channel Physics of Fluids
Y Y Renardy The thin layer e ect and interfacial stability in a two
layer Couette ow with similar liquids Physics of Fluids
Y Y Renardy Stability of the interface in two layer Couette ow of
Upp er Convected Maxwell liquids Journal of Non Newtonian Fluid
Mechanics
Y Y Renardy Spurt and instability in a two layer Johnson Segalman
liquid Theoretical and Computational Fluid Dynamics
Y Y Renardy Weakly nonlinear b ehaviour of p erio dic disturbances
in two layer plane channel ow of Upp er Convected Maxwell liquids
Journal of Non Newtonian Fluid Mechanics
R S Rivlin The hydrodynamics of non Newtonian uids Proceedings
of the Royal Society A
BIBLIOGRAPHY
R S Rivlin and J L Ericksen Stress deformation relations for isotropic
materials Journal of Rational Mechanics and Analysis
W J Sang Interfacial instabilities in plane Poiseuille ow of two strat
i ed visco elastic uids with heat transfer Evolution equation and
stability analysis Journal of Fluid Mechanics
G Schleiniger and R J Weinacht Steady Poiseuille ows for a Giesekus
uid Journal of Non Newtonian Fluid Mechanics
W R Schowalter The b ehaviour of complex uids at solid b oundaries
Journal of Non Newtonian Fluid Mechanics
E S G Shaqfeh Purely elastic instabilities in viscometric ows Annual
Review of Fluid Mechanics
E S G Shaqfeh R G Larson and G H Fredrickson The stability
of gravity driven visco elastic lm ow at low to mo derate Reynolds
number Journal of Non Newtonian Fluid Mechanics
E S G Shaqfeh S J Muller and R G Larson The e ects of gap
width and dilute solution prop erties on the visco elastic Taylor Couette
instability Journal of Fluid Mechanics
J D Shore D Ronis L Piche and M Grant Sharkskin texturing in
stabilities in the ow of p olymer melts Physica A
BIBLIOGRAPHY
J D Shore D Ronis L Piche and M Grant Theory of melt fracture
instabilities in the capillary ow of p olymer melts Physical Review E
B
N Sombatsompop and A K Woo d Flow analysis of natural rubb er in
a capillary rheometer Flow patterns and entrance velocity pro les
in the die Polymer Engineering and Science
N A Sp enley M E Cates and T C B McLeish Nonlinear rheology of
wormlike micelles Physical Review Letters
H B Squire On the stability for three dimensional disturbances of vis
cous uid ow b etween parallel walls Proceedings of the Royal Society
A
C W Stewart Wall slip in the extrusion of linear p olyole ns Journal
of Rheology
C W Stewart R S McMinn and K M Stika A mo del for predicting
slip velocity during extrusion with uorop olymer pro cessing additives
Journal of Reinforced Plastics and Composites
Y Y Su and B Khomami Stability of multilayer p ower law and second
order uids in plane Poiseuille ow Chemical Engineering Communi
cations
Y Y Su and B Khomami Interfacial stability of multilayer visco elas
tic uids in slit and converging channel die geometries Journal of
Rheology
BIBLIOGRAPHY
Y Y Su and B Khomami Purely elastic interfacial instabilities in su
p erp osed ow of p olymeric uids Rheologica Acta
T Takaki and D C Bogue The extensional and failure prop erties of
p olymer melts Journal of Applied Polymer Science
K C Tam C Tiu and T N Fang Remarks on the shear thickening
b ehaviour of dilute p olymer solutions Polymer Plastics Technology
and Engineering
R I Tanner Engineering Rheology Oxford Press New York
R I Tanner and M Keentok Shear fracture in cone plate rheometry
Journal of Rheology
G I Taylor Stability of a viscous liquid contained b etween rotating
cylinders Philosophical Transactions of the Royal Society of London
Series A
V Tirtaatmadja and T Sridhar A lament stretching device for mea
surement of extensional viscosity Journal of Rheology
G Tlapa and B Bernstein Stability of a relaxation type visco elastic
uid with slight elasticity Physics of Fluids
J P Tordella An instability in the ow of molten p olymers Rheologica
Acta
B Tremblay Sharkskin defects of p olymer melts the r ole of cohesion
and adhesion Journal of Rheology
BIBLIOGRAPHY
C Tzoganakis B C Price and S G Hatzikiriakos Fractal analysis of the
sharkskin phenomenon in p olymer melt extrusion Journal of Rheology
Milton Van Dyke Perturbation Methods in Fluid Mechanics Academic
Press New York and London
C Venet and B Vergnes Exp erimental characterization of sharkskin in
p olyethylenes Journal of Rheology
G V Vinogradov Visco elasticity and fracture phenomenon in uni
axial extension of high molecular linear p olymers Rheologica Acta
G V Vinogradov and L I Ivanova Wall slippage and elastic turbulence
of p olymers in the rubb ery state Rheologica Acta
G V Vinogradov A Y Malkin Y G Yanovskii E K Borisenkova B V
Yarlykov and G V Berezhnaya Visco elastic prop erties and ow of nar
row distribution p olybutadienes and p olyisoprenes Journal of Polymer
Science A
S Q Wang P A Drda and Y W Inn Exploring molecular origins of
sharkskin partial slip and slop e change in ow curves of linear low
density p olyethylene Journal of Rheology
N D Waters The stability of two strati ed p ower law uids in Couette
ow Journal of Non Newtonian Fluid Mechanics
BIBLIOGRAPHY
N D Waters and A M Keeley The stability of two strati ed non
Newtonian liquids in Couette ow Journal of Non Newtonian Fluid
Mechanics
C B Weinberger G F Cruz Saenz and G J Donnelly Onset of draw res
onance during isothermal melt spinning A comparison b etween mea
surements and predictions American Institute of Chemical Engineering
Journal
J L White and A B Metzner Development of constitutive equations
for p olymeric melts and solutions Journal of Applied Polymer Science
G M Wilson and B Khomami An exp erimental investigation of in
terfacial instabilities in multilayer ow of visco elastic uids Incom
patible p olymer systems Journal of Non Newtonian Fluid Mechanics
G M Wilson and B Khomami An exp erimental investigation of inter
facial instabilities in multilayer ow of visco elastic uids Elastic and
nonlinear e ects in incompatible p olymer systems Journal of Rheology
G M Wilson and B Khomami An exp erimental investigation of interfa
cial instabilities in multilayer ow of visco elastic uids Compatible
p olymer systems Journal of Rheology
BIBLIOGRAPHY
H J Wilson and J M Rallison Short wave instability of co extruded
elastic liquids with matched viscosities Journal of Non Newtonian
Fluid Mechanics
W T Wong and C C Jeng The stability of two concentric non
Newtonian uids in circular pip e ow Journal of Non Newtonian Fluid
Mechanics
C S Yih Stability of liquid ow down an inclined plane Physics of
Fluids
C S Yih Instability due to viscosity strati cation Journal of Fluid
Mechanics
J Y Yoo and Y Na A numerical study on the planar contraction
ow of a visco elastic uid using the SIMPLER algorithm Journal of
Non Newtonian Fluid Mechanics
H S Yu and E M Sparrow Exp eriments on two component strati ed
ow in a horizontal duct Journal of Heat Transfer
G Zeichner Spinnability of visco elastic uids Master s thesis Univer
sity of Delaware Newark
A Ziabicki and R Takserman Krozer Formation and breakage of liquid
threads I Mechanism Roczniki Chemii
A Ziabicki and R Takserman Krozer Formation and breakage of liquid
threads I I Cohesive break of a steady liquid jet Roczniki Chemii
BIBLIOGRAPHY
A Ziabicki and R Takserman Krozer Formation and breakage of liq
uid threads I I I Capillary break up of a steady viscous jet Roczniki
Chemii
A Ziabicki and R Takserman Krozer Formation and breakage of liquid
threads IV E ect of rheological b ehaviour on the length of liquid
threads Roczniki Chemii
A Ziabicki and R Takserman Krozer Formation and breakage of liq
uid threads V Range of o ccurrence of the individual break pro cesses
General conclusions Roczniki Chemii
A Ziabicki and R Takserman Krozer Formation and breakage of liquid
threads VI Following the break pro cess with high sp eed photography
Roczniki Chemii
A Ziabicki and R Takserman Krozer Mechanism and breakage of liquid
threads to the problem of spinnability of liquids Kol loid Zeitschrift
und Zeitschrift f ur Polymere
A Ziabicki and R Takserman Krozer E ect of rheological factors on
the length of liquid threads Kol loid Zeitschrift und Zeitschrift f ur
Polymere
Index
channel ow advected form of equations see equa
tion transformation
co extrusion see interfacial in
asymptotic expansion
stability
high concentration see Upp er
generalised uid see co extrusion
Convected Maxwell uid
instability
long waves
review of theory
shear thinning uid see shear
low concentration
thinning instability
narrow inner uid
co extrusion instability
short waves
criterion for existence
strong elasticity
interfacial see interfacial insta
strong shear thinning
bility
single uid
weak elasticity
cone and plate
bulk instability
birefringence
edge fracture
Boger uid
fracture
cone and plate ow
constitutive equations
Taylor Couette ow
general form
b oundary layers
constitutive instability
INDEX
in cone and plate see cone and contractions
plate Couette ow see asymptotic expan
melt see gross melt fracture
sion short waves
review of theory
gross melt fracture
curved streamline instability
helical distortions see wavy insta
bility
in contraction
in extension
in nite lo op see lo op in nite
instability
Dean ow see Taylor Couette in
co extrusion see co extrusion in
stability
stability
Doi Edwards uid
exp erimental observations
de nition
extension thinning
interfacial see interfacial insta
shear banding
bility
draw resonance see melt spinning
shear thinning see shear thin
dumbbell mo dels
ning instability
equation transformation
theory in literature
extrusion
interfacial instability
theory see channel ow
bre breakage see melt spinning
Lagrangian co ordinate
lm blowing
are
linear stability theory
fracture
long waves see asymptotic expan
sion edge see cone and plate
INDEX
Poiseuille ow see channel ow lo op in nite see in nite lo op
ribbing see wavy instability melt spinning
draw resonance
second order uid see slow ow ex
bre breakage
pansion
necking
sharkskin
multi layer ows see interfacial in
shear banding see constitutive in
stability
stability
shear thinning instability necking see melt spinning
short waves see asymptotic expan Newtonian uid
sion de nition
slow ow expansion draw resonance
spatial stability theory nonlinear stability theory
Squire s theorem nth order uid see slow ow expan
stagnation p oint ows sion
numerics
Taylor Couette instability
Taylor Dean ow see Taylor Couette
instability Oldroyd derivative see upp er con
temp oral stability theory vected derivative
timescale to distinguish mo dels Oldroyd B uid
co extrusion see interfacial in
UCM see Upp er Convected Max
stability
well uid
de nition
upp er convected derivative
generalised
Upp er Convected Maxwell uid
de nition plate and plate see cone and plate
INDEX
interfacial Couette ow
long wave analysis
short wave analysis
wake ows
wavy instability
White Metzner uid
channel ow see shear thinning
instability
de nition
generalised
winding number