Nonmonotonic Constitutive Laws and the Formation of
Shear-Banded Flows
N. Spenley, X. Yuan, M. Cates
To cite this version:
N. Spenley, X. Yuan, M. Cates. Nonmonotonic Constitutive Laws and the Formation
of Shear-Banded Flows. Journal de Physique II, EDP Sciences, 1996, 6 (4), pp.551-571.
<10.1051/jp2:1996197>. <jpa-00248317>
HAL Id: jpa-00248317
https://hal.archives-ouvertes.fr/jpa-00248317
Submitted on 1 Jan 1996
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
J.
Phys.
France
II
6
(1996)
Spenley
N-A-
(~)Cavendish Laboratory,
(~)Max-Planck-Institut
and
Laws
Yuan
X-F-
(~>~>~),
1996,
APRIL
Constitutive
Flows
Nonnlonotonic
Shear-Banded
of
551-571
(~>~)
and
Fornlation
the
Cates
M.E.
551
PAGE
(~~~)
Madingley Road, Cambridge CB3 OHE,
Grenzflichenforschung,
Kolloidund
U.K.
55,
Germany
(~)H. H. Wills Physics Laboratory, University of Bristol, Royal Fort, Tyndall Avenue,
Bristol
BSB ITL, U-k(~)Department of Physics, University of Edinburgh, King's Buildings, Mayfield Road,
Edinburgh EH9 3JZ, U-Kfur
Kantstrasse
Teltow,
14513
(Received
September
18
1995,
received
jn
final
form
accepted
1995,
November
29
January
3
1996)
PACS.47.50.+d
Non-Newtonian
PACS.83.20.Hn
Abstract.
We
which
stress
flow
is
is
(the
stochastic
plane
The
is
show
be
the
for
are
the
modified
to
models
function
shear
the
flow for
that
strongly
must
constitutive
of the
method
integral
an
of
constitutive
fluid
to
solve
those
dynamics
shear
a
combined
with
use
shear
simulation
Ottinger's
two-dimensional
and
one-
a
novel
We
rate.
predict a
homogeneous
which
that
models
of
micelles.
equation describing entangled wormlike
(with a differential
equation).
model
constitutive
of a 'toy'
actually consists
and
preliminary
our
allow for spatial
variations
with
shear
the
equation)
known
It is
rate.
with
for
behaviour
viscoelastic
shear
decreases
stress
constitutive
compared
steady
inhomogeneous,
results
We
consider
Lagrangian-Eulerian
method
Couette
and
when
flows
phase changes
Structural
nonmonotonic
a
unstable
technique
fluid
(non-linear fluid)
dynamics
Fluid
PACS.83.10.Ji
of
state
in
bands
of
different
that
results
indicate
the
viscoelastic
shear
the
rate.
Such
constitutive
a
flow
equation
stress.
Introduction
1.
Several
important
that
steady
the
models
shear
of
stress
viscoelastic
decreases
with
behaviour,
shear
rate
in
~,
polymers
and
other
materials,
predict
large enough. For instance,
~
model, at high shear rate, the polylonger generate a shear stress. Some
when
is
theory Ill shows this behaviour; in their
aligned
become
along the flow direction, and can no
mers
phenomenological models have this property too, such as the
Maxwell (CAM)
corotational
model [2].
On physical grounds, we do not
behaviour
expect this
to persist
to indefinitely
high shear rate. For example, if a Newtonian
solvent is present, the
Newtonian
viscosity might
eventually dominate.
Alternatively, the models may fail, as short
relaxation-time
in
processes
the polymer itself become
important [3], causing the stress to increase. However, these effects
comparatively weak, so that there may still be a range of shear rates
the
shear
where
are
and
Doi
(*)
©
Author
Les
Edwards'
for
flditions
correspondence
de
Physique
(e-mail:
1996
spenleyslmpikg-teltow.mpg.de)
JOURNAL
552
PHYSIQUE
DE
II
N°4
m
m
©
#
$~
~d
©
',
',
~
m
',
',
«~ «~
«~
«~
shear
Fig.
Schematic
I.
decreasing, Figure
is
stress
conditions
steady
nonmonotonic
This
I.
«~ «~
rate
stress/shear
shear
significant,
is
,'
rate
because
a
curve
steady flow
unstable
is
these
under
[4].
materials?
shown by real
The
indication
important
experimental
most
effect',
which
'spurt
shown
by
linear
polymers
flow.
On
is
in
pipe
comes
many
gradient P' beyond a critical value, the volume throughput
increasing the driving pressure
discontinuously.
When P' is reduced, the throughput
decreases gradually~
is seen
increase
to
showing hysteresis. For reviews, see [5,6].
This
behaviour
has been explained in terms of a
shear
nonmonotonic
stress [7-9]. The spurt
effect
from a classical
flow, to a 'plug'flow, in which a
Poiseuille-like
transition
to be a
seems
gently sheared central cylinder (the plug) is surrounded by an outer layer of strongly sheared
An
alternative
material.
explanation, in terms of wall slip, is possible, but in this paper we
condition is valid.
will
that the no-slip boundary
assume
Rheometrical
experiments are often done in cone-and-plate or cylindrical Couette
apparatus.
In these geometries,
expects the flow to have the same shear rate everywhere, but this is
one
impossible if the applied shear rate ~~ppiied lies in the region of decreasing stress [4]. What is
the analogy of plug flow here, or is a stable, steady flow possible at all? Cates,
McLeish
and
(CMM) [lo] noted that, in simple shear, the shear stress is a constant
Marrucci
throughout
the material, so that tyTo shear
coexist if they correspond to the
rates, ~A and ~B, can
same
S(~B). They suggested that the flow takes a banded form,
S(~A)
steady shear stress:
containing regions with shear rate ~A~ and regions with ~B, with volume
fractions
and zB
xA
such that
their
shear rate equals whatever
rate is applied:
average
~~ppjied
XAKA + XBKB
this
Is
behaviour
from
the
"
"
(with
+
xA
xB
I).
=
possible (although
If
banded
a
for
curve
at
until
the
and
KB
as
long
K3
"
~3.
The
argued
from
possible.
so
of the
~~ppiied is
hallmark of this
between
shear
is
stress
that
the
the
entire
only
is
not
is
stress
flow
when
~appiied
~i
is
is
still
(see Fig. I),
~3
remains
on
low-shear
the
reached, a small
further, the
more.
at
amount
this
In
The
flow is
short, it is
assumed
rate.
~~ppjied,
the
is
homogeneous
shear
flow
may
banded
a
and
~i
but
stable
of
that
between
lie
can
and
branch
material
now
flo~v" is
~2).
history,
proportion
that, in the
shear-banded
land equals Sm~x). Once the flow is in the
behaviour
then
if ~~ppiied lies between
def.ermined by ~~ppiied,
determined
by the flow
increased
being sheared
increased
and
~o
forbidden
Then,
As
sample
if
independent of ~~ppiied
regime, then, on decreasing
is
zero,
l'top-jumping').
whole
remains
as
the
present,
CMM
increased
is
shear
a
flow is
Smjn and Smax.
~~ppiied
If ~~ppiied lies
uniform flow is
that
if
of the
begins
to
of ~3 increases~
uniform again,
~A
"
~i
and
regime, the shear stress
uniform high shear rate
persist until ~2 is reached,
FORMATION
N°4
only
and
then
behaviour
Such
shear-thinning
ever,
the
are
hysteresis
observe
to
bands
the
stress.
not
appears
l'bottom-jumping').
reformed
have
to
higher
at
occurs
surfactant
a
shear
in
but
rates,
(cetylpyridinium
solution
for
observed
been
shear
FLOWS
SHEAR-BANDED
OF
the
shear
this
In
case,
would
we
expect
Typically, some
gently. Howsodium
salicylate in water) shows
surfactant
forms highly elongated
polymers.
ordinary
continues
stress
and
chloride
553
rise
to
predicted by CMM Ill]. This
wormlike micelles, which can entangle, and
to share
many of the rheological features of
appear
equation [13] for them,
polymers (for a review, see [12]). Cates has formulated a constitutive
Edwards.
Using this equation, together with the shear-banding ideas
based on that of Doi and
the
able to predict, with good
McLeish
of reference [lo], Spenley, Cates and
accuracy,
were
evidence for shear-banding
cetylpyridinium system [14]. Direct
observed
shear
in the
stress
(birefringence experiments) has recently been found [15] in another entangled micellar system
(aqueous cetyltrimethylammonium chloride). Shear-banding has also been reported in refer[16,17], although here the behaviour is complicated by the fact that the high-shear band
ences
(metastable) nematic phase [18-20].
shear-induced
is a
theoretically the shear-banding conjecture, by modelling the
The aim of this
paper is to test
mathematical
of
shear
flow
viscometric
experiment. In Section 2, we set up the
in
start-up
a
Section 4 analyses linear
model, and in Section 3 we try to predict some of its properties.
numerical
presented in
calculations
perturbations on a uniform shear flow. Preliminary
are
Section 7,
two-dimensional
model.
In
and in Section 6 for a
Section 5 for a one-dimensional,
bands
describe
the
interface
modification
of
model
suggested
between
the
constitutive
is
to
a
accurately.
more
the
constant
plateau
stress
Formulation
Mathematical
2.
problem
The
assumed,
is
2-dimensional
shear
We
begin
with
the
(or
Newtonian)
with
nian
solvent
model is the
(so
assumed
is
very
some
It
P
where
v(r)
is
the
two
shear
have
+
cases.
stress
If the
S
=
?v)
v
stress
a~~
+
we
is the
is
time
i«
viscoelastic
the
to
if
axial
only.
of two
variables
free
there
are
no
The
surfaces).
that
symmetry along
variable). For simplicity, we
one
models.
+
material
in
adequately
be
viscoelastic
a
modes
that
n?v
Pi
can
a
Newtomodelled
ii
part of the
stress,
and p is its density. Later, we
consider a completely general
will
control
qbv~ lay
equation for
?
that
except
same,
would
surfaces
,
is the
solvent
plates parallel
of
2-dimensional
=
fluid
free
infinity, so
2-dimensional, except
to
functions
are
relaxation
velocity field, ~r(r)
rheometer
two
fields
short
pressure,
q is the viscosity of the
constitutive
equation, but at first
We
the
as
same
the
(I+I)-dimensional and
generalised
Navier-Stokes
only the
consider
will
as
direction
(extending
Couette
the
functions
are
model
3-dimensional
with
essentially
be
velocity fields
and
A
cone-and-plate),
would
model
stress
planar
is
'(I+I)-dimensional'
The
the
model
the
so
sophistication.
(e.g.
geometry
'(2+1)-dimensional'
A
levels of
several
at
experimental
whole
appropriate.
symmetry
modelled
be
can
the
include
parameter,
separated by
then the boundary
x-axis,
the
p is
will
hat,e
hydrostatic
specify a
to
behaviour.
constitutive
distance
condition
L.
We
is
that
distinguish
the
total
fixed, thus
?~y + ~/b~~
lay
"
sapplied
for y
#
o, L.
(2j
JOURNAL
554
On
is
the
shear
v(x,
v(x,
gives
This
shear
average
an
(I+I )-dimensional
The
the
is
rate
control
of
rate
~
condition
y
y
o, z)
=
=
U/L.
v(r)
=
=
L, z)
is that
)
Pll
The
normal
any
information
=
(3)
U&.
(4)
~(y)& (so
=
and
by
normal
rheometer
the
boundary conditions).
(in fact, they began with
able
In
between
such
system,
such
system
Constitutive
introduce
We
a
a
an
as
effect
the
in
polymer
viscoelastic
shear
This
as
model
section,
but
time
function
a
we
use
form
The
total
stress
assume
this
can
difference.
stress
S
=
to be
corotational
the
work.
~/laxwell
equa-
single relaxation time,
replaced by the CAM equation
state is a plug flow.
be
local
present
Gg
=
a
of micelles,
composition.
In
elucidate
any
case,
there
it
be
may
This
does
some
discussed
is
arise
not
in
in
the
shear
rate
it is
this is
that
a
(5)
(6)
a
steady
for
The
shear
in
both
viscoelastic
here
modulus.
written
show
can
the
is
subscript
in
not
of equation
behaviour
~~)
~9
Here
elastic
the
T
equation
response.
that
in
help
to
tensor
viscoelasticity (I.e. a
(denoted
shear
stress
simplicity),
function
g is
while
the
is
T
the
viscoelastic
flow, and may be chosen freely.
form, but for the purposes of
irrelevant.
in the
studied
context
of slit flow [23].
Following
reference
[23],
for g:
§~~~
always
solution
7
G is
and
Equation (6) has already been
this
aqueous
and the
drop the
we
of the
deficient
is
can
steady
~
+
constitutive
nonlinear
(I+I )-dimensional modelling,
will
obtain
we
melt.
model
and
relaxation
stress
toy
possible
previous
the
normal
can
nor
experiment,
real
a
[24] with
this
the
properties
constitutive
time)
relaxation
in
a~y
the
across
Model
simple
simplest
the
is
In
was
model
final
that
~t
finite
relation
problem,
~'
This
dynamics,
the
in
first
the
constitutive
found
mechanical
the
one-component
Toy
can
straight
are
change
(5)
plates.
the
Johnson-Segalman
their
for
[25]. We neglect this
reference
A
that,
Their
the
generality). They
two-component
a
coupling
3.
show
to
loss of
streamlines
the
at
different
were
role
no
on
calculate
to
velocity
studied by Malkus~
to pipe flow, that of flow down a slit channel, was
University of Wisconsin [9, 21-23]. They used a (I+I)-dimensional ap(and
identical
equation
equation (5) apart from a driving pressure
to
term
Pego and others
proach, with an
without
play
exerted
used
and
+
and
force
all the
the
problem
similar
A very
the
about
that
n()
a~v
=
eliminated
been
have
stresses
measured
but
condition
0
parallel, the flow everywhere is in the x-direction,
flow, but not along it). Equation (I) reduces to
tion
boundary
the
then
parameter,
and
be
N°4
II
velocity
the
on
hand, if the
other
PHYSIQUE
DE
a
+
true.
i~b~
lay
is
then
i
~~~
j2'
nonmonotonic,
provided i~/GT
<
1/8,
which
we
will
FORMATION
N°4
SHEAR-BANDED
OF
~l
Q
FLOWS
555
L
2
L
~
max
Fig.
2.
Schematic
phase diagram
The
and
parameter.
L3 are lines
of
begin
We
p
solid
shear
constant
the
with
the
and
viscoelastic
indicate
stress
stable
and
with
a,
the
total
equilibria
unstable
S
stress
the
as
respectively.
control
Li, L2 and
rate.
Assuming
regime.
stress-controlled
o), equations (2)
=
for
lines
dashed
creeping flow (I.e.
the
density
fluid
(5) give
lay
~)
+
~
=
(8)
SaPPied.
equation (6), leaving an equation in the single variable
derivative b lay has disappeared.
Notice
that the spatial
This
parameter.
a, with Sappiied as a
different
(in other words) that
that
the
evolve
independently,
positions
at
stresses
means
or
This
allows
the
streamlines
eliminate
to
us
have
been
Equation (6)
trivial.
Via)
where
may
b~
decoupled
be
has
from
another.
one
in the
written
of the
behaviour
The
system is
now
form
is
Via)
V
from
if Smin
minima
two
<
=
~~
ja~
+
l
ln
Sappiied
IS
+
2
Smax land i~/GT
<
(lo)
a)~)
~l
1/8),
<
otherwise
has
it
one.
for S~ppjjed < Smin and Sappjied > Sm~x, a has one
equilibrium point. For
This is
summarised
Smin < S~ppjied < Sm~x, there are two stable equilibria and one
unstable.
Branch A corresponds to low shear rates, where
in Figure 2.
most of the
stress is due to the
viscoelastic
material, and branch B to high shear rates, where the polymer (or whatever the
It
follows
material
is
that
has
is
strongly aligned
been
and
can
dealing with a macroscopic system,
(I+I)-dimensional approximation).
stress-controlled
regime, but
position in the
We
(in
different
of
longer generate
a
parts
of the
the
attraction,
We
now
take
2.
and
the
the
By
rest
shear
the
into
rate
as
the
principle
a
would
this is
necessarily
a
not
streamlines
suitable
a
Fluctuations
branch.
unstable
different
flow
is in
Normally,
and
are
the
phase diagram, Figure
on
no
stress,
of the
most
so
stress
Newtonian.
then
will
be
on
carry
some
be
parameter.
the
with
stable
may
streamlines
will
be
into
yield
respect
branches
taken
one
a
to
possible for
It is
case.
different
y-coordinate
of the
constant
system
subsequent evolution
Again with
other, and
control
to
of S~ppiied, the
control
function
a
to
of the
'banded'
of the
a
point
basins
flow.
creeping flow, equation (5)
becomes
~
~
by
~~
'
~ ~
by
~~ ~~
JOURNAL
556
The
total
shear
S
stress
qb~/by
a+
=
°j)~~
a(Y)
where (a)
b~(y) lay,
the
again
therefore
is
defined
is
but
long
streamlines
the
value
j/
(I IL)
be
to
now
average
of the
+
a(y)dy.
diverge from
viscoelastic
stable
two
Using
space.
boundary
the
stress
(a).
state
branches
A
The
that
so
shear
at
ei>ery
a
line S
the
<
~
total
~2
>
~
coupled
S
stress
eliminate
to
to
another
one
with
varies
now
point is equal to la), the
i~~. In Figure 2, Li, L2 and
If the applied shear rate
~2.
always remaining
in time,
evolves
the
+
a
=
and
used
be
may
are
equilibrium
manifold
and
the
on
attains
line
final
a
unstable
regime, then the system
rate is in the
fluctuations
Instead, the
and the
L2.
streamlines
grow,
and from L2. (A more
precise discussion of this is given in the next
reached only when every
streamline
is now
is on one or other of the
other
each
i12)
+ il~,
Again, this equation
independent; they
remains
along the
evolve
steady
A
section.
the flow
as
simply
cannot.
in
constant
quite
not
are
(a)
=
homogeneous,
state of the system is represented by a point on
L3 are such lines for, respectively, ~ < ~i, ~i <
is on one of the stable branches, then the system
(Li or L3), and moving along it until it reaches
If the applied shear
stable (homogeneous) state.
As
time.
N°4
II
(3)-(4),
conditions
via
PHYSIQUE
DE
B,
or
so
line
the
that
flow is
banded.
closely related to flow through a pipe. The difstress
ference is that each point on the radius of the pipe has a different
(determined by the
stress
driving pressure gradient), whereas in Couette flow, the stress is constant
throughout the flow.
Since, in creeping flow, the
streamlines
fundamental
impormay be decoupled, this is not of
(unless
flow
In
Couette
the
will
normally
homogeneuus
the applied stress
be
tance.
geometry,
manipulated in a very careful way), and in the pipe, the flow will be either
is deliberately
Poiseuille-like
take a simple plug form.
or
controlled
be fully decoupled,
By contrast, in the shear-rate
regime, the streamlines
can
never
and the
distribution
of a
the
streamlines
be
explicitly
taken
With
into
must
account.
across
flow,
final
steady
determined
by
applied
banded
the
the
shear
To
is
stress
not
create
rate.
a
the bands, a
translational
be broken, but (unlike the pipe) each
streamline
symmetry
must
is in principle equivalent. The symmetry
must be broken by fluctuations, implying that the
of
the
bands
how
bands
thickness
they have) is not
(I.e.
there
and what
pattern
many
are,
determined
by the shear rate, but rather by thermal
fluctuations
other
perturbations.
In the
or
section, we will discuss these issues at the level of linear perturbations on a homogeneous
next
flow
Couette-like
controlled
with
is
flow.
In
section,
this
tion, with
flow
of
Analysis
Linear
4.
was
shear
as
by
whether
the
xA
and
In
xB,
stability analysis provides
linear
control
Yerushalmi
flow.
shear
types of band,
two
ask
rate
considered
unstable
an
we
et al.
[4].
particular,
we
this
and
Consider
shear
stress
a
uniform
a~~(~),
shear
where
are
like
to
discover
stands
the
take
law
constitutive
the
volume
S.
stress
the
to
into
band
of
uniform
subsequent
the
in
equivalent to knowing the
perturbation grow preferentially in
flow, with
'ss'
we
would
interested
which is
wavelengths of an initial
bands begin to form.
We will
is how
the fluid density p to be small, but
certain
Here
insights
stability
any
question of linear
The
parameter.
shear
evolution
of the
possible
regime,
completely
linear
be
fractions
It is
forma-
and
that
that
general,
non-zero.
U/L, producing a viscoelastic
steady shear rate of ~
for 'steady shear'.
The fluid velocity ~ is given by
a
=
~lv)
=
~Y
(13)
OF
FORMATION
N°4
whilst
total
the
rate
and
~
independent of position,
stress,
Aa(y, t)
perturbations
it
write
SHEAR-BANDED
A~(y,t).
and
a~~(~) +i~~.
is
Linearise
FLOWS
the
/~ dt'K(t'IT; ~) ~~~~~'~
G
=
G
The
the
is
kernel
memory
/h~
as
~j~
t
Fourier
a
K
~
+
y
material,
the
on
eliminate
to
~~
i~
=
of
depend
must
Using equation (5)
p
Write
modulus
characteristic
jf Kit /T)dt.
G
around
~'~
~l/
o
time.
(I+I )-dimensional
steady shear
consider
equation
constitutive
as
Aa(t)
where
Now
557
(14)
,
and
is
T
the
shear
rate
the
stress
leaves
its
Note
K.
=
equation for A~:
this
)G /~ dt'K(t'IT; ~) ~~~~l'
y
(terminal)
relaxation
that da~s(~)/d~
~'~
(15)
y
o
series;
~j ~n(t)
cz
/h~(j,t)
=
(")~)
sin
(16)
_~
Here
we
have
assumed
could
be
done
with
In
same.
/h~(L)
non-slip boundary conditions, /h~(o)
linear boundary condition, and the final
equation (15) becomes
components,
=
terms
p~~"
Taking
be the
to
vno
~n(t)
(~°~)~
L
-q
=
dt
s~n(s)
Now, the problem
it
initial
=
=
the
to
=
viscoelastic
dimensionless
After
forces.
perturbation
The
where
that
it
(d/d~)(a~~(~)
Here,
rate.
perturbation
part of the
we
sum
shown
was
a
any
that
to
most
position
finite
out
value
a
n.
sufficient
on
how
number
the
be
t') IT; K)~n(§, t').
Laplace
transforming
iii
K(sT; K)~~(s)
[~
of
ratio
which
~
(17)
in t:
i18)
Weissenberg number We
~/GT. It is useful
q~
the
parameters,
=
viscosities
=
the
measures
inertia
of
the
~T,
to
mode
nth
+~~~~(8T( ~)
+ i1~ + K
(ST; ~) has
formalism
The
calculation
would
rearrangement,
8Tp(
sTp[
if
of
and
condition
above
for
is
a
~~~~
zero
in
equivalent
instability
~~)
+
need
grow
find
must
of
we
unstable
is
a
dt'K((t
the
conclusion
o
~n(s)
pLU/GT, and the
(p/GT~)(L/nor)~,
~"~~~
complex plane, for
~
/~
G
it),
of ~n
=
three
contains
Reynolds number Re
Re/We(nor)~
define p[
relative
o) value
=
-j iii
~no
the
(~°~)~
L
o, but
=
any other
of the
Fourier
the
to
right-hand
that
of
~* + K(s
decreasing function
is
that
half of the
reference
=
o)
<
[4j,
o, I-e-
the steady shear
of the shear
is a
stress
< o
of the
know, given the
of
instability,
which
wavelengths
an
presence
real
rapidly. The growth rate of any given Fourier
is
the
component
K(sT;
~),
of
and
the complex plane of the rightmost
sTp[
q*
+
+
zero
this
of
varies
as
a
function
of
n.
Assume
that
K(t IT)
can
be
written
as
the
exponentials
Q
Klt/T)
=
~j Kq
q=1
exPl-cat/T)
120)
JOURNAL
558
PHYSIQUE
DE
II
N°4
that
so
~
~~
~j
K(sT)
=
q=I
This
is
not
naturally.
flow
serious
a
For
Differential
restriction.
example, the
geometry, the shear
spectrum
can
corotational
and
stress
first
equations
only two
difference,
constitutive
normal
stress
likely
are
has
model
Maxwell
the
(21)
+ Cq
ST
variables
satisfy
to
in
the
(after linearising)
so
it
present
the
only two decay rates. By contrast, an integral equation, like
number of variables,
because to define
the
Doi-Edwards
equation, has, in effect, an infinite
of the system at any given time, we must specify an entire probability
distribution
the
state
(that of tube segment orientations).
This implies an
infinite
number
of
relaxation
times.
function to a finite
distribution
number of variables (e.g.
However, it is possible to reduce the
by discretising it, or by expanding it in basis functions to finite order), and this should not
change the qualitative behaviour; such a change would be unphysical. We infer that, to an
arbitrary degree of accuracy, the relaxation
be replaced by a finite
number
of
spectrum
can
relaxation
discrete
We
relaxation
are
have
times.
interested
thus
the
in
roots
8Tp(
equation
of the
ij
q*
+
~~
+
(22)
0.
=
+ Cq
ST
q"1
lo~ kg m~~, and
extremely small. For example, the density of water is p
are
cetylpyridinium system studied in reference ill], the modulus G and
relaxation
time
lo~~
of
The
order
Pa
and
respectively.
of
the
plates
L
be
30
10
separation
5
x
are
s
may
lo~s/n~, and it is enough to consider the limit p[ Therefore, p[
(p/GT~)(L/nor)~
Rearranging,
The
p[
=
for
the
o.
=
=
Q
p~(8T)~~~
+
~
P~(8T)~
Q-i
Cq +
~*(8T)~
~
+
the ap and bp
functions
are
singular perturbation problem.
(8T)~(P~ap
+
Q*bp
+
fp)
=
(23)
0
p=0
q=i
where
T
m.
of the cq and
limit p[
In the
the
-
fp
are
functions
o, there is
one
of the cq and Kq.
root s~~"~T
singular
This is
=
-~~
a
/p[
therefore
Q further
which do not depend on p[. s~~"~ is necessarily negative, and
roots
corresponds to a decaying solution.
None of the
other
depends on n. We conclude
roots
flow grow at the
uniform
that, when a uniform shear flow is unstable,
perturbations of that
of
length-scale,
long
they
If a banded
remain
in
linear
irrespective
regime.
rate,
as
a
same
flow
that
have
all
length-scales,
other
words,
this
implies
it
in
must
structure
at
emerges,
or
This
that there are bands of indefinitely
width.
presumably
unphysical.
There
is
are
narrow
characteristic
band width
perturbations have
three
alternatives:
either a
emerges only after the
I-dimensional
model here is inadequate, and the
essential phealready become large; or the
model is deficient, and
take place in 2 (or 3) dimensions; or the underlying
constitutive
nomena
length-scale. These hypotheses
be modified
intrinsically define a
characteristic
to
must
so
as
Sections 5, 6, and 7 respectively.
discussed
in
are
and
5.
In
Numerical
this
uniform.
section,
This
Analysis
we
must
will
be
of
the
calculate
done
(I+I
the
)-Dimensional
flow
numerically,
without
so
we
will
Model
making
use
the
the toy
assumption
constitutive
that it is nearly
equation (6).
FORMATION
N°4
SHEAR-BANDED
OF
FLOWS
559
©
O
+
li
~
(
(
°'~
o-I
fi
o.ooi
0
0
2.5
5
lo
7.5
timeh
Fig.
The
3.
different
Figure
time,
steady
solutions
start-up
(without loss of generality),
stress), and the Newtonian
dimensionless
value
of the
(5)
equation
of
steady
of
shear flow, using the
equations (6), (7),
model,
constitutive
toy
and
density.
fluid
of the
shows
3
after
of
start-up
values
shear
shear
the
the
flow
rate
~
order
U/L
=
viscosity ~
of lo~~
Here
solvent
=
o.ol.
=
use
we
felt by the plates,
of p. We take
stress
several
with
as
G
values
=
T
=
L
of
=
I
regime of decreasing
In the cetylpyridinium
system, p has a
larger
values,
and
also
the case of
some
5
(well
function
a
inside
the
creeping flow, p
o.
Velocity and stress are defined on separate grids (offset by one half lattice spacing), and the
parabolic equation (5) is solved by a fully-implicit method. The lattice spacings are /hj
o.02
(this
for
and /ht
initial
condition is ~
only
o.ol.
The
everywhere
is
relevant
nonzero
~j
p), while a, at each value of y, is initially an independent
variable, with
Gaussian
random
10~~ (if the initial
standard
deviation
condition
does not
contain
noise,
0 and
mean
some
the
simulation
shear flow, which we know to be physically
is liable to predict a homogeneous
unstable). For creeping flow, there is no need to solve a partial differential equation. Equation
(12) is used in place of (5).
As can be seen from Figure 3, the results
virtually independent of the presence or absence
are
=
=
=
of
a
small
of
a
banded
relaxation
The
=
inertial
flow,
onto
the
viscoelastic
Notice
term.
the
because
steady
part
the
toy
This
overshoot.
equation
can
only
directly
arises
ever
describe
from
the
monotonic
development
(exponential)
state.
of
the
stress
is
shown
as
a
function
of
both
position
and
time
in
o). This is related to the local shear rate by equation (12),
(p
corresponds to a lower shear rate. In both figures, by time 2, the
stress
fluctuations
have
started to become significant (of course, in the linear regime, their actual size
initial values), and by time 5, two
distinct
phases have developed. At
is proportional to the
relaxation
the steady state has begun, and by time 20, it has essentially
time lo, the
toward
reached.
The
of a small
of inertia
been
makes no difference; in any case, the
amount
presence
bands have a thickness of the order of the
numerical grid spacing.
This
apparently unphysical result is not a failing of the numerical
technique, but seenls to
be
inherent
Equation (5) is a diffusion equation, with diffusion coefficient ~/p.
in the model.
When p is very small, the time taken to diffuse
the space
between
the plates becomes
across
arbitrarily small. Therefore, each streamline is as strongly coupled to distant
streamlines
as it
o.ool)
Figures 4 (p
viscoelastic
so a larger
=
and
5
=
JOURNAL
560
DE
OOOOOOOO
PHYSIQUE
OOOOOOOOO
OOOOOOOOOtOOOOOOO
I
I
o
II
OO
COO
COO
OOOO
°o
°
N°4
O
n
i
°
~
O
5
~
~~
V
20
°
o
°
~
_
~
Oo
O-I
%
fl
~~
v
@K~
2~~~~
jio
z
~
~
0.25
0
0.5
0.75
position
Fig.
The
4.
viscoelastic
stress
density is
plate (position 1).
shear.
The
second
fluid
as
function
a
The
o.ool.
of
horizontal
mm
o
m
z£OO
~
09
position,
axis
~
o
o
~
~
several
at
extends
from
times
the
after
the
onset
of
(position o)
plate
first
steady
to
the
ommcoJo
~aunmt
Q
tD
tD
9
%
i
~
zm~fiizi1°z8zn~zz'
~
°a
~
tD
0.25
0
Fig.
As
5.
is to its
so,
on
that
would
each
value
course,
can
It
a.
no
I
o
2
a
also
separate
result
was
0.5
o
~
z
z
°
~
a
lo
V
20
0.75
position
inertia.
change only on the time-scale of the viscoelastic
relaxation
time,
it is effectively fixed.) This is why equation (12) gives 0~/0j in
neighbouring
is
same
with
(a(j)
average
require
of j
the
but
time-scale,
diffusive
of the
terms
Of
Figure 4,
neighbours.
the
i
°
implies
dynamical
obtained
be produced,
because
macroscopic bands
cannot
be
fact
correlated,
when
each
in
streamline
to
system, coupled to the others only tin equation (12).
the previous
section by linear analysis,
where it ~vas
that
streamlines
in
preferred length scale for bands. The numerical
calculations
have an
was
no
length scale, the grid spacing, and select this as the band width.
extra
It appears
that it is not possible to obtain a realistic flow pattern
when the full nonlinear
even
dynamics are
accounted
for, at the (I+I)-dimensional level of analysis. Leaving aside this
difficulty for the moment, we will ask what
behaviour
the model does have.
An important
found
that
there
FORMATION
N°4
OF
FLOWS
SHEAR-BANDED
561
0.80
0.70
~'~~
~9
~
0.50
©
i~
0.40
i
0.30
~
A
.
~
~'
0.20
0.
°'°°
io
i
shear
Fig. 6.
equation)
the
The
model
with
homogeneous
question
final
and
the
other
steady
band
stress
and
~ is
the
in
simulated
toy
banded
in
in
Figure
conditions
The
each
after
material
shown
initial
o.01
as
flow
6
for
(using
the
equation (6), and
line is the steady
equation,
equation (26).
viscosity
is
the
unmodified
constant,
simulations
values
parameter
in
stress
for the
Newtonian
The
final
from
are
diffusion
stress
a
shear
circles
The
flow.
is
stress
steady
final
(points).
ioo
rate/z-1
solid
toy
constitutive
triangles for
assuming
stress
the
case.
the
(for
banded
a
range
flow
of
developed.
has
shear
rates,
with
The
p
=
0,
Fig. 3).
have been done by Espafiol et al. for a different
constitutive
model (that
Segalman, with a single relaxation time), with a very similar result [26j. Clearly,
the stress
associated
with banding is near the
minimum
in the underlying
constitutive
curve,
whereas, in [10], the assumption was made that the stress would be near the maximum of the
These
simulations
done by starting from
shear
stress, and applying a
constant
curve.
were
zero
It may be that the final stress is dependent on the previous
history, so that a
shear-rate
rate.
banded
maximum-stress
state, as postulated in [loj, could be created by using an appropriate
of
shear
It is plausible that this can be done by slowly increasing the shear
rates.
sequence
from
This
would
that, by the time the unstable
shear-rate
regime is entered,
rate
zero.
mean
the
material is already in a state in which the stress has the desired
value.
This also
simulates
viscometric
experiments in which the shear rate is slowly increased, with a
being
measurement
of
Similar
calculations
Johnson
and
made
at
each
shear
rate.
idea,
Figure 7 the effect of a slowly increasing shear rate on the
after the critical
Shortly
shear rate is passed, the stress drops
toy
dramatically as a banded flow develops near the minimum stress. As the shear rate is further
increased, the
band-structure
invariant
the proportion of each type of band
remains
remains
the same, but the shear
and
Eventually the maximum
increase.
rates
stresses
stress is reached
with a higher proportion of the
again, the stress drops, and a new
band-structure
is
created
high-shear phase. This happens several times. A similar result is obtained if the fluid is given
inertia by setting p
0.001.
some
To
test
this
equation,
show
we
without
in
inertia.
=
This
shear
behaviour
rate
and
can
stress
be
of
a
understood
whole
band,
as
follows.
than
It is
it is to
easier
change
to
the
make
size
a
of the
small
change
band.
in
To do
the
that
requires the boundary between bands to be moved, so that the material on the boundary has
be subjected to a different
times) to bring it
shear
relaxation
rate (for a period of several
steady state.
McLeish
into a new
and Ball argued that this
be
achieved by a small
cannot
to
JOURNAL
562
(
DE
PHYSIQUE
II
N°4
0.40
w
fi
$
~
m
#
0.20
w
~~~0.0
1000.0
2000.0
3000.0
time/z
Fig.
The
7.
shear
using
stress,
smoothly
homogeneous flow, at the
increased
is
rate
from
the
o
at
shear
same
perturbation,
because
part of the
infinitesimal)
amount
[7j. Instead,
therefore
As the
shear
A
'cure'
for
should
above.
be
this
tested,
material
the
=
=
change its stress by
changes throughout the
would
stress
a
finite
(as opposed
material.
When
the
to
low
reached
to
EQUATION.
REPTATION-BREAKING
see
whether
shear-banding
Since
=
until
grow
rate is
MICELLE
THE
5.I.
rate.
the
unstable
maximum
against fluctuations,
stress, it becomes
parts of the low shear bands have changed into the high-shear phase.
increased, this happens several times, giving rise to the sawtooth
pattern.
physically implausible behaviour is proposed in Section 7.
phase has
shear-rate
which
equation, as a function of time (heavy line). The shear
o, at o.01T~~. The thin line shows the steady stress for
/hy
o.001, /ht
o.ol, ~
o.01.
toy
time
behaves
it
has
been
in the
claimed
same
for
a
integral
An
way
as
micelle
wormlike
equation
constitutive
differential
the
models
system,
we
considered
will
use
the
materials.
equation proposed for these
constitutive
Entangled polymers are described using the tube model, in which each chain is assumed to
be confined to a tube by the surrounding
molecules.
Doi and
Edwards
devised a
constitutive
micelles
entangle in the same way as polymers, the
equation using this idea Iii. Wormlike
can
difference being that they can break and reform reversibly. Cates adapted the
important
most
model of Doi and
Edwards to include the effect of these breaking
reactions [13,27j. The resulting
of
similar
that
Doi
and
Edwards,
and a brief description of it
equation is very
constitutive
to
the
Appendix.
is given in
simulations
of banded flow, similar to those with the toy equation, in which
We have done
method of 0ttinger [28] (see
solve the micelle
reptation-breaking model using the stochastic
we
Appendix).
EQUATION.
Figure 8 shows the startup of
equation and a range of shear rates (without
inertia). A Newtonian
viscoelastic
falls away
viscosity of 0.001 is now used, since the
stress
(6).
high
shear
for
the
model,
slowly
than
Bands
equation
toy
at
rate
present at
are
more
shear
the
SIMULATIONS
BAND
5.2.
flow
higher
similar
steady
to
USING
(from equilibrium)
shear
stress
rates.
is
The
THE
MICELLE
the
distribution
Figures 4, 5, but
shown in Figure 9.
of
that
with
rather
micelle
of the
noisy
viscoelastic
because
stress
of the
(I.e.
stochastic
band structure) is
algorithm. The final
the
FORMATION
N°4
OF
FLOWS
SHEAR-BANDED
563
Q~
Q~
tll
tll
j
~
(
(
l$
l$
~
~
~
~d
©
~d
©
~
~
0
3
2
5
4
a)
0'
£
i~
~
l$
i
~i
~
~
~
~
0
0
3
time/~
c)
Start-up
of
The
flow
shear
solid
lines
2
equation
for
the
micelle
the
simulated
model),
and
at
banded
the
3
time/~
d)
show
reptation-breaking
constitutive
homogeneous flow
micelle
5
j
03
8.
4
tll
)
id) 8.5v~~
5
©
o~
and
4
time/z
b)
O
Fig.
3
2
0
time/~
la) 1.275v~~, (b) 2.125v~~, (c) 5.1T~~
(solution of Eq. (12) coupled to the
flow
dotted
lines
show
the
solution
of
the
same
assuming
model
0.25
(
m
0.20
g
i~
~
#
O
0,15
o-lo
w
o.05
o.oo
io
i
shear
Fig.
open
Final
9.
circles
are
steady
for
stress
(1+1)-
after
and
start-up
steady
of
2-dimensional
ioo
rate/z
shear
calculations
iooo
-1
flow, for the
micelle
model.
The
filled
and
respectively.
of bands
formation
overshoot
shear
in the
toy model, the
stress, which
causes
an
of the micelle
otherwise.
In
the
direct
solution
constitutive
present
contrast,
equation for homogeneous shear flow (Fig. 8, dashed lines) already shows an overshoot, so
might expect this to be enhanced by the banding. In fact, this does not happen, and the
we
overshoot is no larger than in the direct
solution.
With
could
If
the
not
we
use
be
a
slowly increasing
shear
rate,
a
result
similar
to
Figure
7 is
obtained.
JOURNAL
564
2-Dimensional
The
6.
PHYSIQUE
DE
II
N°4
Model
performed numerical
calculations
2-dimensional
model for micelles using a
the
on
novel
Lagrangian-Eulerian method. This uses a mesh which moves and deforms with the fluid,
flows [29, 30j. It enables fairly high Weissenberg
and has been used for shearing and
extrusion
numbers to be reached, but, at present, is restricted
to creeping flows (no inertia).
We
have
The
works
method
about
300
this
in
follows.
as
work.
The
The
the
stores
program
defined
is
stress
points,
time-step,
Then the positions of the
the Lagrangian part of the
used to update the
stress.
of
some
points.
At
coordinates
each
at
these
of
Eulerian
fashion.
and velocity fields are solved for, in
updated by convecting them with the fluid flow. This is
The velocity gradient at each point is also known, and
method.
This procedure is repeated for each time-step, Figure 10.
and velocity, a grid must be laid
To solve for the
pressure
over
the
pressure
points
are
Voronoi
mesh
element
one
point
than
to
Thus, the
other.
any
follows.
as
the
at
of
the
to
grid
of the
points.
material
point is
closer
space
centre
connected
element
the
material
of the
all
is
be
all of the
with
Each
contains
point
material
finding the pressure, is deemed to
neighbouring elements. If any
of
purpose
suitable, and it is defined
of the mesh, and
that
element
is
material
every
The
associated
that
to
with
material
element, and, for the
material
points associated
an
becomes
distorted
too
the
as
reference [29j.
move,
are given in
found
boundary
The
and
velocity
be
subject
the
conditions,
the
requirement
must
to
pressure
of incompressible flow, V v
Vp
0, and the creeping flow condition, V («~~~~° +«~~~~
0.
points
material
the
reconnected.
mesh is
Voronoi
More
details
=
«~~~~° is the
viscoelastic
The
is
solution
contribution
"
the
solutions
parameters.
calculate
To
equations
model
vectors
are
finite
Voronoi
was
in
in
it
required
the
relation
work
required.
is
[29j.
Here
pi
tolerance.
we
Both
the
use
and fl2
this
efficient
relaxation
are
and integral
reptation-breaking
differential
micelle
stochastic
method.
For each
using the
Ours is the first work
time-step.
every
(with
the
stress
II.
steady
errors
arise
relaxation
viscoelastic
combine
[28j of implementing
method
point,
material
to
the
integral
an
regime,
This
shows
state.
The
the
as
the
taken
time
condition
initial
the
as
time),
of
unit
equilibrium,
was
and
the
and
Newtonian
shear
constant
a
rate
on.
steady
final
The
from
The
in
the
local
points
state
was
banded
a
flow
for
(I+I )-dimensional problem.
the
are
numerical
shear
rate
as
over
averages
Notice
noise.
(I+I)-dimensional calculation.
(or, equivalently, the shear stress) appears
unlike
jjjj
was
RESULTS.
The
constitutive
solve
~~
i~/ v)
within
to
updated
procedure with
had viscosity o.ool.
instantaneously
switched
the
~~
«Newt
+
conditions
decreasing
in Figure
in
+
and
solvent
6.I.
~
(~ («~"~°
of planar
Couette
flow. The plate separation
simulation
L, with
a
was
specified by equations (3)-(4). The
simulation
extended
distance
space
the longitudinal (~-) direction, with periodic boundary
conditions.
The time-step used
0.01
was
~ ~~
previous
and
element
calculation
boundary
L/2
Newtonian.
is the
=
equation.
constitutive
Our
a
used
above,
stored,
described
1000
stress,
been
~)
converged
have
the
have
=
~(Vv+VVT
and «~~~~
stress,
by iterating
found
~~~
until
the
to
The
to
a
function
time
that
and
the
position of the
be
determined
shear
of
the
transverse
broad
is
band
by
band
longitudinal
the
band
in
rates
typical
A
the
is
and
random,
shear
rate.
the
profile
unstable,
is given
coordinate
coordinate
j,
z.
macroscopic,
but
The
its
width
edges
of
N°4
FORMATION
~
FLOWS
565
Start
Iteration
Euleriar
~
SHEAR-BANDED
OF
Lagrangian Scheme
given flow geomewy,
the Voronoi grid
For
generate
boundary
and set
Viscoelastic
may be
stresses
computed
~~(#~~n~e~~~~~~$#~
results
from
of molecules
deformation
the
conditions
~~ P~Y~~~~
~~
~
whicll
history
physical quantities
grid,
Vv, V.v, V.~i and Vp
Calculate
tile
on
The
grids
and
then
as
r(~+A~)
the
Voronoi
moved
are
adapted by
e.g
r(~) + A~v ;
=
rule
N
~
~
~
~
~"
~~
~
Vp(£e
V.~i
~~
(V.vi£8
y
N
~~~~
Stop
Fig.
Flowchart
10.
band
undulations
final shear
the
for
blurred,
are
the
in
stress
is
the
Lagrangian-Eulerian
because
and
noise,
of the
(implying
interface
method
shown in Figure 9, and
for
the
used
~
~l[~'~i
P~
+
~2[V°V)
unchanged
~P]
two-dimensional
whether
unsteady flow)
almost
this is
"
unclear
it is
somewhat
a
P~~~
~~
or
this
from
is
due
numerical
to
the
calculations.
to
physical
effects.
The
(1+1)-dimensional
case.
Our
calculations
in
the
(I+I )-dimensional.
We
note
Segalman
that
the
equivalent
fluid [26j, with
2-dimensional
The
final
calculations
results
model
steady stress,
have
similar
to
given
have
as
been
those
a
carried
here.
realistic
band
than
pattern
rate, is not nluch changed.
Johnsonout by Espafiol et al. for the
function
a
more
of shear
JOURNAL
566
0
PHYSIQUE
DE
II
0.5
0.25
N°4
0.75
position
Fig.
11.
The
local
Band
shear
time
in
average
deviation.
Interface
7.
We
width.
found
some
take
E;r~terface
2-dimensional
a
is
state
shear
the
on
c~
the
start-up
steady
of
coordinate
bars
shear
some
are
bands.
the
shear
lov~ ~.
at
points are an
corresponding
standard
The
y.
We
of
character
the
that
Recall
the
to
be
order
an
will
have
and
(evtovit
thickness
in
Section
new
Notice
of
the
3,
term,
because
regarded
model.
order
context
we
may
as
a
is the
the
of the
this
to
linear
eliminate
equation (9)
Again
altered.
have
the
removed
Equation (26)
channel flow [33].
has
and
from
(Ref. [32]
of
0~/0j
of
cost
stress. It
to the total
material
and therefore has
viscoelastic
dimensions
becomes
consider
energy
an
cut
contribution
non-equilibrium
system is far
(~fT)~/~.
of slit
be
to
effects
is
of
powers
term
that
~f0E;r~terface/0a, giving
form
of the
every
structure
Expansions of the free energy in
parameter.
gradient-squared term [31j, and it is just this
phenomenological.
that ~f has
model
interfacial
viscoelastic
configuration
approach
usual
that
that, like
expect
(though possibly quite small)
(I+I )-dimensional model, in
non-zero
constitutive
suppose
now
microscopic
the
over
therefore
microscopic
the
between
have
must
requires the
effects
of these
dj(0a/0j)~.
similar
As in
and
of
lateral
at
rigorously justified here,
With
of
motivated
is
parameter usually include a
expression for E;r~terface.
use
as
our
Now include in equation (6) a term
should
calculation
function
a
interface
order
is
as
(~-) direction,
this
equation (6), and
J/
something of
This
given
flow
nature,
structure
account
average
an
in
discussion
toy model,
the
the
attention
our
found
we
To
an
along
from
by our
calculations
with the
indefinitely short length-scales. Presumably,
microscopic length, which would then be the interface width.
Our
off at
we
and
interface
which
is
in the
Effects
focus
now
other
arising
steady
pattern
rate
a
thermodynamics,
which is not
equilibrium. Our derivation
obtains
already
replace the
a
similar
coefficient,
diffusion
been
'force'
g
studied
with
the
so
result
from
the
interface
a
by Brunovskf
'potential'
V.
FORMATION
N°4
steady
the
In
0a/0t
state,
0, and
=
SHEAR-BANDED
OF
equivalent
therefore
is
the
variable
time
and
Now,
as
banded
a
flow
through
pass
we
as
Therefore,
Sn~;r~ < S < Sr~~ax.
has
position of
the
to
~fT
~)
=
mass).
Assume
-V
has
maxima.
two
through
followed by
the
constant
a
interface,
particle sitting
128)
particle moving
a
the
the
567
have
we
~fT~i
a
FLOWS
we
first
band,
different
a
in
potential
a
have
that
by
followed
in
constant
(with
-V
controlled
rapid
a
second
the
g
and
stress,
as
that
transition
band.
This
(for an extended period of
corresponds to the
maxima
top of one of the
on
time), then rolling down through the minimum, coming to rest on top of the other maximum,
Of course,
and remaining
there.
because of energy
this is only possible if the
conservation,
have the
height, and this will only be true for one particular applied stress. We
maxima
same
deduce
shear bands
For the toy model, with potential
that only for this one
coexist.
stress
can
This is
0.251.
V given by equation (10) and G
o.ol, this value is S
I and ~
T
=
in
have
we
exist,
could
flow
shown
for
motion
What
each
might
a
=
would
impossible
be
not
ut).
3,
This
in
have
the
gives
us
'~Q
equation
~
between
stress
flow is
banded
a
=
discussed
term,
any
actually
To
Section
does
f(j
that
is
case
even
stress
principle, for
in
in
=
=
interfacial
the
system.
model
if the
form
the
least
at
discussed
and
stress,
without
case
in
our
already
as
the
to
contrast
create
require
practice.
careful
coexistence
value?
~~$
Sn~;r~
stationary
a
3, where
Section
Of
and Sn~ax.
a
banded
what
course,
equations of
flow in the
controlled-stress
regime,
and
deliberate
manipulation of the
banded
a
in
Postulate
of
"
of the
solution
a
time-varying
of
solution
°.
~~~~
particle, moving in the same potential, but with a friction
constant
the
begin
higher
of
the
travel
through
the
valley
maxima,
two
can
now
on
UT.
and
before,
though
have
the
other
exactly
the
maxima
maximum,
to
rest
come
on
as
even
different heights.
during
This
that
the
The surplus
is
dissipated
the
journey.
energy
means
f has the same
function
albeit
qualitative form as the stress profile in the
coexistence
case,
travels at a
with
Thus, a transient
band
The
interface
is possible.
structure
non-zero
now
u.
speed u, determined by the height difference between the maxima, and therefore by the applied
Eventually, the interface will reach one of the rheometer plates and disappear, leaving
stress.
flow as the final steady state.
homogeneous
a
Olmsted
Goldbart
and
studied
similar
phenomena in the context of the
shear-induced
isotrosurface
which
pic-to-nematic
transition
arise
[34]. Their equations included
terms
natenergy
urally from the model of a nematogen used, but are similar in essence to the one here. They
studied the propagation of the interface
between
non-equilibrium phases, and located the
two
coexistence
A similar
point by finding the stress for which the interface no longer
moves.
technique has been used for the driven diffusive lattice gas by Ilrug et al. [35].
This
With
way.
shear
Now
changes.
The
describes
our
particle
The
the
The
results
rate
final
of
density p
/hj
o.ool,
size
decreasing stress
The
=
=
as
stress
the
control
simulated
o
parameter,
adjusts, according
steady state would
and
other
regime,
the
parameters
the
then
=
(a)
be
experiments
start-up
(creeping flow),
anticipate
we
to S
coexistence
stress
have
the
stress,
+ i1~,
banded
a
as
interface
that
the
the
proportion
flow
at
the
coexistence
(in (1+1)-dimensions) are
lo~~,
diffusion
constant
~f is
same
S
=
values
o.25,
as
is
stress.
in
Figure
(Recall
that
this
6.
grid
numerical
previously. As expected,
selected.
same
type of band
shown
the
the
in
moves
of each
in
the
is
the
JOURNAL
568
I
PHYSIQUE
DE
II
N°4
0.40
w
i~
$
~
m
#
0.20
w
~~~0.0
2000.0
1000.0
3000.0
time/z
Fig.
heavy
The
12.
toy
the
constitutive
homogeneous flow
starting at 30v~~ at
of S for
value
high
one
or
broad
two
If the
shear
corresponding
each
In
the
low-shear
until
the
The
model
line
shows
side
hand
upper
behaviour
the
to
of the
effect
of
3000
at
The
into
is
to
plateau,
the
whilst
mathematical
immediately
after.
prior
This
is in
to
the birefringence
sense). Thus, the
formation
band
to
contrast
the
shear
rate,
in the
stresses
(not shown)
pattern
low
contains
as
Figure
in
term
(heavy line).
is
given
instability
of
remains
stress
12
constant
flow is
them
to
recovered
the
(so the
observed;
rate
is
the
shear
is
which
high
axis
all of the
the
uniform
branch.
reads
now
rate
possible
shear-rate
time
7.
banded
a
Eventually,
move.
at
~i,
as
The
Figure
in
from
persists
flow
met.
be
much
increases
of the
are
realistic
the
than
that
with
shear
rate
in
of
measurements
and
Figure 7, although
same
shown
[11,15].
references
smoothly
then
goes
but is
phase shows a kink
continuous
optical properties of the homogeneous phase
those of the low-shear
band immediately
as
immediately
in Figure 12, where the
stress
low~shear
and
stress
behaviour
more
rheological
not
a
(in
decreasing
a
viscoelastic
threshold
between
uniform
a
~2
stress
types of band is changing,
two
quite reproduce the experimental
experiments show a stress which initially
does
the
past
decreasing the shear
the left). Hysteresis
stability threshold
in Figure 12
seems
the
are
(interfacial energy)
interface
the
and
to
Figure 7, on
corresponding
for
as
the
left.
behaves
stress
model, the
new
allows
the
increased
proportion
eliminated,
is
right
the
o at
it
modified
rate,
line is
corresponding flow
diffusive
is
the
the
at
aB
shear
thin
corresponds
line
zero
and
aA
The
rate.
the
rate
dotted
The
shear
with
The
material
dashed
shear
However,
increased.
because
The
the
where
continuously,
without
(Eq. (26)).
reaching
and
increasing
steadily
a
The
rate.
side
V(aB),
each
at
swept
is
when
case,
further
bands
=
calculation
develops.
flow
is
hand
of
term
respectively.
bands
rate
shear
same
V(aA)
rate
effect
the
diffusion
a
right
the
which
shear
with
the
at
to
and
shows
line
model
(as defined in Fig. I) and the stress
afterwards
This
is the
coexistence
stress.
possible
explanations.
Firstly,
the
bands
could
be
forming as soon as the coexissuggests two
though the system is not yet mechanically unstable, by nucleation
tence
stress is reached,
even
before
of
a
is Sn~ax
high
close
shear-rate
phases,
nematic
to
the
overshoot
We
have
interfacial
stress
(For a discussion
[17]). Alternatively,
band.
Ref.
see
at
which
the
mechanical
of
the
nucleation
coexistence
instability
occurs
in
the
stress
of
context
shear-induced
might happen
(I.e. Sn~ax),
Figure 12 is present but too small to detect.
attempted to calculate the properties of a
two-dimensional
not
surface
clear
that
this
will
However,
it
is
term
create
term.
a
to
that
so
be
the
very
stress
in
model
tension
in
with
the
the
interface
new
N°4
bands,
which
increase
the
between
tend
to
8.
Conclusions
We
shown
have
shear
that
width.
Numerical
whether
this
of the
is
unaltered
rate
is
new
physics
added
into
the
this
with
produce
bands,
model,
broad
steady
flow in
indefinitely
have
but
it
narrow
clear
yet
not
high shear rate).
equilibrium, there is
from
the
if
material.
is
term
a
if the
Even
small, the behaviour of the system is qualitatively
macroscopic bands, and the final stress after startmodel allows the position of the interface
between
very
therefore
is
stress
avoided
are
for the
constitutive
new
the
is
term
bands
unphysical
narrow
inhomogeneity in
the
allow
to
produces broad,
The
and
move,
and
phases (low
removed
dimensions
two
model
modification
different.
to
in
banded
a
bands
the
2-dimensional
simply an artefact
steady-state property of the
or
algorithm. The final steady shear stress after start-up of shearing at a given
from the (I+I )-dimensional problem, which may be an
that no
indication
constitutive
The
also
is
bands
analysis. However,
real
associated
changed.
produces
relation
constitutive
level of
(I+I )-dimensional level,
the
coefficient
calculations
presumably
therefore
and
flow.
banded
nonmonotonic
a
569
interface,
the
in
FLOWS
involved.
is
at
Even
up
a
numerical
fluctuations
suppress
stability of the
(I+I )-dimensional
the
at
will
SHEAR-BANDED
OF
FORMATION
by
selected
Although we
close analogy
a
the
two
only systems in states
equilibrium phase coexistence.
far
criterion
coexistence
a
have
between
considered
with
Acknowledgments
We
grateful
are
Olmsted
D.
P.
cial
Max
by
of
support
a
Planck
from
XCT
Appendix
Here
we
stochastic
every
the
EPSRC
04
a
is
contained
breaking
inside
or
within
Every
those
chains.
of
not
a
by position in the
position has a different
segments
micelle
The
The
along
moves
l'reptation').
segments
them
research
at
stress
of
a
by
part by
funded
been
funded
R.
S.
A.
form
the
has
was
N.
in
reptation-breaking
Ball,
C.
Espafiol,
P.
acknowledges
CASE award)
finan-
the
and
of
the
project
the
DTI
Link
the
EC
Network
and
contract
are
rate
is
The
supposed
B.
The
related
to
be
is in
distribution
the
position
lengths,
statistically
This
contour
breaking
segment
of the
tube
in
couple
destroyed at a
random
orientations
«
=
G~
range
a
and
we
its
the
micelle
Every
tube.
of
Now,
changes.
segments.
into
segment
positions
different
need
not
polymers,
unbreakable
one-dimensional
a
to
rate
reptation
D, and
orientations
u
and
micelles
distinguish
where
every
orientations.
reactions
have
at
equivalent,
to
the
within
been
and
contrast
of
continually
wormlike
imagine to be divided
place, the length of
of
range
is
for
takes
well-defined
a
segment
tube.
we
reaction
have
whole
segments
the segment
new
to
which
model
[13, 27, 28j.
references
see
tube,
a
recombination
does
motion
(in
of X. F. Y.
This
Greco,
F.
discussions.
Research
give a short description of the
implementation. For details,
time
McLeish,
B.
86.
Therefore, a segment
has belonged to chains
tube
work
BBSRC.
the
94
Shell
and
The
C.
T.
interesting
for
A
micelle
The
Drasdo
D.
Gesellschaft.
grant
ERBCHR
Marrucci,
G.
to
and
with
in
new
an
curvilinear
such
ones
way
created
a
isotropic
Brownian
that
to
tube
replace
distribution.
by
juuj
(A.ij
JOURNAL
570
~vhere
the
each
tube
PHYSIQUE
DE
material.
In the
segment in the
average (. .) is taken
over
every
constitutive
equation
segment is deformed affinely. The following
t
«(t)
This
is
integral
t
~cz Bit')
Ge
=
dt"D(t")
exp
survives
all
over
created
by the
t'.
at
past
t'.
times
This
The
contribution
is
viscoelastic
affine
relaxation
Assuming,
compensate.
In the
time.
of the
deformation
isotropic
the
the average is taken
over
B and D have yet to be found.
as
an
the
is
u
increase
that
B
both
are
is
(All
+
=
tube
a
equal to I IT, where T is
In particular,
longer
true.
no
the total
tube length, so D
increases
is unchanged, we have
flow, they
flow, this
a
(Vv)T
and
segment,
the
(uu)
(A.5)
(.
average
is
taken
over
all
segments
an
ensemble
material.
0ttinger
Following
of N
of
orientation
is
iA.3)
=
D
the
a
of
approximation,
B
where
of
presence
tends to
segments
ij
distribution.
absence
the
In
number
t.
l~
-
where
the
time
~~°~'iillli'~.~~
QiEtt)
in
(A.2)
=
entanglements per unit
given in terms of it, by
to
Q(Et,t) dt'.
degree of polymerisation per entanglement (so c/Ne is
volume). Et,t is the deformation
from time t' to
tensor
of
the
flow,
a
results:
and N~ is the
monomers,
the
of
presence
from
integrand is the
contribution
to the
stress
equal to the stress per segnlent, Q, multiplied
created at t', Bit'), multiplied by the probability that a segment
number of segments
3(c/Ne)kBT, where c is the
from t' to t (the exponential). Also, Ge
concentration
an
segments
of
N°4
II
vectors,
arbitrary,
according to
is
[28j, this
each
of
we
take
so
this
which
it
to
model
be
be
may
represents
unity.
On
stochastically.
solved
tube
one
each
segment.
time-step,
The
every
We
total
maintain
length of
vector
is
the
deformed
vectors
ailinely,
equation
r
r
~
+
/ht(Vv)~
(A.6)
r.
/ht/T
of the
eliminated, and replaced by new
random
with
vectors
vectors
are
relaxation
length I/N; this simulates the reptation-breaking
occurring
process
I/T. All the remaining vectors are then scaled by a constant factor, chosen so as to
at
rate
of the
enhanced
the total length to I, and this takes
destruction
account
segment
restore
rate
of a flow. This procedure gives the
results as the model
described
above.
in the
same
presence
The
is given by
stress
tensor
fraction
A
orientation
and
«
where
for
was
the
every
loco
sum
is
taken
vectors
over
grid
numerical
per
all the
element
element.
in
vectors
a
=
~jrr
Ge
in
the
simulation.
(A.7)
ensemble.
For all
A
the
separate
calculations
is required
presented here, N
ensemble
FORMATION
N°4
OF
SHEAR-BANDED
FLOWS
571
References
[1]
Doi
M.
Edwards
and
S-F-, The Theory of Polymer
Dynamics (Clarendon Press, Oxford,
1986).
[2]
[3j
[4j
[5j
[6j
[7]
[8]
[9]
[lo]
[ll]
[12j
[13j
[14j
[lsj
[16]
(Butterworth,
R-G-,
Solutions
Constitutive
Equations for Polymer Melts and
Stoneham, 1988).
Spenley N-A- and Cates M-E-,
Macromolecules
27 (1994) 3850.
Yerushalmi
Katz
S.
Shinnar
Chem.
J.,
and
R.,
Eng. Sci. 25 (1970) 1891.
Denn M-M-, Ann. Ret.
Fluid Mech. 22 (1990) 13.
Larson R-G-, Rheoi. Acta 31 (1992) 213.
McLeish
T-C-B- and Ball R-C-, J. Poiym. Sci. Part B-Poiym. Phys. 24 (1986) 1735.
McLeish T.C.B., J. Poijm. Sci. Part B-Poiym. Phys. 25 (1987) 2253.
Malkus D-S-, Nohel J-A- and Plohr B-J-, J. Camp. Phys. 87 (1990) 464.
Cates M-E-, McLeish
T-C-B- and
Marrucci G., Europhjs. Lett. 21 (1993) 451.
Rehage H. and Hoffmann H., Mol. Phys. 74 (1991) 933.
Cates M-E- and
Candau S-J-, J. Phys.:
Condens.
Matter 2 (1990) 6869.
Cates M-E-, J. Phys. Chem. 94 (1990) 371.
Spenley N-A-, Cates M-E- and McLeish T-C-B-, Phys. Ret. Lett. 71 (1993) 939.
Makhloufi R., Decruppe J. P., Ait-ali A. and Cressely R., Europhys. Lett. 32 (1995) 253.
Decruppe J-P-, Cressely R., Makhloufi R. and Cappelaere E., Coiioid Poiym. Sci. 273
Larson
(1995)
[17]
[18j
[19j
[20j
[21j
[22j
[23]
[24]
[25]
[26j
[27j
[28j
[29j
[30j
[31]
346.
J.-F., Roux D-C- and Porte G., J. Phys. II France 4 (1994) 1261.
Schmitt V., Lequeux F., Pousse A. and Roux D.,
Langmuir10 (1994) 955.
J.-F., Roux D-C-, Porte G. and Lindner P., Europhys. Lett. 25 (1994) 521.
Berret
Cappelaere E., Cressely R., Makhloufi R. and Decruppe J-P-, Rheoi. Acta 33 (1994) 431.
Malkus D-S-, Nohel J-A- and Plohr B-J-, SIAM J. Appi. Math. 51 (1991) 899.
Nohel J-A- and Pego R-L-, SIAM J. Math.
Anal. 24 (1993) 911.
Nohel J-A-, Pego R-L- and
Tzavaras A-E-, Proc. Roy. Sac. Edinburgh
l15A (1990) 39.
Johnson
M-W- and Segalman D., J.
Fluid Mech. 2 (1977) 255.
Non-NewtSchmitt V., Marques C-M- and Lequeux F., Phys. Ret. E 52 (1995) 4009.
Espafiol P., Yuan X-F- and Ball R-C-, submitted to J. Non-NewtFluid Mech.
Berret
Cates
M-E-,
Macromoiec~ies
20
(1987)
2289.
Ottinger H.C., J. Chem. Phys 91 (1989) 6455.
Yuan X-F-, Ball R-C- and
Edwards S-F-, J.
Non-NewtFluid Mech. 46 (1993) 331.
Yuan X-F-, Ball R-C- and
Edwards S-F-, J.
Non-NewtFhid Mech. 54 (1994) 423.
Rowlinson
J.S. and ividom B., Molecular Theory of Capillarity (Clarendon Press, Oxford,
1989).
[32]
[33j
[34j
El-Kareh
A-W-
Brunovskf
Olmsted
[35j Krug J.,
and
P.
and
P-D-
and
Lebowitz
Lea'
G-L-, J.
Non-Newt-
Fhid
Mech.
33
(1989)
(evbovib D., Q~art. Appi. Math. L2 (1994) 401.
Goldbart P-M-, Phys. Ret. A46 (1992) 4966.
J-L-, Spohn H. and Zhang M-Q-, J. Stat. Phys.
44
257.
(1986)
535.