Turning polymeric liquids into theorems

1
Turning polymeric liquidsinto theorems

• Michael Renardy
• Department of Mathematics
• Virginia Tech
• Blacksburg, VA 24061-0123, USA

2

• I came to New York because I heard the streets
  were paved with gold.
• When I got here, I learned three things
• The streets are not paved with gold.
• They are not paved at all.
• I am expected to pave them.
• Objective
• Why should mathematicians be interested in
  viscoelastic flows?
• What are some of the major open questions?
• What challenges arise that are qualitatively
  different from what everybody
• is familiar with?
• Mathematical issues in viscoelastic flows
• Existence
• Flow stability
• The high Weissenberg number limit

3
A quick review of models of non-Newtonian
fluids Balance Laws Conservation of mass and
momentum
v velocity T extra stress tensor
p pressure ? density
4
Constitutive Laws How is the stress tensor
related to the motion? Linear models Newtonian
fluid Linear viscoelasticity G Stress
relaxation modulus (usually assumed completely
monotone) Viscosity
5
Maxwell model Take G(s)? exp(-? s). Then we
have the alternative formulation Similarly,
if G is a linear combination of exponentials, we
can formulate a differential system. Jeffreys
model T is a linear combination of a Maxwell and
Newtonian term. 1/? is called a relaxation
time. Weissenberg or Deborah number Ratio of
the relaxation time to a time scale typical of
the flow.
6
(No Transcript)
7
Nonlinear models Generalized Newtonian fluid
Viscosity of polymeric fluids in shear flows
generally decreases strongly with increasing
shear rate. But Behavior in shear is different
from behavior in elongation, and model misses out
on normal stresses.
8
Integral models Nonlinear generalization of
Boltzmann theory Relative deformation gradient
Let y(x,t,s) be the position at time s of the
particle that occupies position x at time t. We
have Define In general, the stress is a
functional of the history of the relative
deformation gradient
9
Material frame indifference Stresses result only
from deformations, they are unaffected by
rotations of the medium. For any rotation Q, we
must have Relative Cauchy strain We
have where the functional is isotropic
10
K-BKZ integral model Motivated by
analogy with elasticity
Here C is defined analogously as above, but with
y(x,t,s) replaced by y(x,t), which is the
position of a particle in the equilibrium
configuration of the medium.
11
Differential models A system of ODEs relates the
stress to the velocity gradient. Easiest models
to analyze or simulate numerically. Examples Up
per convected Maxwell (UCM) model Phan-Thien
Tanner Giesekus Johnson-Segalman
12
The upper convected Maxwell model has the
alternative integral form In general,
differential models cannot be transformed to
integral form and vice versa.
13
Molecular theories
The time has come, the penguin said,
To speak of many things
Of flowing macromolecules,
And little beads and springs
That join together into chains
Or even stars or rings.
Robert Byron Bird
(with help from Lewis Carroll)

• Dilute solution theories
• Network theories
• Reptation theories.

14

• Polymer molecule is modeled as two spherical
  beads connected by a spring.
• Forces acting on beads
• Spring force.
• Drag from surrounding fluid.
• Random force due to Brownian motion.

15
Force balance on each bead With Rr2-r1,
this leads to If the stochastic forces are
described by a Wiener process with magnitude kT,
we have an equivalent Fokker-Planck equation for
a probability density ?(R,x,t)
16
Stress tensor With n denoting the number density
of dumbbells, the contribution of the tensions in
the springs to the stress is Hookean
dumbbells F(R)HR. Use the notation Let C
is known as a conformation tensor. For Hookean
dumbbells, we find
17
For STp-nkTI, this yields the upper convected
Maxwell model Nonlinear dumbbells
F(R)?(R2)R. Peterlin approximation
F(R)?(ltR2gt)R. With the Peterlin
approximation, we get
18

• Approximations
• Type A There is an identified small parameter ?.
  If the equations are correct up to
• some power of ?, we can expect that the solutions
  are also correct up to some power
• of ?.
• Type B The equations are too difficult. But our
  advisor want us to finish a thesis,
• our personnel committee wants us to publish a
  paper, our dean wants us to get a
• grant, etc. We cannot just say we are stuck. So
  we replace the equations with
• simpler ones that might still reproduce essential
  aspects of the problem.
• The Peterlin approximation is Type B. But it can
  become Type A in two limiting cases
• If ? is approximately constant. This is a
  reasonable assumption for R small.
• If R is approximately constant. This is a
  reasonabe assumption for steady
• homogeneous flows with large strain rates.

19
More issues Dumbbells ? chains. Hydrodynamic
interaction. Effect of the other polymers in the
surrounding fluid.
20
Existence problems for non-Newtonian flows
21

• Types of constitutive theories
• Differential models
• Integral models
• Microstructure models
• Partly solved
• Local existence for initial value problems
• Stability of the rest state and global existence
  for small data
• Existence of steady flows for small data
• Remaining issues
• Microstructure and integral models are less
  completely studied than differential
• models.
• Behavior of distribution function at the limiting
  extension.
• Characterization of decay rates (when they are
  not exponential).
• Conditions at infinity for unbounded flows.

22

• But what about global existence?
• Existence proofs for initial value problems have
  two parts
• An argument for local existence, typically based
  on proving convergence
• of some approximation scheme.
• 2. A priori estimates showing solutions do not
  blow up and can be continued.
• The Newtonian case
• Assume, for simplicity, periodic boundary
  conditions.
• If we multiply by v and integrate we find

23
This is enough to guarantee global existence (but
not uniqueness) of a weak solution. In two
dimensions, we can do more. Take the curl of the
equation of motion, and let ? be the vorticity.
We find and hence This suffices to
prove global existence of smooth solutions.
24

• Do non-Newtonian fluids help?
• It was once widely believed that non-Newtonian
  fluids might have better existence
• results. This is the case only in very limited
  instances, e.g.
• Shear thickening generalized Newtonian fluids.
• The thermodynamic second order fluid
• Great theorems if ?1gt0 and ?22?1. But ?1lt0
  in real fluids.
• 3. Equations based on cutting corners (e.g.
  keeping the inertial nonlinearities
• and neglecting the constitutive ones).
• In general, however, the situation for
  non-Newtonian fluids is far less understood than
• the Newtonian case. The reason for this is the
  lack of a priori estimates.

25
The corotational Jeffreys fluid (Lions and
Masmoudi)
Energy estimate Multiply momentum equation by v,
and constitutive equation by T/(2?) and
integrate. The result is
26
Lions and Masmoudi prove that this a priori bound
can be used to prove global existence of weak
solutions. But this only works because of a
miracle Compare the upper convected Maxwell
model Note that The correct energy
estimate is but this is much too weak to
infer existence of weak solutions.
27
Why is this an a priori estimate at all? Note
that It follows that positive definiteness
of T? I is preserved. Only solutions with
positive definite stresses are physically
relevant. The Challenge The only global
existence results of any kind for realistic
models of polymeric fluids are for
one-dimensional shear flows and they are based on
treating the viscoelastic terms as a controlled
perturbations of a Newtonian problem.
28
A remark on the side
If you feel nervous about drinking a glass of
water while global well-posedness of
Navier-Stokes is unresolved
you might consider that the water must form a
free surface jet before it gets into the glass.
Free surface jets break into droplets. Asymptotic
solutions for breakup have been studied for
Newtonian as well as viscoelastic
fluids. Heuristic arguments and numerical
evidence suggest there should be general theorems
linking the initial shape of the jet to the
eventual asymptotics of the breakup (or, if no
breakup occurs, the asymptotics for t??). No such
theorems have been proved, even for
one-dimensional approximations of the Newtonian
case.
29

• Stability problems in viscoelastic flows
• Viscoelastic flows have been found to show many
  instabilities, even at zero
• Reynolds number.
• Examples
• Shear flows with curved streamlines.
• Entry flows.
• Extrudate instabilities.
• Elastic mechanisms for jet breakup.
• Interfacial instabilities.

30
Taylor cells
31
Cone and plate flow
32
Coextrusion interfacial waves
33
Melt fracture
34

• Usual analysis of instabilities
• Linearize about a known base state.
• Look for eigenvalues crossing the imaginary axis
  at critical values of the
• relevant parameter (e.g. Reynolds or
  Weissenberg number)
• Do a bifurcation analysis.
• All of this has been justified rigorously for the
  Navier-Stokes equations.
• However, there are major unresolved problems for
  viscoelastic flows.

35
Abstract framework C0 semigroups Consider an
evolution problem with solution For
matrices, we know that the eigenvalues of exp(At)
correspond to those of A, but in infinite
dimensions the situation is more complicated. In
general, we can split the spectrum of A and of
exp(At) into isolated eigenvalues of finite
multiplicity and the rest, known as the
essential spectrum. The isolated eigenvalues
cause no problem, but we cannot always infer the
essential spectrum of exp(At) from that of A. It
is actually possible for exp(At) to have an
essential spectrum even if A does not. But does
this happen in real problems?
36
Example with 2?-periodic boundary
conditions. The natural setup is to set and
look for (u,v) in the function space H1? L2. The
eigenvalues of A are purely imaginary, and there
is no essential spectrum. However, the essential
spectrum of exp(At) has radius exp(t/2).
Newtonian flows Essential spectra exist only
in unbounded geometries. Even then, linear
stability of the flow can be inferred from the
spectrum of A. Viscoelastic flows Essential
spectra always exist, they are notoriously
difficult to compute, and we have no rigorous
connection between stability and spectrum.
37
Advective equations (R. Shvydkoy) These are
problems of the form where q satisfies
periodic boundary conditions and A is a
pseudodifferential operator of order zero
(something like, for instance the inverse of a
second order differential operator times another
second order differential operator). Shvydkoys
result does not say stability is determined by
the spectrum of but instead it characterizes
what else we need to determine stability.
38
Pseudodifferential operators of order zero have
the property that where A0 is homogeneous of
order zero For instance, we could have
39

• To determine the stability of an advective
  equation, we have to do two things
• Find the discrete eigenvalues.
• Study the stability of the amplitude b in the
  following ODE system
• The good news Creeping flows of fluids with
  differential constitutive laws fit into
• this framework.
• The bad news At this point, this only works for
  periodic problems.

40
The high Weissenberg number limit Analogy high
Reynolds number limit for Newtonian flow
Problem pour a pint of beer
We ignore the difficult free surface flow and
concentrate on the flow inside the tap, which we
idealize as a straight pipe.
41
Attempt 1 Poiseuille flow solution. This is a
solution at any Reynolds number, but is not
observed at high Reynolds number. Attempt 2
Euler equations. Allow for any parallel flow
profile, and numerous other solutions. Too many
solutions to predict anything. Attempt 3 Assume
approximately uniform flow away from the walls,
and try to match the boundary conditions with
boundary layers. Textbook example of boundary
layers Boundary layers in fluids are sort
of like this example, but only sort of. There
is the well known Blasius solution. But how does
a boundary layer that looks like this fit
into a tap that looks like that?
42
The actual description of this seemingly simple
problem is a patchwork of heuristics, formal
approximation, truncated models etc. We really
cannot expect the situation to be any better in
the high Weissenberg number limit of viscoelastic
flows. Theorems will necessarily be limited in
scope, addressing only specific aspects of the
overall problem. Excerpt from careerplanner.com

Skill Requirements for "Bartender" Active
Listening -- Giving full attention to what other
people are saying, taking time to understand the
points being made, asking questions as
appropriate, and not interrupting at
inappropriate times.Speaking -- Talking to
others to convey information effectively.Social
Perceptiveness -- Being aware of others'
reactions and understanding why they react as
they do.Mathematics -- Using mathematics to
solve problems.
43
High Weissenberg Number Asymptotics The upper
convected Maxwell model and the Euler
equations Steady creeping flow of the
UCM If we formally set W?, we get
44
If we assume T is of rank 1, i.e. T? qqT, where
? is normalized in such a way that then we
obtain the equations The first two
equations are like the compressible Euler
equations with q replacing the velocity and ?
replacing the density. However, there is no
equation of state linking p to ?.
45
Analogue of potential flow Assume a
two-dimensional flow where q is parallel to v,
and ? is constant along streamlines. Then we
obtain precisely the incompressible Euler
equations. All solutions of these equations, in
particular potential flows, reappear as
solutions of the high Weissenberg number
viscoelastic problem.
46
Boundary layers Model problem Suppose f is
periodic in x. If we formally let W ? ?, u
becomes equal to the average of f, but for y0,
we must have uf(x). There is a boundary layer
when y is of order 1/W. In fact, the solution
is In viscoelastic fluids, similar boundary
layers arise in the stress, due to the
presence of the term
47
(No Transcript)
48
From H.-S. Dou and N. Phan-Thien, JNNFM 87
(1999), 47-73.
49
Boundary layer for the UCM fluid Governing
equations
50
The boundary is at y0. If we assume a shear rate
of order 1 at the wall, then T12 is of order 1,
and T11 is of order W there. If we balance, for
instance, the term u(? T11/? x) against T11/W, we
find that the term becomes significant when u
(i.e. y) is of order 1/W. This motivates the
scaling
51
Boundary layer equations
52
Similarity solutions It is possible to find
self-similar solutions of the boundary layer
equations which depend only on a combination of
the form x-?z. These solutions are analogous to
the Blasius solution for Newtonian boundary
layers. They play a role in describing corner
behavior.
53
Other models Boundary layers can similarly be
analyzed for other constitutive models. Because
the viscometric behavior changes, we have other
balances to be considered. For the PTT or
Giesekus models, we have to balance Viscomet
ric stresses are T11? W-1/3 for the PTT and T11?
W-1/2 for the Giesekus model. Hence the boundary
layer thickness is W-1/3 for PTT and W-1/2 for
Giesekus.
54
Corner Singularities
55
Newtonian case Stokes flow Note If the
velocity behaves like r?, then viscous stresses
behave like r?-1, while Reynolds stresses behave
like r2?. As long as ?gt-1, viscous stresses
dominate near the corner. Stream
function Leads to the equation
56
Separate variables The biharmonic equation
becomes with boundary conditions Solution
57
This leads to the eigenvalue problem We want
finite velocity at the corner, i.e. Re(?)gt1. The
smallest eigenvalue is less than 2 if ?gt? and
greater than 2 if ?lt?. Note ?lt2 means that
viscous stresses are infinite at the
corner. Implication for non-Newtonian flows If
?lt?, we can expect Stokes behavior, nonlinear
terms are a perturbation. But if the corner is
reentrant (?gt?), the corner behavior is not
determined by linear terms!
58
Reentrant corner flow of the UCM
fluid Assumptions ? No separating
streamlines ? Flow away from walls given by
potential flow solution of the Euler
equations. ? Similarity solution of the boundary
layer equations applies near the
wall. Remark Some experiments do show
separating streamlines (lip vortex). Corner
asymptotics with lip vortex remains to
be understood. We shall focus on the 270 degree
corner. We can scale out the Weissenberg number,
due to the self-similar geometry.
59
(No Transcript)
60
Potential flow solution for a 270 degree
corner The stream function is given
by and the stress is given by
61
On the other hand, we must have viscometric
stresses at the wall. The transition occurs when
the stretch rate is of order 1, i.e. when ?
is of order r(6-2n)/(3n-3). At this point
the shear rate is of order and the
corresponding viscometric first normal stress is
of order
62
On the other hand, the stress from the Euler
solution is of order To enable a transition,
we want Hence
This potential flow breaks down near the walls.
It can be matched to a similarity solution of the
boundary layer equations. These similarity
solutions satisfy a nonlinear system of ODEs and
must be found numerically.
63
The time dependent case Consider the system of
equations It can be checked that any flow
of the form is a solution. Thus there is no
well-posed initial value problem associated with
these equations. What do you do when your
equations cannot predict the future evolution?
64

• The best way to predict the future is to invent
  it. (Alan Kay, 1971)
• But the future may not turn out to be as you
  invented it
• We must invent the future, not just accept it.
  (Walter Mondale, 1984)
• Rigid body Motion completely determined,
  stresses unknown.
• Vacuum Stresses completely determined, motion
  unknown.
• The viscoelastic fluid at infinite W with rank
  one stress behaves like an elastic
• medium with a nonzero modulus in only one
  direction.
• Challenges
• Can we characterize the known and unknown
  part of the evolution in a
• meaningful way?
• What can we say about the asymptotics of
  singularly perturbed problems

65
Shear flow stability at high Weissenberg number
Dimensionless equations We are
interested in the case where R and W are both
large. We shall set them equal to infinity. Since
the largest stress component in shear flow is of
order W rather than order 1, we also scale the
stresses with an additional factor W. The
resulting equations involve the
combination EW/R, known as the elasticity number.
66
The reduced equations are The
boundaries are either walls v(x,0,t)v(x,1,t)0,
or free surfaces p(x,0,t)-T22(x,0,t)p(x,1,t)-T2
2(x,1,t)0.
67
Linearization at parallel flows We consider two
dimensional flows. Any steady flow of the
form with arbitrary functions U(y) and S(y)
is a solution. Since stresses for the UCM fluid
must satisfy a positive definiteness condition,
we assume S0. We now linearize the equations
for small perturbations
68
The resulting linear system can be reduced to the
single equation The associated boundary
conditions are q(0)q(1)0 for walls and
q(0)q(1)0 for free surfaces.
69
Howards semicircle theorem
It can be shown that for nonreal c we must
have This is the equation of a circle. For
E0, this is Howard's semicircle theorem.
When the circle disappears, and unstable
eigenvalues cannot exist.
Hughes and Tobias (2001) found the same result in
the context of ideal magnetohydrodynamics (same
equations). Challenge Can a rigorous proof of
stability be given?
70

• Remarks
• The result above can be interpreted in terms of
  wave speeds. Inviscid instabilities
• are suppressed if the range of fluid speeds
  is less than twice the elastic wave
• speed. This precludes a resonant
  interaction between forward and backward
• traveling waves.
• 2. For free surface flows, and E0, all
  nonconstant velocity profiles are unstable.

71
Are viscoelastic flowsor out of control?
under control
TexPoint fonts used in EMF. Read the TexPoint
manual before you delete this box. AAAAAA
72
(No Transcript)
73
Controllability

• Can f(t) be chosen such that x(T) assumes a given
  value?
• In the context of PDEs, f is usually restricted
  to a subdomain or to the boundary.
• For elasticity and Newtonian fluid mechanics,
  controllability (with the control being
• a given body force or given boundary data) is
  widely studied.
• Usual approach
• Show that the linear problem is controllable.
• Use a perturbation method for the nonlinear
  problem.
• New issues in viscoelastic flows
• We can (pretend to) control the equation of
  motion, but not the interaction
• between flow and microstructure. This
  precludes full controllability.
• 2. Linear results do not tell us what happens in
  the nonlinear problem.

74
Example Linear Maxwell fluid
We can add a control to the momentum equation,
but we cannot control the constitutive law!
75
Can we control the stresses in addition to the
velocities? No!
We have no control on R through anything we put
into the equation of motion! The invariance of
the subspace Y does not persist if nonlinearities
are included!
76
Nonlinear Problems We consider a restriction to
simple flows (Assumption homogeneous velocity
field)
77
Shear flows of the upper convected Maxwell fluid
Special case of homogeneous shear flow
78
We have ?0 and
Hence
with equality possible only if ? is identically
zero.
Note that ??-?2 is the determinant of the matrix
On the other hand, if we make the shear rate
large, we can move rapidly along the Manifold
??-?2const. Note For inhomogeneous shear
flows, a pointwise constraint of the form ()
suffices to ensure accessibility of a final state
if there is control on the entire flow domain,
but not if the control is only on a subinterval.
79
Generalizations Other constitutive
models Several relaxation modes Three dimensional
homogeneous flows Open problem What can we say
about inhomogeneous flows?
80
(No Transcript)
81
Questions?