Turning polymeric liquids into theorems

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

“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:

1. Existence 2. Flow stability 3. The high Weissenberg number limit 4. Controllability

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

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:

Maxwell model: Take G(s)=  formulation exp(  s). Then we have the alternative 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.

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.

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:

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:

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.

Differential models

A system of ODEs relates the stress to the velocity gradient. Easiest models to analyze or simulate numerically.

Examples: Upper convected Maxwell (UCM) model: Phan-Thien Tanner: Giesekus: Johnson-Segalman:

The upper convected Maxwell model has the alternative integral form In general, differential models cannot be transformed to integral form and vice versa.

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)

1. Dilute solution theories 2. Network theories 3. Reptation theories.

Polymer molecule is modeled as two spherical beads connected by a spring.

Forces acting on beads: 1. Spring force.

2. Drag from surrounding fluid.

3. Random force due to Brownian motion.

Force balance on each bead: With

R

=

r

2 -

r

1 , 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):

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

)=H

R

.

Use the notation Let

C

is known as a conformation tensor. For Hookean dumbbells, we find

For

S

=

T

p -nkT

I

, this yields the upper convected Maxwell model: Nonlinear dumbbells:

F

(

R

)=  (|

R

| 2 )

R

.

Peterlin approximation:

F

(

R

)=  (<|

R

| 2 >)

R

.

With the Peterlin approximation, we get

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: 1. If  is approximately constant. This is a reasonable assumption for R small.

2. If R is approximately constant. This is a reasonabe assumption for steady homogeneous flows with large strain rates.

More issues: Dumbbells → chains.

Hydrodynamic interaction.

Effect of the other polymers in the surrounding fluid.

Existence problems for non-Newtonian flows

Types of constitutive theories:

1. Differential models 2. Integral models 3. 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.

But what about global existence?

Existence proofs for initial value problems have two parts: 1. 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

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.

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.

1. Shear thickening generalized Newtonian fluids.

2.

The “thermodynamic” second order fluid: Great theorems if  1 >0 and  2 =2  1 . But  1 <0 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.

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

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.

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.

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.

Stability problems in viscoelastic flows

Viscoelastic flows have been found to show many instabilities, even at zero Reynolds number.

Examples: 1. Shear flows with curved streamlines.

2. Entry flows.

3. Extrudate instabilities.

4. Elastic mechanisms for jet breakup.

5. Interfacial instabilities.

Taylor cells

Cone and plate flow

Coextrusion: interfacial waves

Melt fracture

“Usual” analysis of instabilities

1. Linearize about a known base state.

2. Look for eigenvalues crossing the imaginary axis at critical values of the relevant parameter (e.g. Reynolds or Weissenberg number) 3. 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.

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?

Example: with 2  -periodic boundary conditions. The natural setup is to set and look for (u,v) in the function space H 1  L 2 . 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.

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). Shvydkoy’s result does not say stability is determined by the spectrum of but instead it characterizes what else we need to determine stability.

Pseudodifferential operators of order zero have the property that where A 0 is homogeneous of order zero: For instance, we could have

To determine the stability of an advective equation, we have to do two things: 1. Find the discrete eigenvalues.

2. 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.

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.

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?

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.

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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

If we assume

T

is of rank 1, i.e.

T

=  that

qq

T , where  is normalized in such a way then we obtain the equations The first two equations are like the compressible Euler equations with

q

the velocity and  replacing replacing the density. However, there is no “equation of state” linking p to  .

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.

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 y=0, we must have u=f(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

From H.-S. Dou and N. Phan-Thien, JNNFM 87 (1999), 47-73.

Boundary layer for the UCM fluid

Governing equations:

The boundary is at y=0. If we assume a shear rate of order 1 at the wall, then T 12 is of order 1, and T 11 term u(  is of order W there. If we balance, for instance, the T 11 /  x) against T 11 /W, we find that the term becomes significant when u (i.e. y) is of order 1/W. This motivates the scaling

Boundary layer equations:

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.

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 Viscometric stresses are T 11  W -1/3 for the PTT and T 11  Hence the boundary layer thickness is W -1/3 W for PTT and W -1/2 -1/2 for the Giesekus model.

for Giesekus.

Corner Singularities

Newtonian case: Stokes flow

Note: If the velocity behaves like r  , then viscous stresses behave like r  -1 , while Reynolds stresses behave like r 2  . As long as  >-1, viscous stresses dominate near the corner.

Stream function: Leads to the equation

Separate variables: The biharmonic equation becomes with boundary conditions Solution:

This leads to the eigenvalue problem We want finite velocity at the corner, i.e. Re(  )>1. The smallest eigenvalue is less than 2 if  >  and greater than 2 if  <  .

Note:  <2 means that viscous stresses are infinite at the corner.

Implication for non-Newtonian flows: If  <  , we can expect Stokes behavior, nonlinear terms are a perturbation. But if the corner is reentrant (  >  ), the corner behavior is not determined by linear terms!

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.

Potential flow solution for a 270 degree corner:

The stream function is given by and the stress is given by

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

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.

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?

“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: 1.

Can we characterize the “known” and “unknown” part of the evolution in a meaningful way?

2. What can we say about the asymptotics of singularly perturbed problems (small inertia or small additional stress components)?

3. How should we interpret solutions of the degenerate set of equations (e.g.

“potential flow” solutions)?

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 E=W/R, known as the elasticity number.

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)-T 22 (x,0,t)=p(x,1,t)-T 22 (x,1,t)=0.

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 S ≥0.

We now linearize the equations for small perturbations:

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.

Howard’s semicircle theorem

It can be shown that for nonreal c we must have This is the equation of a circle. For E=0, 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?

Remarks:

1. 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 E=0, all nonconstant velocity profiles are unstable.

Are viscoelastic flows

under control or out of

contr

o l ?

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: 1. Show that the linear problem is controllable.

2.

Use a perturbation method for the nonlinear problem.

New issues in viscoelastic flows: 1. 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.

Example: Linear Maxwell fluid

We can add a control to the momentum equation, but we cannot control the constitutive law!

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!

Nonlinear Problems: We consider a restriction to simple flows (Assumption: homogeneous velocity field)

Shear flows of the upper convected Maxwell fluid

Special case of homogeneous shear flow:

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   2 = const.

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.

Generalizations:

Other constitutive models Several relaxation modes Three dimensional homogeneous flows Open problem: What can we say about inhomogeneous flows?

Questions?