27/11/2007 Bonn, Institut fuer Numerische Simulation 1 A century of viscoelastic modelling: from...

27/11/200727/11/2007 Bonn, Institut fuer Numerische SimulatiBonn, Institut fuer Numerische Simulationon

11

A century of viscoelastic modelling: A century of viscoelastic modelling: from Maxwell to the eXtended Pom-from Maxwell to the eXtended Pom-

Pom moleculePom molecule

Giancarlo RussoGiancarlo RussoCardiff School of MathematicsCardiff School of Mathematics


Outline

• Different models for different problemsDifferent models for different problems -Disperse polymer solutions -Upper Convective Maxwell -Oldroyd-B -Finite Extensible Nonlinear Elasticity

• A few words about my research project - Motivation - Results achieved - Coming next (hopefully…)

• Viscoelastic matter: a brief description and main featuresViscoelastic matter: a brief description and main features

- Concentrated polymer solutions - Phan Tien – Tanner - Tube model - Pom-Pom model - eXtended Pom-Pom model


Stress-Strain relation Stress-Strain relation

(constitutive equation) (constitutive equation)

Modelling continuum mechanics Modelling continuum mechanics

( ( ) )q

q q p gt

( ) 0q

qt

2 &

Field Equation of MomentumField Equation of Momentum

Continuity EquationContinuity Equation

Newtonian FluidsNewtonian Fluids

Non-Newtonian FluidsNon-Newtonian Fluids

Viscoelastic FluidsViscoelastic Fluids

2 ( : ) & &&

??


The relaxation time, aka The relaxation time, aka “will I run out of patience before the flow ?”“will I run out of patience before the flow ?”

About 10d-12 for water; more than a day for About 10d-12 for water; more than a day for glass.glass.

Relaxation Time and Deborah NumberRelaxation Time and Deborah Number

The The DeborahDeborah number, aka number, aka

““I’ve been waiting enough !!!”I’ve been waiting enough !!!”

Setting the time of the experiment will Setting the time of the experiment will determine the behaviour of the matter. determine the behaviour of the matter.

Solution relating shear stress to rate of Solution relating shear stress to rate of strain: the stress at any time depends on strain: the stress at any time depends on the whole strain history; the further back the whole strain history; the further back

in time, the more memory fades in time, the more memory fades

Maxwell Model: Young modulus and Maxwell Model: Young modulus and viscosity gathered togetherviscosity gathered together

1 1xy xy xy

d

G dt

&

21 1

( ) exp( ( )) ( )t

xy xy

t st s ds

&

1 G

““Memory” of the fluidMemory” of the fluid

1DeT


Some (not so obvious…) effects of elasticity Some (not so obvious…) effects of elasticity

1, solid-likeT

High extensional High extensional viscosity/shear viscosity viscosity/shear viscosity

ratio: the ratio: the open siphon effectopen siphon effect

Releasing tension along the Releasing tension along the streamlines: the streamlines: the Die SwellDie Swell

Supporting tension along Supporting tension along the SL: the SL: Rod ClimbingRod Climbing

Visualizing different Deborah Visualizing different Deborah numbers: 9 to 1 mixture of numbers: 9 to 1 mixture of

cornstarch and watercornstarch and water

1, fluid-likeT


……leads to the leads to the Upper Convective Upper Convective Maxwell Maxwell model for the constitutive model for the constitutive equation.T is total extra-stress and equation.T is total extra-stress and

Gamma-dot is the rate of strain.Gamma-dot is the rate of strain.

Modelling disperse polymer solutions I: UCM and Kelvin (late Modelling disperse polymer solutions I: UCM and Kelvin (late XIXth century)XIXth century)

1( ( ) ( ) ) 2TDTT u T T u

Dt &

1 1xy xy xy

d

G dt

&

The Maxwell model for shear flows…The Maxwell model for shear flows…

The The upper convective derivative upper convective derivative takes into account takes into account the deformation induced by the rate of strain, a the deformation induced by the rate of strain, a

feature which is typical of elastic fluids, and adds to feature which is typical of elastic fluids, and adds to the usual material derivative describing the flux. the usual material derivative describing the flux.

The Maxwell model above is one of The Maxwell model above is one of two basic ways of “mixing” the two basic ways of “mixing” the

Young modulus with the Newtonian Young modulus with the Newtonian viscosity. Another one is the Kelvin viscosity. Another one is the Kelvin

modelmodel

( )xy xy xyt G &

Let’s combine them…Let’s combine them…


Modelling disperse polymer solutions II: Oldroyd-B (1950)Modelling disperse polymer solutions II: Oldroyd-B (1950)

1 2ˆˆ 2 ( )T T & &

ˆ 2(1 )p pWe &

……and we find the Oldroyd-B modeland we find the Oldroyd-B model

2

1

1

solv

tot

UWe

L

Physically the pure elastic molecules are replaced by bulks of particles and fluid, and Physically the pure elastic molecules are replaced by bulks of particles and fluid, and these will be the clusters swimming in the solvent. these will be the clusters swimming in the solvent.

This replacement leads to the splitting of the This replacement leads to the splitting of the extra-stress into its extra-stress into its solventsolvent and and polymericpolymeric

contributions. The former is plugged as diffusive contributions. The former is plugged as diffusive term into the balance of momentum; the latter is term into the balance of momentum; the latter is what is practically computed in the constitutive what is practically computed in the constitutive

equation, here in its dimensionless form.equation, here in its dimensionless form.

Solvent vs Total Solvent vs Total viscosity ratioviscosity ratio

Weissenberg Weissenberg number, measure number, measure of elasticity of the of elasticity of the

fluidfluid


The equations of motions for the bulks The equations of motions for the bulks include some Brownian forces; include some Brownian forces;

these Brownian forces depends on a these Brownian forces depends on a probability density function, say probability density function, say qq;;

A Fokker-Planck equation (diffusion A Fokker-Planck equation (diffusion equation for equation for q) q) is derived.is derived.

Modelling disperse polymer solutions III: Oldroyd-B at the microscopeModelling disperse polymer solutions III: Oldroyd-B at the microscope

m2

m1

sT n k Temp I n RF &

r1

r2

F1Fluid

Bulk

R = r2 - r1

F2

A common expression relating stress and A common expression relating stress and the product RF is due to Kramers: the product RF is due to Kramers:

Combining this and the FP equation we Combining this and the FP equation we find the so called find the so called Giesekus expr.Giesekus expr.

1 2

( )2(1/ 1/ )s

nT UCD RR

&

The whole point is the The whole point is the choice for Fchoice for F : :F F hookean hookean meansmeans Oldroyd-B. Oldroyd-B.

Unfortunately this leads to an infinite Unfortunately this leads to an infinite extensibility of the dumbbells!!!extensibility of the dumbbells!!!

Another problem is a discontinuity in the Another problem is a discontinuity in the extensional viscosity.extensional viscosity.


Modelling disperse polymer solutions IV: FENE modelsModelling disperse polymer solutions IV: FENE models

2 201 ( / )

hRF

R R

r

2 201 /

hRF

R R

r

1

1( ) ( ( ) ( ))

ktUCD RR I f R UCD RR

h

FENE FENE

1

1( ) ( ) ( ( ))

ktUCD RR f R I UCD RR

h

FENE – P (Peterlin)FENE – P (Peterlin) FENE – CR (Chilcott - Rallison)FENE – CR (Chilcott - Rallison)

Limiting the dumbbells extensibility through bounding functions; overcoming the Limiting the dumbbells extensibility through bounding functions; overcoming the discontinuity in the extensional viscosity. Mainly suitable for extensional flows.discontinuity in the extensional viscosity. Mainly suitable for extensional flows.

Combining the Kramers, Giesekus and the Combining the Kramers, Giesekus and the FENE-P we obtainFENE-P we obtain

The quantity RThe quantity R0 0 is is the maximum the maximum extension the spring extension the spring can reach. can reach. It predicts constant It predicts constant shear flow, a non-shear flow, a non-zero 1zero 1stst normal normal stress difference and stress difference and the extension is the extension is bounded.bounded.

The model is shear-The model is shear-thinning; the thinning; the extensional extensional viscosity exhibit viscosity exhibit continuous continuous dependence on the dependence on the extensional rate. extensional rate. Being shear Being shear thinning is suitable thinning is suitable for shear flows.for shear flows.

The FENE-CR replaces the expression above The FENE-CR replaces the expression above by the following: by the following:

It presents the same features as Old-B but the It presents the same features as Old-B but the extensibility is bounded. extensibility is bounded.


Modelling concentrated polymer solutions: the PTT model Modelling concentrated polymer solutions: the PTT model (1977)(1977)

Lt r

1 2( ) ( ) ( )f f f

v L C C D g h ft r

In describing these melts, is essentials to represent in a proper way the In describing these melts, is essentials to represent in a proper way the entanglement between molecules. The PTT looks at the polymer entanglement between molecules. The PTT looks at the polymer molecules and their interaction as a network.molecules and their interaction as a network.

Strands are linked through rigid junctions. Strands are linked through rigid junctions.

Slip is the cause of dynamics of the strands, modelled as follows:Slip is the cause of dynamics of the strands, modelled as follows:

Shear thinning; extensional bounded; 1Shear thinning; extensional bounded; 1 stst normal stress predicted. normal stress predicted. This makes PTT a suitable model for shear flows of polymer melts. This makes PTT a suitable model for shear flows of polymer melts. Stress overshoots at high strain rates in elongational flows are also fairly reproduced. Stress overshoots at high strain rates in elongational flows are also fairly reproduced. The absence of 2The absence of 2ndnd NSD is the main limit for extensional simulations. NSD is the main limit for extensional simulations.

11 1ˆ (1 ( )) ( ) p

p

tr

& & &

Multiplying bothMultiplying both sides by rho and sides by rho and averaging, we obtain the Const. Eq.averaging, we obtain the Const. Eq.

f is the probability distribution of the junctions; f is the probability distribution of the junctions; here is its rate of change balance, with g and h here is its rate of change balance, with g and h rate of creation and destruction of the junctions. rate of creation and destruction of the junctions.


Modelling concentrated polymer solutions II: the tube model (1978)Modelling concentrated polymer solutions II: the tube model (1978)

2

2

( ,0; ) ( )

( , ; ) ( , ; ) 0

cDt

s s

s t s s L t s

31

22

DE

R

T K N

T K N

Developed by Doi and Edwards, translates the interactions between molecules as Developed by Doi and Edwards, translates the interactions between molecules as topological constraints. The presence of other chains surrounding a test molecule topological constraints. The presence of other chains surrounding a test molecule will confine the allowed configurations within a tube of a certain diameter.will confine the allowed configurations within a tube of a certain diameter.

The primitive chain AB The primitive chain AB reptatesreptates; part of it leaves ; part of it leaves the original tube for another.the original tube for another.This is measured by a probability distribution This is measured by a probability distribution function, say function, say thetatheta: :

2

( ) ( ) exp( )p d

p tt C p

T

( ) ( )GQ E t

Averaged (in space) solution Averaged (in space) solution Disentanglement Disentanglement time comparisontime comparison

Stress-orientation tensor Q(E) Stress-orientation tensor Q(E) relationrelation

Psi Psi is the key; the reptation dynamics is responsible for the change of conformation when elastic effects is the key; the reptation dynamics is responsible for the change of conformation when elastic effects are taken into account.. are taken into account..


Modelling concentrated polymer solutions III: the Pom-Pom model (1998)Modelling concentrated polymer solutions III: the Pom-Pom model (1998)

'' '' 1 ( , ') ( , ')( ) exp( )

( ') ( '') ( , ') ( , ')

tt

b bt

dt dt E t t uE t t uS t

T t T t E t t u E t t u

1( : ) ( 1),

s

K S qt T

1( )( : ) ,

2 2 ( )c b

ca c

s sq s K S q

t T x

McLeish and Larson “mounted” arms at the end of McLeish and Larson “mounted” arms at the end of primitive chain, which became the “backbone” of primitive chain, which became the “backbone” of the “Pom-Pom molecule”. These arms will be then the “Pom-Pom molecule”. These arms will be then “released” and “withdrawn” by the backbone, “released” and “withdrawn” by the backbone, but but only when the BB is fully stretched.only when the BB is fully stretched.

It describes very accurately Low It describes very accurately Low Density PolyEthylenes dynamics, Density PolyEthylenes dynamics, whose irregular branches give raise to whose irregular branches give raise to high level of shear thinning and strain high level of shear thinning and strain hardening.hardening.

Evolution equation for the orientation tensor Evolution equation for the orientation tensor

2 2 ( )( ( ) ) ( )

2 2 2b b c

a b a b a b

s s qs tC t S t

qs s qs s qs s

Evolution equation for the backbone stretch Evolution equation for the backbone stretch

Evolution equation for the arms motion Evolution equation for the arms motion Derivation of the stress tensor Derivation of the stress tensor

Three main problems affect this model:Three main problems affect this model: 1) discontinuities in Grad u in steady flows; 1) discontinuities in Grad u in steady flows;

2) the orientation tensor is unbounded at high strain rates; 2) the orientation tensor is unbounded at high strain rates; 3) 23) 2ndnd NSD is not predicted NSD is not predicted


Modelling concentrated polymer solutions IV: the XPP model (2001)Modelling concentrated polymer solutions IV: the XPP model (2001)

10ˆ ( ) 2G D

1 1 1 10

0 0

1( ) ( ( ) [ ( ) 1] )

b

f I G fG

12 2

0 0 0

1 2 1 1( ) (1 ) (1 )

3b s b

If

G

Viscoelastic StressViscoelastic Stress

00

21 , exp( ( 1))

3 s

I

G q

Backbone stretch and stretch relaxation timeBackbone stretch and stretch relaxation time

Extra FunctionExtra Function

Relaxation time tensorRelaxation time tensor

Blackwell modification: a smoother Blackwell modification: a smoother approach to the maximum BB stretch by approach to the maximum BB stretch by mean of withdrawing arms before such mean of withdrawing arms before such

maximum stretch is achievedmaximum stretch is achieved

Tackling Tackling issue # 1issue # 1


High strain rates means the 1High strain rates means the 1stst term totally term totally outweighs the 2outweighs the 2ndnd one; the result is that the one; the result is that the stretching effect is predominant enough to stretching effect is predominant enough to avoid unbounded orientation.avoid unbounded orientation.The macroscopic dependence of the slip The macroscopic dependence of the slip tensor on the stress is the physical reason. tensor on the stress is the physical reason.


The Giesekus - like 2The Giesekus - like 2ndnd order term is order term is responsible for predicting 2responsible for predicting 2ndnd NSD. NSD.

ARE WE HAPPY ???ARE WE HAPPY ???

SADLY NOT…SADLY NOT…


Model Model Fluids describedFluids described FeaturesFeatures DrawbacksDrawbacks Viscosity Viscosity

Oldroyd-BOldroyd-B Boger fluids; Polymer Boger fluids; Polymer Solutions (molecules are Solutions (molecules are far enough from each far enough from each other so that their other so that their interactions can be interactions can be neglected)neglected)

Analytical Solutions Analytical Solutions Available for both Available for both transient and steady flow transient and steady flow (Channel); direct (Channel); direct derivation from F - P eq.; derivation from F - P eq.; 11stst NSDiff predicted NSDiff predicted

Infinite extensibility of Infinite extensibility of dumbbells allowed dumbbells allowed (unphysical); no shear (unphysical); no shear thinning predicted; no 2thinning predicted; no 2ndnd NSDNSD

Shear : constant over a Shear : constant over a wide range of shear wide range of shear rates;rates;

Ext.: possibly Ext.: possibly unbounded. unbounded.

FENE modelsFENE models Polymer solutionsPolymer solutions General agreement with General agreement with experiments in experiments in extensional flows. extensional flows. Maximum extension Maximum extension bounded. bounded.

No 2No 2ndnd NSD predicted. NSD predicted. Poor representation of Poor representation of shear flows.shear flows.

FENE-P is shear FENE-P is shear thinning. The others thinning. The others reflects much better reflects much better extensional properties.extensional properties.

PTT (Phan TPTT (Phan T

hien-Tanner)hien-Tanner)

Polymer Melts (molecules Polymer Melts (molecules are not so dispersed are not so dispersed anymore) anymore)

Extension of dumbells Extension of dumbells bounded; shear thinning bounded; shear thinning predicted; 1predicted; 1stst NSDiff NSDiff predicted .predicted .

No 2No 2ndnd NSD predicted NSD predicted Shear thinning, Shear thinning,

Extensional bounded. Extensional bounded.

Pom PomPom Pom Polymer Melts (molecules Polymer Melts (molecules are not so dispersed are not so dispersed anymore) anymore)

Vortex prediction in Vortex prediction in contraction flows due to contraction flows due to branches effect; shear branches effect; shear thinning and strain thinning and strain hardening predicted.hardening predicted.

Derivatives of ext. Derivatives of ext. viscosity discontinuous; viscosity discontinuous; no 2no 2ndnd NSD. Orientation NSD. Orientation tensor possibly tensor possibly unbounded.unbounded.

Shear thinning, constant Shear thinning, constant plateau without plateau without Blackwell’s mod.; Blackwell’s mod.;

Tension Thickening at Tension Thickening at low strain rates. low strain rates.

XPP (eXtended XPP (eXtended Pom Pom)Pom Pom)

Polymer Melts (molecules Polymer Melts (molecules are not so dispersed are not so dispersed anymore)anymore)

Derivatives of ext. Derivatives of ext. viscosity are smooth; 2viscosity are smooth; 2ndnd NSD also predicted by NSD also predicted by Giesekus term.Giesekus term.

Ext. unbounded and Ext. unbounded and shear approaching shear approaching constant plateau if ETA constant plateau if ETA is switched off; possible is switched off; possible unbounded stress and unbounded stress and imaginary stretch.imaginary stretch.

Shear thinning.Shear thinning.

Tension Thickening at Tension Thickening at low strain rates. low strain rates.

ReviewReview


1515

NUMERICAL1) A code on the shelf has been extended to include almost all the models

mentioned; tests are positive2) The MATRIX-LOGARITHM formulation has been implemented and tested

for the flow past a cylinder. Higher We calculation achieved for OLD-B.

EXTRUDATE SWELLEXTRUDATE SWELL A viscoelastic fluid is extruded A viscoelastic fluid is extruded from a pipe. The from a pipe. The Stress gradient at the die is Stress gradient at the die is responsible responsible for the extensional swelling. for the extensional swelling. Capturing this gradient, as wellCapturing this gradient, as wellas the free surface’s behaviour, as the free surface’s behaviour, is crucial. Extrusion processes in is crucial. Extrusion processes in food and manufacturing food and manufacturing industries industries are the main industrial are the main industrial applicationsapplications

FILAMENT STRETCHINGFILAMENT STRETCHING

A viscoelastic fluid is confined A viscoelastic fluid is confined between two plates. When these between two plates. When these plates are pulled apart, the fluid plates are pulled apart, the fluid stretches. stretches. Tracking the free surface and Tracking the free surface and describing the necking effect at describing the necking effect at the centre are the main the centre are the main challenges. This phenomenon is challenges. This phenomenon is common in fibre spinning and common in fibre spinning and industrial processes involving industrial processes involving thin films. thin films.

Velocity extrapolated on the FS Velocity extrapolated on the FS along the vertical gridlines. Nodes along the vertical gridlines. Nodes shifted. New transfinite mapping shifted. New transfinite mapping built. Moving forward in time. built. Moving forward in time.

PressureHorizontal Velocity

Vertical Velocity

My research project in few words My research project in few words

1) Discretization in space: the Spectral Element Methods + transfinite techniques 1) Extending the code to the real problems1) Extending the code to the real problems

2) Discretization in time: usually 1st order + OIFS for the material derivative

RESULTS ACHIEVED

THEORETICAL1) Stability estimate for the stress tensor in the 3 fields Stokes problem;

2) Existence and uniqueness of a steady state weak solution for the 3fields die swell Stokes problem.

Ongoing and doming next (hopefully…)

2) Improve the MATRIX-LOG version ( problem with some stress 2) Improve the MATRIX-LOG version ( problem with some stress profiles) and make it model independent (so far just Oldroyd-B)profiles) and make it model independent (so far just Oldroyd-B)

3) Gather the two approaches3) Gather the two approaches


[1] PHILLIPS T.N., OWENS. R., Computational Rheology, Imperial College Press,

2002.[2] KEUNINGS R, On the Peterlin approximation for finitely extensible dumbbells ,

Journal of Non-Newtonian Fluid Mechanics, 68-1:85-100, 1998.[3] PHAN-TIEN N., TANNER R.I., A new constitutive equation derived from

network theory, Journal of Non-Newtonian Fluid Mechanics, 2:353-365, 1977.[4] DOI M.,EDWARDS S.F., The theory of polymer dynamics , Oxford University

Press, 1988.[5] MCLEISH T.C.B., LARSON R.G., Molecular constitutive equations for a class

of branched polymers: the pom-pom polymer , Journal of Rheology, 42: 81-110, 1998.

[6] VERBEETEN W.M.H., PETERS G.W.M., BAAJIENS F.P.T., Differential constitutive equations for polymer melts: the eXtended Pom Pom model, Journal Rheology, 45-4: 823-843, 2001

[7] PHAN-TIEN N., Understanding Viscoelasticity, Springer, Berlin, 2002.

ReferencesReferences

27/11/2007 Bonn, Institut fuer Numerische Simulation 1 A century of viscoelastic modelling: from...

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