Done with Inelastic models (GNF). · 2020. 3. 4. · Newtonian contribution contribution from t0 seconds ago contribution from 2t0 seconds ago ~ 1.8 ~ (t) t0 2t0 t 2.4 ~ The memory

Generalized LVE CM4650 3/4/2020

1

© Faith A. Morrison, Michigan Tech U.1

Summary: Generalized Newtonian Fluid Constitutive Equations

•A first constitutive equation

•Can match steady shearing data very well

•Simple to calculate with

•Found to predict pressure-drop/flow rate relationships well

•Fails to predict shear normal stresses

•Fails to predict start-up or cessation effects (time-dependence, memory) – only a function of instantaneous velocity gradient

•Derived ad hoc from shear observations; unclear of validity in non-shear flows

PRO:

CON:

Done with Inelastic models (GNF).

Let’s move on to Linear Elastic models

Summary:

Rules for Constitutive Equations

•Must be of tensor order

•Must be a tensor (independent of coordinate system)

•Must be a symmetric tensor

•Must make predictions that are independent of the observer

•Should correctly predict observed flow/deformation behavior

The stress expression:

Innovate and improve constitutive equations

What we know so far…

Chapter 8: Memory Effects: GLVE


f

Dtotal

initial stateno force

final stateforce, f, resists

displacement

Maxwell’s model combines viscous and elastic responses in series

dashpotspringtotal DDD

Displacements are additive:

Spring (elastic) and dashpot (viscous) in series:

Professor Faith A. Morrison

Department of Chemical EngineeringMichigan Technological University

CM4650 Polymer Rheology


2

Fluids with Memory - Chapter 8

© Faith A. Morrison, Michigan Tech U.

We seek a constitutive equation that includes memory effects.

calculates the stress at a particular time, 𝑡

2 Constitutive Equations so far:

So far, stress at 𝑡 depends on rate-of-deformation at 𝒕 only

3

Fluids with Memory – Chapter 8

𝜏 𝑡 𝑓 𝛾, 𝐼 , 𝐼𝐼 , 𝐼𝐼𝐼 , material info

𝜏 𝜇𝛾 𝑡𝜏 𝜂 𝛾 𝛾 𝑡 𝛾 𝛾


Newtonian

Generalized Newtonian

Neither can predict:

•Shear normal stresses - this will be wrong so long as we use constitutive equations proportional to 𝛾

•stress transients in shear (startup, cessation) - this flaw seems to be related to omitting fluid memory

Current Constitutive Equations

We will try to fix this now; we will address the first point when we discuss advanced constitutive equations

4


𝜏 𝜇𝛾 𝑡𝜏 𝜂 𝛾 𝛾 𝑡 𝛾 𝛾


3

Startup of Steady Shearing


Figures 6.49, 6.50, p. 208 Menezes and Graessley, PB soln 5


𝑣 ≡𝜍 𝑡 𝑥

00

𝜍 𝑡 0 𝑡 0𝛾 𝑡 0

𝜂 ≡ 𝜏 𝑡𝛾

Ψ ≡ 𝜏 𝜏𝛾

Cessation of Steady Shearing


Figures 6.51, 6.52, p. 209 Menezes and Graessley, PB soln

6



00

𝜍 𝑡 𝛾 𝑡 00 𝑡 0

𝜂 ≡ 𝜏 𝑡𝛾

Ψ ≡ 𝜏 𝜏𝛾


4

How can we incorporate time-dependent effects?

First, we explore a simple memory fluid.


Let’s construct a new constitutive equation that remembers the stress at a time t0 seconds ago

“Newtonian” contribution

contribution from fluid memory

𝜼 is a parameter of the model (it is constant)

This is the rate-of-deformation tensor 𝑡0seconds beforetime 𝑡

7


𝜏 𝑡 𝜂𝛾 𝑡 0.8𝜂 𝛾 𝑡 𝑡


What does this model predict?

Steady shear

??

?

2

1

Shear start-up

?)(?)(?)(

2

1

ttt

8


𝜏 𝑡 𝜂𝛾 𝑡 0.8𝜂 𝛾 𝑡 𝑡Simple memory fluid


5


What does this model predict?

Steady shear

??

?

2

1

Shear start-up

?)(?)(?)(

2

1

ttt

9


Let’s try.

𝜏 𝑡 𝜂𝛾 𝑡 0.8𝜂 𝛾 𝑡 𝑡Simple memory fluid

Imposed Kinematics:


00

Steady Shear Flow Material Functions

Material Functions:

Viscosity


𝜍 𝑡 𝛾 constant

𝜂 𝛾 ≡ �̃�𝛾𝜏𝛾

Ψ 𝛾 ≡

Ψ 𝛾 ≡

Material Stress Response: 𝜏 𝑡

𝜏𝑡0

𝑡0

𝛾

𝜍 𝑡

𝑡0

𝛾

𝛾 0, 𝑡

𝑁 𝑡

𝑁 ,𝑡0

First normal-stress coefficient

Second normal-stress coefficient


6


00

Start-up of Steady Shear Flow Material Functions

Shear stress growth

function


𝜍 𝑡 0 𝑡 0𝛾 𝑡 0

Ψ 𝑡, 𝛾 ≡

Ψ 𝑡, 𝛾 ≡

𝑡0

𝛾

𝜍 𝑡

𝑡0

𝛾

𝛾 0, 𝑡

𝜏 𝑡

𝑡0

𝑁 𝑡

𝑡0

First normal-stress growth coefficient

Second normal-stress growth coefficient

𝛾 ,𝛾 ,𝛾 ,

𝛾 ,𝛾 ,𝛾 ,

Imposed Kinematics:

Material Functions:

Material Stress Response:

𝜂 𝑡, 𝛾 ≡ �̃� 𝑡𝛾𝜏 𝑡𝛾

Predictions of the simple memory fluid


Steady shear

0

~8.1

21 The steady viscosity reflects

contributions from what is currently happening and contributions from what happened t0 seconds ago.

Shear start-up

0)()(

00

~8.1

~0

)(

21

0

0

tt

tttt

tt

12




7


Shear start-up

0)()(

00

~8.1

~0

)(

21

0

0

tt

tttt

tt

~ ~8.1

)(t

t

Figures 6.49, 6.50, p. 208 Menezes and Graessley, PB soln

13





Shear start-up

~ ~8.1

)(t

t

0 t

)(21 t

What the data show:

What the GNF models predict:

0 t

)(21 t

Increasing 𝛾

Increasing 𝛾

What the simple memory fluid model predict:

t0

14



For all rates, 𝛾



8

Increasing 𝛾

Increasing 𝛾


~ ~8.1

)(t

t

0 t

)(21 t

What the data show:

What the GNF models predict:

0 t

)(21 t

What the simple memory fluid model predict:

t0

Adding that contribution from the past introduces the observed “build-up” effect to the predicted

start-up material functions.

15



For all rates, 𝛾

Shear start-up


We can make the stress rise smoother by adding more fading memory terms.


Newtonian contribution

contribution from t0

seconds ago

contribution from 2t0

seconds ago

~ ~8.1

)(t

tt0 2t0

~4.2

The memory is fading

16


For all rates, 𝛾

𝜏 𝑡 𝜂𝛾 𝑡 0.8𝜂 𝛾 𝑡 𝑡 0.6𝜂 𝛾 𝑡 2𝑡


9


)(t

t

The fit can be made to be perfectly smooth by using a sum of exponentially decaying terms as the weighting functions.

0 1.001 0.372 0.143 0.054 0.02

/0pte0pt

(𝑡 /𝜆 scales the decay)17


start-up prediction:

For all rates, 𝛾

𝜏 𝑡 𝜂 𝛾 𝑡 0.37 𝛾 𝑡 𝑡0.14 𝛾 𝑡 2𝑡 0.05 𝛾 𝑡 3𝑡 ⋯

𝜂 ∑ 𝑒 / 𝛾 𝑡 𝑝𝑡


This sum can be rewritten as an integral.

)'()('

'

0

tptexiaftpptxi

txtx

ta

tp

NabxxxiafdxxfI

N

i

b

aN

,)(lim)(1

18

New model:

~ ~8.1

)(t

tt0 2t0

~4.2

(Actually, it takes a bit of renormalizing to make this transformation actually work.)

In the current formulation, 𝜂 𝑡 grows as N goes to infinity.


𝜏 𝑡 𝜂 ∑ 𝑒 𝛾 𝑡 𝑝𝑡

→ 𝑒 𝛾 𝑡 𝑝Δ𝑡′


10


Maxwell Model (integral version)

Relaxation time 𝜆 - quantifies how fast memory fades

Zero-shear viscosity 𝜂 – gives the value of the steady shear viscosity

Two parameters:

19

With proper reformulation, we obtain:

Steps to arrive here:•Add information about past deformations•Make memory fade


Often we write in terms of a modulus parameter, 𝐺

𝜏 𝑡 𝜂𝜆 𝑒 𝛾 𝑡 𝑑𝑡′


We’ve seen that including terms that invoke past deformations (fluid memory) can improve the constitutive predictions we make.

This same class of models can be derived in differential form, beginning with the idea of combining viscous and elastic effects.

20



11

Hooke’s Law for elastic solids


Newton’s Law for viscous liquids

The Maxwell Models

The basic Maxwell model is based on the observation that at long times viscoelastic

materials behave like Newtonian liquids, while at short times they behave like elastic solids.

Fluids with Memory – Maxwell Models

𝜏 𝜂𝛾

𝜏 𝐺𝛾

f

Dtotal

initial stateno force

final stateforce, f, resists

displacement


Maxwell’s model combines viscous and elastic responses in series

dashpotspringtotal DDD

Displacements are additive:

Spring (elastic) and dashpot (viscous) in series:

22



12


In the spring:

In the dashpot:

dtdD

dtdD

dtdD

DDD

dashspringtotal

dashspringtotal

dtdDf

DGf

dash

springsp

dtdD

dtdf

Gf

fdtdf

G

total

sp

sp

11

23


(proportional to displacement)

(proportional to rate)


dtdD

dtdf

Gf total

sp

By analogy:shear

all flows

G0 Relaxation timeTwo parameter model:

0 Viscosity

24


𝜏 𝜂𝐺𝜕𝜏𝜕𝑡 𝜂 𝛾


Also by analogy:(we’re taking a leap here)


13


The Maxwell Model

G0 Relaxation timeTwo parameter model:

0 Viscosity

25




How does the Maxwell model behave at steady state? For short time deformations?

26



The Maxwell Model


14

Example: Solve the Maxwell Model for an expression explicit in the stress tensor



To solve first-order, linear differential equations:

Use an integrating function, 𝑢 𝑥

xdxaexu )()(


Recall:

𝑑𝒚𝑑𝑥 𝑎 𝑥 𝒚 𝑏 𝑥 0


15

Example: Solve the Maxwell Model for an expression explicit in the stress tensor


Let’s try.



Maxwell Model (integral version)

30

We arrived at this equation following two different paths:

•Add up fading contributions of past deformations•Add viscous and elastic effects in series


𝜏 𝑡 𝜂𝜆 𝑒 𝛾 𝑡 𝑑𝑡′

This is the same equation we “made up” when starting from the simple memory model!


16


What are the predictions of the Maxwell model?

Need to check the predictions to see if what we have done is worth keeping.

Predictions:

•Steady shear

•Steady elongation

•Start-up of steady shear

•Step shear strain

•Small-amplitude oscillatory shear

31


What now?


What are the predictions of the Maxwell model?

Need to check the predictions to see if what we have done is worth keeping.

Predictions:

32


What now?

Let’s get to work.

•Steady shear






17

Imposed Kinematics:


00


Material Functions:

Viscosity




Ψ 𝛾 ≡

Ψ 𝛾 ≡


�̃�

𝑡0

𝑡0

𝛾

𝜍 𝑡

𝑡0

𝛾

𝛾 0, 𝑡

𝑁 𝑡

𝑁 ,𝑡0



Predictions of the (single-mode) Maxwell Model


Steady shear

0210

Fails to predict shear normal

stresses.

Fails to predict shear-thinning.

Steady elongation

Trouton’s rule

34

�̅� 3𝜂


𝜏 𝑡 𝜂𝜆 𝑒 𝛾 𝑡 𝑑𝑡𝜏𝜂𝐺𝜕𝜏𝜕𝑡 𝜂 𝛾


18

Steady shear viscosity and first normal stress coefficient

0.1

1

10

100

0.1 1 10 100

stress, Pa

1, sFigure 6.5, p. 173 Binnington and Boger; PIB soln

sPa ,

21, sPa


There are some systems with a constant viscosity but still start-up effects.

BOGER FLUIDS

35


𝛾, 𝑠

0.1

1

10

100

1000

0.01 0.1 1 10 100

stea

dy s

hear

mat

eria

l fun

ctio

ns

1 (Pa s2)

(Pa)

2 (Pa s2)

PS/TCP squaresPS/DOP circles

1, s

Steady shear viscosity and first and secondnormal stress coefficient

Figure 6.6, p. 174 Magda et al.; PS solns


BOGER FLUIDS

36


𝛾, 𝑠


19

Step Strain Shear Flow Material Functions

Relaxation modulus



00

𝜍 𝑡 lim→0 𝑡 0

𝛾 /𝜀 0 𝑡 𝜀0 𝑡 𝜀

𝐺 𝑡, 𝛾 ≡ �̃� 𝑡, 𝛾𝛾𝜏 𝑡, 𝛾𝛾

𝐺 𝑡, 𝛾 ≡

𝐺 𝑡, 𝛾 ≡

𝑡0

𝛾 0, 𝑡

𝜏 𝑡

𝑡0

𝑁 𝑡

𝑡0

First normal-stress relaxation modulus

Second normal-stress relaxation modulus

𝛾 ,𝛾 ,𝛾 ,

𝛾 ,𝛾 ,𝛾 ,

𝛾𝜀

𝜀

.

.

.

𝜀

𝑡0

𝜍 𝑡

𝜀

Imposed Kinematics:

Material Functions:


𝛾 𝛾𝜀

Predictions of the (single-mode) Maxwell Model


Shear start-up

0)()(

1)(

21

/0

ttet t Does predict a gradual

build-up of stresses on start-up.

Step shear strain

0

)(

21

/0

GG

etG t Does predict a reasonable

relaxation function in step strain (but no normal stresses again).

38


𝜏 𝑡 𝜂𝜆 𝑒 𝛾 𝑡 𝑑𝑡𝜏𝜂𝐺𝜕𝜏𝜕𝑡 𝜂 𝛾


20

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 2 4 6 8 10

o

tG

t

0.001

0.01

0.1

1

0.01 0.1 1 10

o

tG

t


Step-Shear-Strain Material Function G(t) for Maxwell Model

39


𝜏 𝑡 𝜂𝜆 𝑒 𝛾 𝑡 𝑑𝑡

10

100

1,000

10,000

1 10 100 1,000 10,000time, s

G(t), Pa


Comparison to experimental data

Figure 8.4, p. 274 data from Einaga et al., PS 20% soln in chlorinated diphenyl

s

Pg

150

25000

Pa

40



21


We can improve this fit by adjusting the Maxwell model to allow multiple relaxation modes

Generalized Maxwell

Model

2𝑁 parameters (can fit anything)41


2𝑁 model parameters: 𝑔 , 𝜆 (constants)

𝜂 𝑔 𝜆


𝜏 𝜂𝜆 𝑒/ 𝛾 𝑡′ 𝑑𝑡′

𝜏 𝑡 𝜏


Generalized Maxwell

Model

42


•Steady shear





Let’s try.

What are the predictions of the Generalized Maxwell model?



22


Generalized Maxwell

Model

43


What are the predictions of the Generalized Maxwell model?

•Steady shear





Let’s try.


Imposed Kinematics:


00


Material Functions:

Viscosity




Ψ 𝛾 ≡

Ψ 𝛾 ≡


�̃�

𝑡0

𝑡0

𝛾

𝜍 𝑡

𝑡0

𝛾

𝛾 0, 𝑡

𝑁 𝑡

𝑁 ,𝑡0




23

Step Strain Shear Flow Material Functions

Relaxation modulus



00

𝜍 𝑡 lim→0 𝑡 0

𝛾 /𝜀 0 𝑡 𝜀0 𝑡 𝜀

𝐺 𝑡, 𝛾 ≡ �̃� 𝑡, 𝛾𝛾𝜏 𝑡, 𝛾𝛾

𝐺 𝑡, 𝛾 ≡

𝐺 𝑡, 𝛾 ≡

𝑡0

𝛾 0, 𝑡

𝜏 𝑡

𝑡0

𝑁 𝑡

𝑡0


Second normal-stress relaxation modulus

𝛾 ,𝛾 ,𝛾 ,

𝛾 ,𝛾 ,𝛾 ,

𝛾𝜀

𝜀

.

.

.

𝜀

𝑡0

𝜍 𝑡

𝜀

Imposed Kinematics:

Material Functions:


𝛾 𝛾𝜀

Predictions of the Generalized Maxwell Model


Steady shear

0211

N

kk

Fails to predict shear normal stresses

Fails to predict shear-thinning

Step shear strain

0

)(

21

1

/

GG

etGN

k

t

k

k k

This function can fit anyobserved data (we show how)

Note that the GMM does not predict shear normal stresses.

46



𝜂 𝑔 𝜆𝜏 𝑡 𝜂𝜆 𝑒 𝛾 𝑡 𝑑𝑡


24

10

100

1,000

10,000

1 10 100 1000 10000time, t

G(t), Pak k gk1 450 5002 100 9003 30 13004 5.0 1600

k=4 k=3 k=2

k=1

4

1k

tk

keg


Fitting 𝐺 𝑡 to Generalized Maxwell Model

Figure 8.4, p. 274 data from Einaga et al., PS 20% soln in chlorinated diphenyl

(s) (Pa)

47



𝜂 𝑔 𝜆


The Linear-Viscoelastic Models

Differential Maxwell (one mode):

Integral Maxwell (one mode):

Generalized Maxwell model (N modes):

48



𝜂 𝑔 𝜆





25

Predictions of the Generalized Maxwell Model


Step shear strain

0

)(

21

1

/

GG

etGN

k

t

k

k k

49


Note: The time-dependent function in the integral of the GMM is the same form as the step strain result:







50



Since the term in brackets is just the predicted relaxation modulus G(t), we can write an even more general linear viscoelastic modelby leaving this function unspecified.




26






Generalized Linear-Viscoelastic Model:

51



𝜏 𝑡 𝐺 𝑡 𝑡′ 𝛾 𝑡 𝑑𝑡




Generalized Maxwell Model

52


What are the predictions of these models?

•Steady shear





Let’s try.


Generalized Linear-Viscoelastic Model 𝜏 𝑡 𝐺 𝑡 𝑡′ 𝛾 𝑡 𝑑𝑡


27

𝛿

Small-Amplitude Oscillatory Shear Material Functions

SAOS stress



00

𝜍 𝑡 𝛾 cos 𝜔𝑡𝛾 𝛾𝜔

, , �̃� sin 𝜔𝑡 𝛿 𝐺 sin 𝜔𝑡 𝐺 cos 𝜔𝑡𝐺′ 𝜔 ≡ cos 𝛿 𝐺′′ 𝜔 ≡ sin 𝛿

𝑡0

𝛾 0, 𝑡

𝜏 𝑡

𝑡0

𝑁 𝑡 𝑁 𝑡 0(linear viscoelastic regime)

Storage modulus

Loss modulus

𝑡0

𝜍 𝑡

𝛾 cos 𝜔𝑡 𝛾 sin 𝜔𝑡

𝜏 sin 𝜔𝑡 𝛿

𝛿 phase difference between stress and strain waves

Imposed Kinematics:

Material Functions:


Predictions of the Generalized Maxwell Model (GMM) and Generalized Linear-Viscoelastic Model (GLVE)


Small-amplitude oscillatory shear

GMM

N

k k

k

N

k k

kk

G

G

12

12

2

)(1)(

)(1)(

GLVE

0

0

sin)()(

cos)()(

dsssGG

dsssGG

54



𝜂 𝑔 𝜆

𝜏 𝑡 𝐺 𝑡 𝑡′ 𝛾 𝑡 𝑑𝑡 𝜏 𝑡𝜂𝜆 𝑒 𝛾 𝑡 𝑑𝑡


28

0.00001

0.0001

0.001

0.01

0.1

1

10

0.001 0.01 0.1 1 10 100 1000

1

1'

G

1

1"

G

10-5

10-4


Predictions of (single-mode) Maxwell Model in SAOS

21

12

1

11

21

211

21

2211

)(1)(1)(

)(1)(1)(

gG

gG

55


Two model parameters: 𝑔 , 𝜆 (constants)

𝜂 𝑔 𝜆


SAO

S m

ater

ial f

unct

ions


Predictions of (multi-mode) Maxwell Model in SAOS

Figure 8.8, p. 284 data from Vinogradov, PS melt

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07

aT, rad/s

G' (Pa)G'' (Pa)

k k (s) gk(kPa) 1 2.3E-3 16 2 3.0E-4 140 3 3.0E-5 90 4 3.0E-6 400 5 3.0E-7 4000

N

k k

k

N

k k

kk

G

G

12

12

2

)(1)(

)(1)(

k 𝜆 (s) gk(kPa)

56



29


Predictions of (multi-mode) Maxwell Model in SAOS

Figure 8.10, p. 286 data from Laun, PE melt

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E-04 1.E-02 1.E+00 1.E+02 1.E+04 1.E+06

aTrad/s

mod

uli

G' (Pa)G" Pa

k gk50 4.42E+025 3.62E+031 4.47E+030.5 3.28E+030.2 9.36E+030.1 1.03E+040.05 1.71E+030.02 1.18E+020.02 2.55E+040.01 3.33E+040.0007 1.83E+05

N

k k

k

N

k k

kk

G

G

12

12

2

)(1)(

)(1)(

57




http://pages.mtu.edu/~fmorriso/cm4650/PredictionsGLVE.pdf

Predictions of the Generalized Linear Viscoelastic Model (the “report card”)


30

0.1

1

10

100

1000

0.01 0.1 1 10 100

stea

dy s

hear

mat

eria

l fun

ctio

ns

1 (Pa s2)

(Pa)

2 (Pa s2)

PS/TCP squaresPS/DOP circles

1, s

Steady shear viscosity and first and secondnormal stress coefficient

Figure 6.6, p. 174 Magda et al.; PS solns


BOGER FLUIDS

59


𝛾, 𝑠

x1

x2

x3

H

W V

v1(x2)

EXAMPLE: Drag flow of a Generalized Linear-Viscoelastic fluid between infinite parallel plates•steady state•incompressible fluid•infinitely wide, long

60




31


Limitations of the GLVE Models

•Predicts constant shear viscosity

•Only valid in regime where strain is additive (small-strain, low rates)

•All stresses are proportional to the deformation-rate tensor; thus shear normal stresses cannot be predicted

•Cannot describe flows with a superposed rigid rotation (as we will now discuss; see Morrison p296)

61



What else can we try?


Jeffreys Model - Mechanical Analog

Maxwell Model - Mechanical Analog

62

Advanced Constitutive Modeling – Chapter 9

𝜏 𝑡 𝜆 𝜕𝜏𝜕𝑡 𝜂 𝛾

𝜏 𝑡 𝜆 𝜕𝜏𝜕𝑡 𝜂 𝛾 𝜆𝜕𝛾𝜕𝑡


32


Jeffreys Model

Now, solving for 𝜏 explicitly we obtain,

Other modifications of the Maxwell model motivated by springs and dashpots in series and

parallel modify 𝐺 𝑡 𝑡’ but do not otherwise introduce new behavior.

(Might as well use the Generalized Maxwell model)

63


𝜏 𝜆 𝜕𝜏𝜕𝑡 𝜂 𝛾 𝜆𝜕𝛾𝜕𝑡

𝜏 𝑡 𝜂𝜆 1𝜆𝜆 𝑒

2𝜂 𝜆𝜆 𝛿 𝑡 𝑡 𝛾 𝑡 𝑑𝑡

Unfortunately, this change only modifies 𝐺 𝑡 𝑡 ;the Jeffreys Model is a GLVE model











Where do we go from here?


33











Where do we go from here?

Let’s talk about this

issue.


fluid x

y

zyx ,,

zyx ,,

Shear flow in a rotating frame of reference

66


𝜏 𝑡 𝐺 𝑡 𝑡′ 𝛾 𝑡 𝑑𝑡GLVE:


34


fluid x

y

zyx ,,

zyx ,,



GLVE:

𝑥,𝑦, 𝑧 coordinate system stationary�̅�,𝑦, 𝑧̅ coordinate system rotating at Ω

The choice of coordinate system should make no difference. Let’s check.

Predict: 𝜂


fluid x

y

zyx ,,

zyx ,,



GLVE:

𝑥,𝑦, 𝑧 coordinate system stationary�̅�,𝑦, 𝑧̅ coordinate system rotating at Ω

𝑣 𝑣 𝑣

𝛾 𝑦�̂� ̅ 𝑣

We write the fluid velocity two ways:

Both ways we should get the same answer for 𝜏 𝜏 𝑡 𝐺 𝑡 𝑡′ 𝛾 𝑡 𝑑𝑡

GLVE:


35



tyybdtyyctxxbtxxa

o

o

o

o

cossinsincos

oyy

t

P

x

y

x

y

oxx

ab

d

c

69


.P



tyybdtyyctxxbtxxa

o

o

o

o

cossinsincos

oyy

t

P

x

y

x

y

oxx

ab

d

c

70


𝑦 𝑦𝑥 𝑥

Location of P: 𝑥 𝑥 ,𝑦 𝑦

.


36



tyybdtyyctxxbtxxa

o

o

o

o

cossinsincos

oyy

t

P

x

y

x

y

oxx

ab

d

c

71


𝑦 𝑦𝑥 𝑥

Location of P: 𝑥 𝑥 ,𝑦 𝑦 𝑦 𝑑 𝑑 𝑏 𝑏

.


Summary: Generalized Linear-Viscoelastic (GLVE) Constitutive Equations

•A first constitutive equation with memory

•Can match SAOS, step-strain data very well

•Captures start-up/cessation effects

•Simple to calculate with

•Can be used to calculate the LVE spectrum

•Fails to predict shear normal stresses

•Fails to predict shear-thinning/thickening

•Only valid at small strains, small rates

•Not frame-invariant

PRO:

CON:

72


𝜏 𝑡 𝐺 𝑡 𝑡′ 𝛾 𝑡 𝑑𝑡GLVE:


37

73

Find the recipe card (has kinematics,

definitions of material functions; 𝑣(t) is known)

Predict a material function for a constitutive equation

For the given kinematics, calc. needed

stresses from the constitutive equation

From stresses, calculate the material function(s) from the

definition on the recipe card

Sketch the situation and choose an appropriate

coordinate system

Calculate a flow field or stress

field for a fluid in a situation

Solve the differential equations (Solve EOM, continuity, and constitutive

equation for 𝑣 and 𝑝 and apply the boundary conditions)

Model the flowby asking yourself questions about the velocity field (are some components and derivatives zero? The constitutive

equationmust be known)

x1

x2

x3

H

W V

v1(x2)

EXAMPLE: Drag flow between infinite parallel plates

•Newtonian•steady state•incompressible fluid•very wide, long•uniform pressure

2

Step Shear Strain Material Functions

00

00

constant

00

0

0lim)(

t

tt

t

Kinematics:

123

2

00)(

xtv

Material Functions:

0

0210

),(),(

ttG

20

22111

G

20

33222

GRelaxation

modulus


Second normal-stress relaxation

modulus


What kind of rheology calculation am I considering?

𝜏 𝑡 𝑓 𝑣constitutive equation


Done with GLVE.Let’s move on to Advanced Constitutive Equations



38

Chapter 9: Advanced Constitutive Models


CM4650 Polymer Rheology

Professor Faith A. Morrison

Department of Chemical EngineeringMichigan Technological University

Documents

Done with Inelastic models (GNF). · 2020. 3. 4. · Newtonian contribution contribution from t0 seconds ago contribution from 2t0 seconds ago ~ 1.8 ~ (t) t0 2t0 t 2.4 ~ The memory