Chapter 7: Generalized Newtonian Fluids

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Generalized Newtonian Fluid CM4650 3/21/2018

Chapter 7: Generalized Newtonian fluids

CM4650 Polymer Rheology Carreau- Michigan Tech Yassuda GNF

log positionposition of break of break on onscale  is determinedscale is determined by by 

o slope is determined by n curvaturecurvature here here is is determineddetermined byby a



log 1

1 © Faith A. Morrison, Michigan Tech U.

Develop Non-Newtonian Constitutive Equations

So Far: • We seek constitutive equations for non- Newtonian fluids. • We start with the Newtonian constitutive equation • We “modify it” and make the Fake-O model. • We checked out the model to see how we did.

2 © Faith A. Morrison, Michigan Tech U.

1 Generalized Newtonian Fluid CM4650 3/21/2018

≡

constant

When we tried the model out with elongational flow material functions, something odd happened …

3 © Faith A. Morrison, Michigan Tech U.

Steady Elongational Flow Material Functions

Imposed Kinematics: 0, ≡ 0 0

constant

Material Stress Response: ̃ ̃

Material Functions: Elongational ̃ ̃ ≡ Viscosity

Alternatively, ̅ 4 © Faith A. Morrison, Michigan Tech U.

2 Generalized Newtonian Fluid CM4650 3/21/2018

2) Predict what Newtonian fluids would do.

1. Choose a material function 2. Predict what Newtonian fluids would do 3. See what non-Newtonian fluids do 4. Hypothesize a 5. Predict the material function 6. Compare with what non-Newtonian fluids do 7. Reflect, learn, revise model, repeat.

2) Predict what Newtonian fluids would do.

5 © Faith A. Morrison, Michigan Tech U.

2) Predict what Newtonian fluids would do.

Trouton viscosity (Newtonian fluids)

• Constant • Three times shear viscosity

6 © Faith A. Morrison, Michigan Tech U.

3 Generalized Newtonian Fluid CM4650 3/21/2018

Investigating Stress/Deformation Relationships (Rheology) 1. Choose a material function 2. Predict what Newtonian fluids would do 3. See what non-Newtonian fluids do 4. Hypothesize a 5. Predict the material function 6. Compare with what non-Newtonian fluids do 7. Reflect, learn, revise model, repeat.

5) Predict the material function (with new )

What does the Fake-O-model predict for steady elongational viscosity?

7 © Faith A. Morrison, Michigan Tech U.

5) Predict the material function

What does the Fake-O-model predict for steady elongational viscosity?

Garbage.

is tied to shear

flow only. 8 © Faith A. Morrison, Michigan Tech U.

4 Generalized Newtonian Fluid CM4650 3/21/2018

What if we make the following replacement?

→ Maybe this “guess” will work in more types of This at least can be written for flows. any flow and it is equal to the shear rate in shear flow. But it assumes a coordinate

system. 9 © Faith A. Morrison, Michigan Tech U.

4) Hypothesize a constitutive equation Fake-O-model

Observations

•The model contains parameters that are specific to shear flow – makes it impossible to adapt for elongational or mixed flows •Also, the model should only contain quantities that are independent of coordinate system (i.e. invariant)

We are identifying constraints on our freedom to “make up” constitutive models.

10 © Faith A. Morrison, Michigan Tech U.

5 Generalized Newtonian Fluid CM4650 3/21/2018

4) Hypothesize a constitutive equation

In Chapter 7, we find a solution:

• take out the shear rate and • replace with the magnitude of the rate-of- deformation tensor (which is related to the second invariant of that tensor).

This is going to work.

11 © Faith A. Morrison, Michigan Tech U.

Chapter 7: Generalized Newtonian fluids

CM4650 Polymer Rheology Carreau- Michigan Tech Yassuda GNF

log positionposition of break of break on onscale  is determinedscale is determined by by 

o slope is determined by n curvaturecurvature here here is is determineddetermined byby a



log 1

12 © Faith A. Morrison, Michigan Tech U.

6 Generalized Newtonian Fluid CM4650 3/21/2018

Current Goal: Develop new constitutive models by trial and error: • Account for rate-dependence • Accommodate known constraints

Known • no parameters specific to flow • only use quantities that are constraints: independent of coordinate system

13 © Faith A. Morrison, Michigan Tech U.

Develop new constitutive models by trial and error: Rate-Dependence

Back to our main goal: Constitutive Equation – an accounting for all stresses, all flows

stress tensor Rate-of- Newtonian fluids: (all deformation tensor flows)

In general: In the general case, f needs to be a non-linear function (in time and position)

What should we choose Answer: something that for the function f? matches the data. Let’s start with the steady shear data.

14 © Faith A. Morrison, Michigan Tech U.

7 Generalized Newtonian Fluid CM4650 3/21/2018

Develop new constitutive models by trial and error: Rate-Dependence

Non-Newtonian, Inelastic Fluids log First, we concentrate on the observation that shear viscosity depends on  shear rate. o

≡ loglog Non-Newtonian ≡ viscosity,

We will design a constitutive equation that shear rate predicts this behavior in shear flow

15 © Faith A. Morrison, Michigan Tech U.

Develop new constitutive models by trial and error: Rate-Dependence

Newtonian Constitutive Equation v   1  For Newton’s experiment (shear flow): v   0   0   123  v   0 1 0      x2   11 12 13    v1    21  22  23    0 0 x      2   31 32 33 123    0 0 0   We could make this equation give 123 the right answer (shear thinning) in steady shear flow if we substituted a function of shear rate for the constant viscosity.

16 © Faith A. Morrison, Michigan Tech U.

8 Generalized Newtonian Fluid CM4650 3/21/2018

Develop new constitutive models by trial and error: Rate-Dependence

Generalized Newtonian Fluid (GNF) constitutive equation

v1 

  SHEAR FLOW v  0   0 0  0   123 00 000

GNF  v v v v v   v   2 1 1  2 1  3   1  v  v ALL FLOWS  x1 x2 x1 x3 x1   2  GNF is    v v v v v  v3  designed in     1  2 2 2 3  2  123 shear flow;  x2 x1 x2 x2 x3  fingers  v1 v3 v3 v2 v3  crossed it’ll   2  x x x x x  work in all  3 1 2 3 3 123 flows.

17 © Faith A. Morrison, Michigan Tech U.

Develop new constitutive models by trial and error: Rate-Dependence

Constitutive Equation – an accounting for all stresses, all flows A simple choice for :

Generalized Newtonian Fluids (GNF)

GNF is designed in 1 shear flow; ≡ : fingers 2 crossed it’ll work in all flows.

18 © Faith A. Morrison, Michigan Tech U.

9 Generalized Newtonian Fluid CM4650 3/21/2018

Develop new constitutive models by trial and error: Rate-Dependence

Generalized Newtonian Fluids (GNF)

What do we pick for ?

•Something that matches the data; •Something simple, so that the calculations are easy

19 © Faith A. Morrison, Michigan Tech U.

Develop new constitutive models by trial and error: Rate-Dependence

In processing, the high-shear-rate behavior is the most important.

+ linear 131 kg/mole branched 156 kg/mole linear 418 kg/mol branched 428 kg/mol

Figure 6.3, p. 172 Piau et al., linear and branched PDMS 20 © Faith A. Morrison, Michigan Tech U.

10 Generalized Newtonian Fluid CM4650 3/21/2018

Develop new constitutive models by trial and error: Rate-Dependence

Power-law In shear flow ≡ model for viscosity

(in shear flow)

On a log-log plot, this would give a straight line: dv log  logmn1 log 1 dx2 Y  B  M X

21 © Faith A. Morrison, Michigan Tech U.

Develop new constitutive models by trial and error: Rate-Dependence

steady Power-law shear Non-Newtonian shear flow viscosity model for viscosity ≡ (GNF) ≡ log

Newtonian , 1

slope = n-1 shear thinning , 1

log 22 © Faith A. Morrison, Michigan Tech U.

11 Generalized Newtonian Fluid CM4650 3/21/2018

Develop new constitutive models by trial and error: Rate-Dependence

Power-Law Generalized Newtonian Fluid (PL-GNF)

m or K = consistency index (for Newtonian) n = power-law index (n = 1 for Newtonian)

≡ (Usually 0.5< n <1)

23 © Faith A. Morrison, Michigan Tech U.

Calculate flow fields with GNF constitutive equations

Let’s use it! EXAMPLE: r Pressure-driven flow z A of a Power-Law cross-sectioncross-section A: A: Generalized Newtonian fluid in a r tube P0 z

•steady state •well developed •long tube L vz(r)

fluid R g

24 © Faith A. Morrison, Michigan Tech U.

12 Generalized Newtonian Fluid CM4650 3/21/2018

Calculate flow fields with GNF constitutive equations

Velocity field Poiseuille flow of a power-law fluid:

1   1   1   RLg  P  P  n R   r  n  v r   o L    1   z 2Lm  1  R     1     n  

25 © Faith A. Morrison, Michigan Tech U.

Calculate flow fields with GNF constitutive equations

Solution to Poiseuille flow in a tube incompressible, power-law fluid

n=1.0 2.0 0.8 0.6 1.5 0.4

z,av 0.2 /v z

v 1.0 0.0

0.5

0.0 0 0.2 0.4 0.6 0.8 1 1.2 r / R 26 © Faith A. Morrison, Michigan Tech U.

13 Generalized Newtonian Fluid CM4650 3/21/2018

Calculate flow fields with GNF constitutive equations

Solution to Poiseuille flow in a tube incompressible, power-law fluid

0.8

0.6 max /v z n= 0.2 v 0.4 n= 0.4 n= 0.6 0.2 n= 0.8 n= 1 0 0 0.2 0.4 0.6 0.8 1 1.2

r/R 27 © Faith A. Morrison, Michigan Tech U.

Calculate flow fields with GNF constitutive equations

EXAMPLE: Pressure-driven flow of a Power-Law GNF between infinite parallel plates

•steady state •incompressible fluid •infinitely wide, long W x2

2H x1 x3 v (x ) x1=0 1 2 x1=L p=Po p=PL

28 © Faith A. Morrison, Michigan Tech U.

14 Generalized Newtonian Fluid CM4650 3/21/2018

Develop new constitutive models by trial and error: Rate-Dependence We can pick any function Generalized we wish for . Newtonian Fluids (GNF)

What do we pick for ?

•Something that matches the data; •Something simple, so that the calculations are easy

When we are focused on Develop new constitutive models by trial and error: Rate-Dependence

the shear-thinning regime, In processing, the high-shear-rate behavior is the most important. we picked the power-law model.

+ linear 131 kg/mole branched 156 kg/mole linear 418 kg/mol branched 428 kg/mol

Figure 6.3, p. 172 Piau et al., There are other choices. linear and branched PDMS

29 © Faith A. Morrison, Michigan Tech U.

Other GNF Carreau-Yasuda viscosity 1 models 0    1  Ellis Model 1     0 4-Parameter Carreau Model (same as CY with 2)

Cross-Williamson Model (same as CY with 1, 0)

. DeKee Model

Casson Model       0  0 Herschel-Bulkley Model DeKee-Turcotte Model See Carreau, DeKee, and Chhabra for complete discussion (Rheology of Polymeric Systems, Hanser, 1997) 30 © Faith A. Morrison, Michigan Tech U.

15 Generalized Newtonian Fluid CM4650 3/21/2018

Develop new constitutive models by trial and error: Rate-Dependence

A model that captures the 1 whole range of is the Carreau- Yasuda GNF. log positionposition of break of break on on scale  is determinedscale is determined by by 

o slope is determined by n curvaturecurvature here here is is determineddetermined byby a



loglog  31 © Faith A. Morrison, Michigan Tech U.

Develop new constitutive models by trial and error: Rate-Dependence

log positionposition of break of break on on scale  Carreau- is determinedscale is determined by by 

o Yassuda GNF slope is determined by n curvaturecurvature here here is is determineddetermined byby a   loglog 

A model with 5 parameters 1

•The viscosity function approaches the constant value of as deformation rate get large

•The viscosity function approaches the constant value as deformation rate gets small • is the time constant for the fluid • n determines the slope of the power-law region

• modifies the sharpness of the transition from to thinning

32 © Faith A. Morrison, Michigan Tech U.

16 Generalized Newtonian Fluid CM4650 3/21/2018

Develop new constitutive models by trial and error: Rate-Dependence

What about shear thickening? 10

1  (Pa s) 0.1

vol % TiO2 47 0.01 42 38 27.2 12 0.001 1 10 100 1000 10000 1 Figure 6.27, p. 188 Metzner and , s Whitlock; TiO2/water suspensions 33 © Faith A. Morrison, Michigan Tech U.

Develop new constitutive models by trial and error: Rate-Dependence

Power-Law GNF The Power-Law GNF model can capture steady shear flow shear thickening.

log shear thickening , 1

Newtonian , 1

shear thinning , 1 log The GNF models are inelastic models. 34 © Faith A. Morrison, Michigan Tech U.

17 Generalized Newtonian Fluid CM4650 3/21/2018

Develop new constitutive models by trial and error: Rate-Dependence

Other Inelastic Fluids

What about mayonnaise?

Mayonnaise and many other like fluids (paint, ketchup, most suspensions, asphalt) is able to sustain a yield stress.

Once the fluid begins to deform under an imposed stress, the viscosity may either be constant or may shear-thin. This type of steady shear viscosity behavior can be modeled with a GNF.

35 © Faith A. Morrison, Michigan Tech U.

Develop new constitutive models by trial and error: Rate-Dependence

Non-Newtonian Fluids, inelastic

For some fluids, no flow occurs when moderate stresses are applied.

Bingham plastic  21 constant slope = mo (mayo, paints, suspensions)

Newtonian

Yield  o stress ≡

36 © Faith A. Morrison, Michigan Tech U.

18 Generalized Newtonian Fluid CM4650 3/21/2018

Develop new constitutive models by trial and error: Rate-Dependence

Non-Newtonian Fluids, inelastic Bingham plastic

PMMA in water

Yield stress

 o

Friend and Hunter, 1971; dispersions of PMMA in water at various z-potentials; ≡ From Larson, p353. 37 © Faith A. Morrison, Michigan Tech U.

Develop new constitutive models by trial and error: Rate-Dependence

Non-Newtonian viscosity, Bingham Plastic GNF steady shear flow ≡ lim lim → → ≡ log log log

 21 log

slope = -1 slope = o

o

y0

≡ loglog 38 © Faith A. Morrison, Michigan Tech U.

19 Generalized Newtonian Fluid CM4650 3/21/2018

Develop new constitutive models by trial and error: Rate-Dependence

Bingham GNF A model with 2 parameters

∞

= viscosity parameter = yield stress

There is no flow until the shear stress exceeds a critical value called the yield stress.

Other GNF Carreau-Yasuda viscosity 1 models 0    1  Ellis Model 1     0 4-Parameter Carreau Model (same as CY with 2)

Cross-Williamson Model (same as CY with 1, 0)

. DeKee Model

Casson Model       0  0 Yield stress plus power- Herschel-Bulkley Model law viscosity behavior DeKee-Turcotte Model See Carreau, DeKee, and Chhabra for complete discussion (Rheology of Polymeric Systems, Hanser, 1997) 40 © Faith A. Morrison, Michigan Tech U.

20 Generalized Newtonian Fluid CM4650 3/21/2018

Develop new constitutive models by trial and error: Rate-Dependence

Address other non- What now? Newtonian phenomena?

•Predict material functions with the Generalized Newtonian Constitutive Equation. Example: Elongational viscosity, etc. •Calculate velocity and stress fields predicted by Generalized Newtonian Constitutive Equations Example: Poiseuille flow, drag flow, etc.

Develop new constitutive models by trial and error: Rate-Dependence

Address other non- What now? Newtonian phenomena?

•Predict material functions with the Generalized We can also calculate Newtonian Constitutive Equation. non-standard flow Example: Elongational viscosity, etc. fields and see if the predictions are •Calculate velocity and stress fields predicted by sensible. Generalized Newtonian Constitutive Equations Example: Poiseuille flow, drag flow, etc.

21 Generalized Newtonian Fluid CM4650 3/21/2018

Step Shear Strain Material Functions EXAMPLE: Drag flow between infinite parallel plates Kinematics:  0 t  0 (t)x2    (t)  lim0 0  t   •Newtonian v   0   0 •steady state  0 t    0  •incompressible fluid  123   0  constant   0 •very wide, long V Predict a Calculate a flow •uniform pressure W Material Functions:     First normal-stress G  11 22 1 2 x2 21(t, 0) relaxation modulus 0 G(t, 0)  v (x ) H  0 1 2     material field or stress Second normal- G  22 33 What kind of Relaxation  stress relaxation 2 2 modulus  0 x1 modulus x3 4 © Faith A. Morrison, Michigan Tech U. function for a rheology calculation field for a fluid 2 constitutive am I considering? in a situation equation

Find the recipe card Sketch the situation (has kinematics, and choose an appropriate definitions of material coordinate system functions; (t) is known)

Model the flow For the given by asking yourself questions about the kinematics, calc. velocity field (are some components from the constitutive and derivatives zero? The constitutive equation equation must be known)

From ,calculate Solve the differential equations the material function (Solve EOM and continuity for and from the definition on the and apply the boundary conditions) recipe card 43 © Faith A. Morrison, Michigan Tech U.

Status: Current Goal: Develop new constitutive models by trial and error: • Account for Rate-Dependence • Accommodate known constraints

Known • no parameters specific to flow • only use quantities that are constraints: independent of coordinate system

22 Generalized Newtonian Fluid CM4650 3/21/2018

Next? Look for more constraints Innovate—improve the predictions of the constitutive equation Status: Current Goal: Develop new constitutive models by trial and error: • Account for Rate-Dependence • Accommodate known constraints

Known • no parameters specific to flow • only use quantities that are constraints: independent of coordinate system

Innovate and improve constitutive equations The steady shear viscosity function can be fit to experimental data to an arbitrarily high precision. Does this mean that Generalized Newtonian Fluid models are okay to use in all situations?

Not necessarily. A constitutive model needs to be able to predict all stresses in all flows, not just shear stresses in steady shearing. We need to check predictions.

For example, does the GNF predict the shear normal stresses?

  0   11 12    21 22 0   0 0    33123

23 Generalized Newtonian Fluid CM4650 3/21/2018

Innovate and improve constitutive equations 2 Generalized Newtonian Fluid (GNF) constitutive 2 equation 2 In Shear Flow: 0 0 v1    v   0  00  0   123 000

No matter what we pick for the function , we cannot predict shear normal stresses with a Generalized Newtonian Fluid.

Innovate and improve constitutive equations shear stress response

What does the GNF What the data show: predict for start-up shear stresses?

 21(t) imposed shear rate increasing  0t

What the GNF models predict:

0 t  21(t)

increasing 

24 Generalized Newtonian Fluid CM4650 3/21/2018

Innovate and improve constitutive equations

Start-up shear stresses What the GNF models predict: What the data show: Correctly captures rate dependence  21(t)  21(t)

increasing  increasing 

0t 0t

misses start-up effects

No matter what we pick for the function , we cannot predict the time-dependence of shear start-up correctly with a GNF.

Innovate and improve constitutive equations elongational stress response

What does the GNF What the data show: predict in steady elongational flow? Trouton’s Rule lim̅ 3 →

imposed deformation (there is limited elongational (steady state) viscosity data available) x3

What the GNF models predict:

x , x 1 2 For all ̅ 3 deformation rates

If a material shear-thins, GNF predicts it will tension-thin. This is not observed. 50 © Faith A. Morrison, Michigan Tech U.

25 Generalized Newtonian Fluid CM4650 3/21/2018

Summary: Generalized Newtonian Fluid Constitutive Equations

PRO: •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 CON: •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

Summary: Generalized Newtonian Fluid Constitutive Equations

PRO: •A first constitutive equation •Can match steady shearing data very well •Simple to calculate with •Found to predict pressure-drop/flow rate We now look to relationships well address this failing of GNF models by •Fails to predict shear normal stresses seeking to CON: incorporate •Fails to predict start-up or cessation effects memory. (time-dependence, memory) – only a function of instantaneous velocity gradient •Derived ad hoc from shear observations; unclear of validity in non-shear flows

26 Generalized Newtonian Fluid CM4650 3/21/2018

Innovate and improve constitutive equations What we know so far…

Rules for Constitutive Equations

, , , , material info

The stress expression:

•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 53 © Faith A. Morrison, Michigan Tech U.

Innovate and improve constitutive equations What we know so far…

Rules for Constitutive Equations

, , , , material info Tensor invariants – scalars The stress expression: associated with a tensor that do not depend on •Must be of tensor order coordinate system •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 54 © Faith A. Morrison, Michigan Tech U.

27 Generalized Newtonian Fluid CM4650 3/21/2018

Innovate and improve constitutive equations

Tensor Invariants

IA  traceA  trA

For the tensor written in Cartesian coordinates: 3 traceA   App  A11  A22  A33 p1 3 3 II A  traceA A  A: A  Apk Akp pk1 1 3 3 3 III A  traceA A A  Apj Ajh Ahp pjh1 1 1

Note: the definitions of invariants written in terms of coefficients are only valid when the tensor is written in Cartesian coordinates.

Done with Inelastic models (GNF). Let’s move on to Linear Elastic models

28 Generalized Newtonian Fluid CM4650 3/21/2018

Chapter 8: Memory Effects: GLVE

CM4650 Polymer Rheology Michigan Tech

Maxwell’s model combines viscous and elastic responses in series

Spring (elastic) and dashpot (viscous) in series: initial state no force

Dtotal

final state force, f, resists displacement Displacements are additive: f

Dtotal  Dspring  Ddashpot