Generalized Newtonian Fluid CM4650 3/4/2020
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Chapter 7: Generalized Newtonian fluids
Β© Faith A. Morrison, Michigan Tech U.
1
CM4650 Polymer Rheology
Michigan TechCarreau-Yassuda GNF
slope is determined by n
log
log
position of break on scale is determined by
o
curvature here is determined by a
position of break on scale is determined by
curvature here is determined by
π π πΎ πΎπ πΎ π π π 1 πΎπ
Β© Faith A. Morrison, Michigan Tech U.
2
β’ We seek constitutive equations for non-
Newtonian fluids.
β’ We start with the Newtonian constitutive equation
β’ We βmodify itβ and make the Fake-O model.
β’ We checked out the model to see how we did.
Develop Non-Newtonian Constitutive Equations
So Far:
π πΎπ πΎ πΎ
ππΎ πΎ πΎ
π π πΎ π»π£ π»π£
Generalized Newtonian Fluid CM4650 3/4/2020
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Β© Faith A. Morrison, Michigan Tech U.
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When we tried the model out with elongational flow material functions,
something odd happened β¦
π π‘ π constant
π£ β‘
π π‘ π₯
π π‘ π₯
π π‘ π₯
Imposed Kinematics:
Steady Elongational Flow Material Functions
Material Functions: Elongational
Viscosity
Β© Faith A. Morrison, Michigan Tech U.4
π π‘ π constant
π π β‘οΏ½ΜοΏ½ οΏ½ΜοΏ½
ππ ππ
Material Stress Response: οΏ½ΜοΏ½ π‘ οΏ½ΜοΏ½ π‘
π‘0
π‘0
π
π 0, π‘
π‘0
π π‘
π
Alternatively, οΏ½Μ οΏ½ π
π£ β‘
π π‘ π₯
π π‘ π₯
π π‘ π₯
Generalized Newtonian Fluid CM4650 3/4/2020
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Β© Faith A. Morrison, Michigan Tech U.
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1. Choose a material function2. Predict what Newtonian fluids would do3. See what non-Newtonian fluids do4. Hypothesize a 5. Predict the material function6. Compare with what non-Newtonian fluids do7. Reflect, learn, revise model, repeat.
2) Predict what Newtonian fluids would do.
2) Predict what Newtonian fluids would do.
?
π ππΎ
Β© Faith A. Morrison, Michigan Tech U.
6
1. Choose a material function2. Predict what Newtonian fluids would do3. See what non-Newtonian fluids do4. Hypothesize a 5. Predict the material function6. Compare with what non-Newtonian fluids do7. Reflect, learn, revise model, repeat.
2) Predict what Newtonian fluids would do.
2) Predict what Newtonian fluids would do.
π 3π
Trouton viscosity (Newtonian fluids)
β’ Constantβ’ Three times shear viscosity
Generalized Newtonian Fluid CM4650 3/4/2020
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Β© Faith A. Morrison, Michigan Tech U.
What does the Fake-O-model predict for steady elongational viscosity?
7
Investigating Stress/Deformation Relationships (Rheology)
1. Choose a material function2. Predict what Newtonian fluids would do3. See what non-Newtonian fluids do4. Hypothesize a 5. Predict the material function6. Compare with what non-Newtonian fluids do7. Reflect, learn, revise model, repeat.
5) Predict the material function (with new π π£ )
π πΎπ πΎ πΎ
ππΎ πΎ πΎ
π π πΎ π»π£ π»π£
Β© Faith A. Morrison, Michigan Tech U.
8
5) Predict the material function
What does the Fake-O-model predict for steady elongational viscosity?
π πΎπ πΎ πΎ
ππΎ πΎ πΎ
π π πΎ π»π£ π»π£
Garbage.
π πΎπ πΎ πΎ
ππΎ πΎ πΎ
π π πΎ π»π£ π»π£
πΎ is tied to shear flow only.
Generalized Newtonian Fluid CM4650 3/4/2020
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Β© Faith A. Morrison, Michigan Tech U.
What if we make the following replacement?
This at least can be written for any flow and it is equal to the shear rate in shear flow.
But it assumes a coordinate system. 9
Investigating Stress/Deformation Relationships (Rheology)
1. Choose a material function2. Predict what Newtonian fluids would do3. See what non-Newtonian fluids do4. Hypothesize a 5. Predict the material function6. Compare with what non-Newtonian fluids do7. Reflect, learn, revise model, repeat.
πΎ βππ£ππ₯
Maybe this βguessβ will work in more types of flows.
Β© Faith A. Morrison, Michigan Tech U.
Observations
10
4) Hypothesize a constitutive equation
β’The model contains parameters that are specific to shear flow β makes it impossible to adapt for elongational or mixed flows
β’Also, the model should only contain quantities that are independent of coordinate system (i.e. invariant)
Fake-O-model
We are identifying constraints on our freedom to βmake upβ constitutive
models.
π πΎπ πΎ πΎ
ππΎ πΎ πΎ
π π πΎ π»π£ π»π£
Generalized Newtonian Fluid CM4650 3/4/2020
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Β© Faith A. Morrison, Michigan Tech U.
In Chapter 7, we find a solution:
β’ take out the shear rate πΎ and
β’ replace with the magnitude of the rate-of-
deformation tensor πΎ (which is related to the
second invariant of that tensor).
11
4) Hypothesize a constitutive equation
π π πΎ π»π£ π»π£
This is going to work.
Chapter 7: Generalized Newtonian fluids
Β© Faith A. Morrison, Michigan Tech U.
12
CM4650 Polymer Rheology
Michigan TechCarreau-Yassuda GNF
slope is determined by n
log
log
position of break on scale is determined by
o
curvature here is determined by a
position of break on scale is determined by
curvature here is determined by
π π πΎ πΎπ πΎ π π π 1 πΎπ
Generalized Newtonian Fluid CM4650 3/4/2020
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Β© Faith A. Morrison, Michigan Tech U.
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Current Goal:Develop new constitutive models by trial and error: β’ Account for rate-dependenceβ’ Accommodate known constraints
β’ no parameters specific to flowβ’ only use quantities that are
independent of coordinate system
Known constraints:
Β© Faith A. Morrison, Michigan Tech U.
Newtonian fluids: (all flows)
In general:
stress tensor
In the general case, f needs to be a non-linear function (in time and position)
Rate-of-deformation tensor
Back to our main goal:Constitutive Equation β an accounting for all stresses, all flows
What should we choose for the function f?
14
Develop new constitutive models by trial and error: Rate-Dependence
π ππΎ
π π π£ πΎ
Answer: something that matches the data. Letβs start with the steady shear data.
Generalized Newtonian Fluid CM4650 3/4/2020
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log
log
o
Β© Faith A. Morrison, Michigan Tech U.
Non-Newtonian, Inelastic Fluids
First, we concentrate on the observation that shearviscosity depends on shear rate.
Non-Newtonian viscosity, πΌ
shear rateWe will design a constitutive equation that predicts this behavior in shear flow
15
Develop new constitutive models by trial and error: Rate-Dependence
log πΎ
πΎ β‘ππ£ππ₯
π β‘
123
2
1
2
1
123333231
232221
131211
000
00
00
x
vx
v
Β© Faith A. Morrison, Michigan Tech U.
Newtonian Constitutive Equation
For Newtonβs experiment (shear flow):
123
1
0
0
v
v
We could make this equation give the right answer (shear thinning) in
steady shear flow if we substituted a function of shear rate for the
constant viscosity.
16
Develop new constitutive models by trial and error: Rate-Dependence
π ππΎ
Generalized Newtonian Fluid CM4650 3/4/2020
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Β© Faith A. Morrison, Michigan Tech U.
Generalized Newtonian Fluid (GNF)constitutive equation
SH
EA
R F
LO
WA
LL
FL
OW
S
1233
3
3
2
2
3
1
3
3
1
3
2
2
3
2
2
1
2
2
1
1
3
3
1
1
2
2
1
1
1
2
2
2
x
v
x
v
x
v
x
v
x
vx
v
x
v
x
v
x
v
x
vx
v
x
v
x
v
x
v
x
v
123
1
0
0
v
v
1233
2
1
v
v
v
v
GNF
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Develop new constitutive models by trial and error: Rate-Dependence
ππ π ππ π ππ π π
π πΎ
0ππ£ππ₯
0
ππ£ππ₯
0 0
0 0 0
πΎππ£ππ₯
πΎ πΎ
π πΎGNF is designed in shear flow; fingers crossed itβll work in all flows.
Β© Faith A. Morrison, Michigan Tech U.
Constitutive Equation β an accounting for all stresses, all flows
A simple choice for π:
Generalized Newtonian
Fluids (GNF)
18
Develop new constitutive models by trial and error: Rate-Dependence
π π πΎ πΎ
π π π£ πΎ
πΎ β‘ πΎ12πΎ: πΎ
GNF is designed in shear flow; fingers crossed itβll work in all flows.
Generalized Newtonian Fluid CM4650 3/4/2020
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Β© Faith A. Morrison, Michigan Tech U.
What do we pick for π πΎ ?
β’Something that matches the data;
β’Something simple, so that the calculations are easy
Generalized Newtonian
Fluids (GNF)
19
Develop new constitutive models by trial and error: Rate-Dependence
π π πΎ πΎ
Β© Faith A. Morrison, Michigan Tech U.
Figure 6.3, p. 172 Piau et al., linear and branched PDMS
+ linear 131 kg/molebranched 156 kg/molelinear 418 kg/molbranched 428 kg/mol
20
Develop new constitutive models by trial and error: Rate-Dependence
In processing, the high-shear-rate behavior is the most important.
Generalized Newtonian Fluid CM4650 3/4/2020
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Β© Faith A. Morrison, Michigan Tech U.
Power-law model for viscosity
On a log-log plot, this would give a straight line:
2
1log1loglogdx
dvnm
Y B M X
(in shear flow)
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Develop new constitutive models by trial and error: Rate-Dependence
π πππ£ππ₯
In shear flow πΎ β‘π πΎ ππΎ
Power-law model for viscosity (GNF)
Β© Faith A. Morrison, Michigan Tech U.
Newtonian
shear thinning
Non-Newtonian shear viscosity
steady shear flow
slope = n-1
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Develop new constitutive models by trial and error: Rate-Dependence
πΎ β‘ππ£ππ₯
π ππΎ ,π 1
π ππΎ ,π 1
π β‘ππΎ
log πΎ
log π
Generalized Newtonian Fluid CM4650 3/4/2020
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Power-Law Generalized Newtonian Fluid (PL-GNF)
m or K = consistency index (π π for Newtonian)n = power-law index (n = 1 for Newtonian)
Β© Faith A. Morrison, Michigan Tech U.
(Usually 0.5< n <1)
23
Develop new constitutive models by trial and error: Rate-Dependence
π π πΎ πΎ
π πΎ ππΎ
πΎ β‘ πΎ
EXAMPLE: Pressure-driven flow of a Power-Law Generalized Newtonian fluid in a tube
β’steady stateβ’well developedβ’long tube
g
zA
r
cross-section A:
r
z
L vz(r)
Rfluid
0P
LP
24
cross-section A:
Calculate flow fields with GNF constitutive equations
Β© Faith A. Morrison, Michigan Tech U.
Letβs use it!
Generalized Newtonian Fluid CM4650 3/4/2020
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Velocity fieldPoiseuille flow of a power-law fluid:
1
11
11
12
nnLo
z R
r
n
R
Lm
PPgLRrv
25
Calculate flow fields with GNF constitutive equations
Β© Faith A. Morrison, Michigan Tech U.
Solution to Poiseuille flow in a tubeincompressible, power-law fluid
0.0
0.5
1.0
1.5
2.0
0 0.2 0.4 0.6 0.8 1 1.2
r / R
vz/v
z,a
v
0.0
0.2
0.4
0.6
n=1.0
0.8
26
Calculate flow fields with GNF constitutive equations
Β© Faith A. Morrison, Michigan Tech U.
Generalized Newtonian Fluid CM4650 3/4/2020
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Solution to Poiseuille flow in a tubeincompressible, power-law fluid
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2
r/R
v z/v
max
n= 0.2
n= 0.4
n= 0.6
n= 0.8
n= 1
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Calculate flow fields with GNF constitutive equations
Β© Faith A. Morrison, Michigan Tech U.
EXAMPLE: Pressure-driven flow of a Power-Law GNF between infinite parallel plates
β’steady stateβ’incompressible fluidβ’infinitely wide, long
x1
x2
x3
W
2H
x1=0p=Po
x1=Lp=PL
v1(x2)
28
Calculate flow fields with GNF constitutive equations
Β© Faith A. Morrison, Michigan Tech U.
Generalized Newtonian Fluid CM4650 3/4/2020
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What do we pick for π πΎ ?
β’Something that matches the data;
β’Something simple, so that the calculations are easy
Generalized Newtonian
Fluids (GNF)
Develop new constitutive models by trial and error: Rate-Dependence
π π πΎ πΎ
Β© Faith A. Morrison, Michigan Tech U.
Figure 6.3, p. 172 Piau et al., linear and branched PDMS
+ linear 131 kg/molebranched 156 kg/molelinear 418 kg/molbranched 428 kg/mol
Develop new constitutive models by trial and error: Rate-Dependence
In processing, the high-shear-rate behavior is the most important.
We can pick any function we wish for π πΎ .
When we are focused on the shear-thinning regime, we picked the power-law model.
There are other choices.
Other GNF viscosity models
Β© Faith A. Morrison, Michigan Tech U.
See Carreau, DeKee, and Chhabra for complete discussion (Rheology of Polymeric Systems, Hanser, 1997)
Ellis Model
4-Parameter Carreau Model (same as CY with π 2)
Cross-Williamson Model (same as CY with π 1, π 0)
DeKee Model
Casson Model
Herschel-Bulkley Model
DeKee-Turcotte Model
1
0
0
1
00
30
π π π π π . π
πππΎ
ππΎ
πππΎ
π π
π πΎ π π π 1 πΎπ
Carreau-Yasudaπ π πΎ πΎ
Generalized Newtonian Fluid CM4650 3/4/2020
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Β© Faith A. Morrison, Michigan Tech U.
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Develop new constitutive models by trial and error: Rate-Dependence
slope is determined by n
log
log
position of break on scale is determined by
o
curvature here is determined by a
position of break on πΎ scale is determined by π
log πΎ
curvature here is determined by π
π πΎ π π π 1 πΎπ
π π πΎ πΎA model that captures the
whole range of πΎis the Carreau-Yasuda GNF.
Carreau-Yassuda GNF
Β© Faith A. Morrison, Michigan Tech U.
β’The viscosity function approaches the constant value of π as deformation rate get large
β’The viscosity function approaches the constant value π as deformation rate gets small
β’ π is the time constant for the fluid
β’ n determines the slope of the power-law region
β’π modifies the sharpness of the transition from π to thinning
A model with 5 parameters
32
Develop new constitutive models by trial and error: Rate-Dependence
π π πΎ πΎ
π πΎ π π π 1 πΎπ
slope is determined by n
log
log
position of break on scale is determined by
o
curvature here is determined by a
position of break on πΎ scale is determined by π
log πΎ
curvature here is determined by π
Generalized Newtonian Fluid CM4650 3/4/2020
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0.001
0.01
0.1
1
10
1 10 100 1000 10000
47 42 38 27.2 12
1, s
)( sPa
vol % TiO2
What about shear thickening?
Figure 6.27, p. 188 Metzner and Whitlock; TiO2/water suspensions
Β© Faith A. Morrison, Michigan Tech U.
33
Develop new constitutive models by trial and error: Rate-Dependence
Power-Law GNF
Β© Faith A. Morrison, Michigan Tech U.
Newtonian
shear thickening
shear thinning
steady shear flow
34
Develop new constitutive models by trial and error: Rate-Dependence
π ππΎ ,π 1
π ππΎ ,π 1
π ππΎ ,π 1
log πΎ
log π
The Power-Law GNF model can capture shear thickening.
The GNF models are inelastic models.
Generalized Newtonian Fluid CM4650 3/4/2020
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Β© Faith A. Morrison, Michigan Tech U.
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
Develop new constitutive models by trial and error: Rate-Dependence
constant slope = mo (mayo, paints, suspensions)
21
o
Bingham plastic
Newtonian
Β© Faith A. Morrison, Michigan Tech U.
Non-Newtonian Fluids, inelastic
Yield stress
For some fluids, no flow occurs when moderate stresses are applied.
36
Develop new constitutive models by trial and error: Rate-Dependence
πΎ β‘ πΎ
Generalized Newtonian Fluid CM4650 3/4/2020
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Β© Faith A. Morrison, Michigan Tech U.
Bingham plasticNon-Newtonian Fluids, inelastic
o
Yield stress
Friend and Hunter, 1971; dispersions of PMMA in water at various z-potentials; From Larson, p353.
PMMA in water
37
Develop new constitutive models by trial and error: Rate-Dependence
πΎ β‘ πΎ
log
log
o
slope = -1
21
y
slope = o
Bingham Plastic GNF
Β© Faith A. Morrison, Michigan Tech U.
steady shear flow
Non-Newtonian viscosity, πΌ
0
38
Develop new constitutive models by trial and error: Rate-Dependence
πΎ β‘ πΎ log πΎ
π β‘ π ππΎ
π π
limβ
π limβ
ππΎ
πππΎ
log π log π log πΎ
π β‘ππΎ
Generalized Newtonian Fluid CM4650 3/4/2020
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π = viscosity parameterπ = yield stress
There is no flow until the shear stress exceeds a critical value π called the yield stress.
Β© Faith A. Morrison, Michigan Tech U.
Bingham GNF A model with 2 parameters
39
Develop new constitutive models by trial and error: Rate-Dependence
π π πΎ πΎ
π πΎβ π π
πππΎ
π π
Other GNF viscosity models
Β© Faith A. Morrison, Michigan Tech U.
See Carreau, DeKee, and Chhabra for complete discussion (Rheology of Polymeric Systems, Hanser, 1997)
Ellis Model
4-Parameter Carreau Model (same as CY with π 2)
Cross-Williamson Model (same as CY with π 1, π 0)
DeKee Model
Casson Model
Herschel-Bulkley Model
DeKee-Turcotte Model
1
0
0
1
00
40
π π π π π . π
πππΎ
ππΎ
πππΎ
π π
π πΎ π π π 1 πΎπ
Carreau-Yasudaπ π πΎ πΎ
Yield stress plus power-law viscosity
behavior
Generalized Newtonian Fluid CM4650 3/4/2020
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What now?
β’Predict material functions with the Generalized Newtonian Constitutive Equation.
Example: Elongational viscosity, etc.
β’Calculate velocity and stress fields predicted by Generalized Newtonian Constitutive Equations
Example: Poiseuille flow, drag flow, etc.
41
Develop new constitutive models by trial and error: Rate-Dependence
Address other non-Newtonian phenomena?
Investigating Stress/Deformation Relationships (Rheology)
1. Choose a material function2. Predict what Newtonian fluids would do3. See what non-Newtonian fluids do4. Hypothesize a 5. Predict the material function6. Compare with what non-Newtonian fluids do7. Reflect, learn, revise model, repeat.
5) Predict the material function (with new )
Β© Faith A. Morrison, Michigan Tech U.
What now?
β’Predict material functions with the Generalized Newtonian Constitutive Equation.
Example: Elongational viscosity, etc.
β’Calculate velocity and stress fields predicted by Generalized Newtonian Constitutive Equations
Example: Poiseuille flow, drag flow, etc.
42
Develop new constitutive models by trial and error: Rate-Dependence
Address other non-Newtonian phenomena?
Investigating Stress/Deformation Relationships (Rheology)
1. Choose a material function2. Predict what Newtonian fluids would do3. See what non-Newtonian fluids do4. Hypothesize a 5. Predict the material function6. Compare with what non-Newtonian fluids do7. Reflect, learn, revise model, repeat.
5) Predict the material function (with new )
We can also calculate non-standard flow fields and see if the predictions are sensible.
Β© Faith A. Morrison, Michigan Tech U.
Generalized Newtonian Fluid CM4650 3/4/2020
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Find the recipe card (has kinematics,
definitions of material functions; π£(t) is known)
Predict a material function for a constitutive equation
For the given kinematics, calc. π π
from the constitutive equation
From π π , calculate the material function 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 and continuity for π£ and πand apply the boundary conditions)
Model the flowby asking yourself questions about the velocity field (are some components and derivatives zero? The constitutive
equation must be known)
x1
x2
x3
H
W V
v1(x2)
EXAMPLE: Drag flow between infinite parallel plates
β’Newtonianβ’steady stateβ’incompressible fluidβ’very wide, longβ’uniform pressure
2
Step Shear Strain Material Functions
00
00
constant
0
0
0
0
lim)(
t
t
t
t
Kinematics:
123
2
0
0
)(
xt
v
Material Functions:
0
0210
),(),(
t
tG
20
22111
G
20
33222
GRelaxation
modulus
First normal-stress relaxation modulus
Second normal-stress relaxation
modulus
Β© Faith A. Morrison, Michigan Tech U.
4
What kind of rheology calculation am I considering?
Β© Faith A. Morrison, Michigan Tech U.
Β© Faith A. Morrison, Michigan Tech U.
44
Current Goal:Develop new constitutive models by trial and error: β’ Account for Rate-Dependenceβ’ Accommodate known constraints
β’ no parameters specific to flowβ’ only use quantities that are
independent of coordinate system
Known constraints:
Status:
Generalized Newtonian Fluid CM4650 3/4/2020
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Β© Faith A. Morrison, Michigan Tech U.
45
Current Goal:Develop new constitutive models by trial and error: β’ Account for Rate-Dependenceβ’ Accommodate known constraints
β’ no parameters specific to flowβ’ only use quantities that are
independent of coordinate system
Known constraints:
Next?Look for more constraintsInnovateβimprove the predictions of the constitutive equation
Status:
Β© Faith A. Morrison, Michigan Tech U.
The steady shear viscosity function π πΎ can be fit to experimental data to an arbitrarily high precision.
Does this mean that Generalized Newtonian Fluidmodels 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?
12333
2221
1211
00
0
0
46
Innovate and improve constitutive equations
Generalized Newtonian Fluid CM4650 3/4/2020
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Β© Faith A. Morrison, Michigan Tech U.
Generalized Newtonian Fluid (GNF) constitutive equation
In Shear Flow:
123
1
0
0
v
v
No matter what we pick for the function π πΎ , we cannot predict shear normal stresses with a Generalized Newtonian
Fluid.
47
Innovate and improve constitutive equations
π π πΎ
2ππ£ππ₯
ππ£ππ₯
ππ£ππ₯
ππ£ππ₯
ππ£ππ₯
ππ£ππ₯
ππ£ππ₯
2ππ£ππ₯
ππ£ππ₯
ππ£ππ₯
ππ£ππ₯
ππ£ππ₯
ππ£ππ₯
ππ£ππ₯
2ππ£ππ₯
πΎππ£ππ₯
ππ π ππ π ππ π π
π πΎ
0ππ£ππ₯
0
ππ£ππ₯
0 0
0 0 0
Β© Faith A. Morrison, Michigan Tech U.
What does the GNF predict for start-up shear stresses?
0 t
0 t
imposed shear rate
shear stress response
)(21 t
What the data show:
What the GNF models predict:
0 t
)(21 t
increasing
increasing
48
πΎ
Innovate and improve constitutive equations
πΎ
πΎ
πΎπ£ π‘π»
Generalized Newtonian Fluid CM4650 3/4/2020
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No matter what we pick for the function π πΎ , we cannot predict the time-dependence of
shear start-up correctly with a GNF.
Β© Faith A. Morrison, Michigan Tech U.
Start-up shear stresses
0 t
)(21 t
What the data show:
What the GNF models predict:
0 t
)(21 t
increasing increasing
misses start-up effects
Correctly captures rate dependence
49
Innovate and improve constitutive equations
πΎπΎ
Β© Faith A. Morrison, Michigan Tech U.
What does the GNF predict in steady elongational flow?
imposed deformation(steady state)
elongational stress response
What the data show:
What the GNF models predict:
21, xx
3x
Troutonβs Rule
For all deformation
rates
(there is limited elongational viscosity data available)
If a material shear-thins, GNF predicts it will tension-thin.
This is not observed.50
οΏ½Μ οΏ½ 3π
limβοΏ½Μ οΏ½ 3π
Innovate and improve constitutive equations
Generalized Newtonian Fluid CM4650 3/4/2020
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Β© Faith A. Morrison, Michigan Tech U.
Summary: Generalized Newtonian Fluid Constitutive Equations
β’A first constitutive equation
β’Can match steady shearing data very well
β’Simple to calculate with
β’Found to predict pressure-drop/flow rate relationships well
β’Fails to predict shear normal stresses
β’Fails to predict start-up or cessation effects (time-dependence, memory) β only a function of instantaneous velocity gradient
β’Derived ad hoc from shear observations; unclear of validity in non-shear flows
PRO:
CON:
51
β’A first constitutive equation
β’Can match steady shearing data very well
β’Simple to calculate with
β’Found to predict pressure-drop/flow rate relationships well
β’Fails to predict shear normal stresses
β’Fails to predict start-up or cessation effects (time-dependence, memory) β only a function of instantaneous velocity gradient
β’Derived ad hoc from shear observations; unclear of validity in non-shear flows
Β© Faith A. Morrison, Michigan Tech U.
Summary: Generalized Newtonian Fluid Constitutive Equations
PRO:
CON:
We now look to address this
failing of GNF models by seeking to
incorporate memory.
52
Generalized Newtonian Fluid CM4650 3/4/2020
27
Β© Faith A. Morrison, Michigan Tech U.
Rules for Constitutive Equations
β’Must be of tensor order
β’Must be a tensor (independent of coordinate system)
β’Must be a symmetric tensor
β’Must make predictions that are independent of the observer
β’Should correctly predict observed flow/deformation behavior
The stress expression:
53
Innovate and improve constitutive equations
π π‘ π πΎ, πΌ , πΌπΌ , πΌπΌπΌ , material info
What we know so farβ¦
Β© Faith A. Morrison, Michigan Tech U.
Rules for Constitutive Equations
β’Must be of tensor order
β’Must be a tensor (independent of coordinate system)
β’Must be a symmetric tensor
β’Must make predictions that are independent of the observer
β’Should correctly predict observed flow/deformation behavior
The stress expression:
54
Innovate and improve constitutive equations
π π‘ π πΎ, πΌ , πΌπΌ , πΌπΌπΌ , material info
What we know so farβ¦
Tensor invariants β scalars associated with a tensor that do not depend on
coordinate system
Generalized Newtonian Fluid CM4650 3/4/2020
28
Β© Faith A. Morrison, Michigan Tech U.
Tensor Invariants
AtrAtraceIA
For the tensor written in Cartesian coordinates:
332211
3
1
AAAAAtracep
pp
3
1
3
1
3
1
3
1
3
1
:
p j hhpjhpjA
p kkppkA
AAAAAAtraceIII
AAAAAAtraceII
Note: the definitions of invariants written in terms of coefficients are only valid when the tensor is written in Cartesian coordinates.
55
Innovate and improve constitutive equations
Β© Faith A. Morrison, Michigan Tech U.
56
Done with Inelastic models (GNF).
Letβs move on to Linear Elastic models
Generalized Newtonian Fluid CM4650 3/4/2020
29
Chapter 8: Memory Effects: GLVE
Β© Faith A. Morrison, Michigan Tech U.
57
CM4650 Polymer Rheology
Michigan Tech
f
Dtotal
initial stateno force
final stateforce, f, resists
displacement
Maxwellβs model combines viscous and elastic responses in series
dashpotspringtotal DDD
Displacements are additive:
Spring (elastic) and dashpot (viscous) in series:
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