Constitutive Modeling

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Constitutive Equations

fluidmechanics

Mathematical function relating stress tensor to velocity field

for a particular fluid


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Constitutive ModelingLooking for appropriate tensorial expressions relating stress

and strain that match observed material behavior.

Constraints:

All terms of a constitutive equation must be of second order

(i.e. have two unit vectors associated with them).

Must be coordinate system invariant. Must not include any

variables that depend on coordinate system. The only scalarfunctions that may be in constitutive equations are functions

of invariants of the vectors or tensors.

Must predict a symmetric stress tensor. (We can achieve this

by using the velocity gradient tensor and its transpose in the

definition of the strain rate tensor.)

Material objectivity is required. The response of the material

to an applied deformation must be the same for all observers.


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Constitutive modeling Many constitutive equations are available

that meet the 4 criteria listed previously

You must choose one that is best suited to

your problem i.e describes the flow

behaviour of the fluid of interest in the flow

of interest to an acceptable level

Dimensionless groups Deborah (De)

number and the Weissenberg number (Wi)


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Types of constitutive equations

Generalized Newtonian

accounts for varying viscosity but not for elasticity

Linear and quasi-linear viscoelastic constitutiveequations

account only for linear viscoelasticity

Non-linear constitutive equations

describe nonlinear viscoelasticity

Choose a constitutive equation by considering the behavior of fluidof interest and the flow of interest. More sophisticated constitutive

models can describe more complex phenomena but often introduce

numerical difficulties. Normally choose the simplest constitutive

equation that will give realistic or useful results.


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Analysis of flow/fluid Deborah number: the

ratio of the relaxation

time of the fluid to thetime scale of a flow

Determines how

important memoryeffects are in the flow

Weissenberg number:

the ratio of the relation

time of the fluid to atime scale of the flow

which relates to non-

linearity Determines how

important non-linear

effects are in the flow

flowtDe

=

'flowtWi

=


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Example: Oscillatory shear

If De


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Example: oscillatory shear and the Pipkin diagram

Wi

De


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Example: Conically converging flow

Q

dz

dR

R

Q

tDe

3

flow

=

=3

1'flow R

Q

tWi =

=


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Generalized Newtonian Constitutive Eqn

Newtonian constitutive Eqn:

Generalized Newtonian

constitutive Eqn:

= &

That constitutive equation only allows for a constant

viscosity, . Since one of the important aspects of non-Newtonian behavior is a shear rate dependant viscosity a

simple modification of this constitutive equation is to include

a shear rate dependant viscosity.

( )

=

=

&&

&&

where

scalar invariant


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Functional forms of the viscosity curvePower law model:

( ) 1nm = &&Carreau-Yasuda model:

( )( )[ ] a

1na

0

1

+=

&&

Bingham model:

( )

stressyieldand

where

y

y

y

0

y

=

=

>+

=&

&


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Limitations of GNF Models

Do not necessarily model the viscosity

curve well.

In general does not represent well non-

shearing flows

Do not predict elastic effects such as normalstresses.

Does not include a dependency on strain

history, therefore cannot predict transient

behavior


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ExampleConsider a hypothetical new material function based on

the following kinematics;

( )( ) ( ){ 0a;0tatexp 0t0twhere00

xt

v

xyz

2

>>


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Example

How would you go about calculating the velocity field

for a specific GNF model?


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EOM, incompressible fluid in cylindricalcoordinates


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Generalized linear viscoelasticconstitutive eqn

( ) ( ) ( ) tdt'ttGtt

=

&

Boltsmann

superposition

principle:

As with the GNF we can consider different forms of

the material function G(t-t). The most common

forms are the Maxwell model and the generalized

Maxwell model.


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Maxwell Model

G

or= /G

=

+ &

t

Differential form:

( ) ( ) tdtttexpGt

= &

Integral form:


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Generalized Maxwell Model

==

=

+N

1ii

ii

ii t &

Differential form:

( )( ) tdt

tt

expG

t N

1i i

i

= = &

Integral form:

N Maxwell elements in

parallel


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Limitations of the GLVE Model

It predicts a constant viscosity = 0. Therefore it is onlyvalid for flows with small shear rates.

Strain is assumed to be additive, therefore it is limited to

small strains.

Predicts zero normal stresses in shear flow.

It is not frame invariant, i.e. it cannot describe flows with a

superimposed rigid rotation.

In chapter 9 we will see how to fix this model

so that it is frame invariant.

Constitutive Modeling

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