# and other funny fluids

Mechanics

Oct 29, 2013 (4 years and 7 months ago)

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Chris Macosko

Department of Chemical Engineering and Materials Science

NSF
-

MRSEC

(National Science Foundation sponsored Materials Research Science and Engineering Center)

IP
RIME

(Industrial Partnership for Research in Interfacial and Materials Engineering)

IMA Annual Program Year Tutorial

An Introduction to Funny (Complex) Fluids: Rheology, Modeling and Theorems

September 12
-
13, 2009

Understanding silly putty, snail slime

and other funny fluids

What is rheology?

rein

(Greek)

=

ta panta rei

=

rheology =

honey and
mayonnaise

rate of deformation

stress=
f/area

to flow

every thing flows

study of flow?, i.e. fluid mechanics?

honey and
mayo

rate of deformation

viscosity =
stress/rate

What is rheology?

rein

(Greek)

=

ta panta rei

=

rheology =

rubber band and
silly putty

time of deformation

modulus
= f/area

to flow

every thing flows

study of flow?, i.e. fluid mechanics?

honey and
mayo

rate of deformation

viscosity

4 key rheological phenomena

fluid mechanician:

simple fluids complex flows

rheologist:

complex fluids simple flows

materials chemist:

complex fluids complex flows

rheology =
study of deformation of complex materials

rheologist fits data to constitutive equations which

-

can be solved by fluid mechanician for complex flows

-

have a microstructural basis

from: Rheology: Principles, Measurement and Applications, VCH/Wiley (1994).

Goal: Understand Principles of Rheology:
(stress, strain, constitutive equations)

stress = f (deformation, time)

Simplest constitutive relations:

Newton’s Law:

Hooke’s Law:

shear thinning (thickening)

time dependent modulus G(t)

normal stresses in shear N
1

extensional > shear stress

u

>

Key Rheological Phenomena

1
-
8

ELASTIC SOLID

1

The power of any spring

is in the same proportion

with the tension thereof.

Robert Hooke (1678
)

f
 
L

L
´

k
1

k
2

L
´

k
1

k
2

f

f

k

L

modulus

stress
t

Young (1805)

strain

1
-
9

Uniaxial Extension

Natural rubber

G=3.9x10
5
Pa

a = area

natural rubber G = 400 kPa

1
-
10

Silicone rubber G = 160 kPa

Goal: explain different results in extension and shear

obtain from Hooke’s Law in 3D

If use stress and deformation tensors

= 0

=
-
0.4

= 0.4

Shear gives different stress response

1
-
11

Stress Tensor
-

Notation

direction of stress on plane

plane stress acts on

Other notation besides
T
ij
:
s
ij

or

P
ij

1
-
12

1. Uniaxial Extension

Rheologists use very simple
T

T
22

=
T
33

= 0

or

T
22

=
T
33

T
22

T
33

T
11
-
T
22

causes deformation

T
11

=
0

1
-
13

Consider only normal stress components

Hydrostatic Pressure
T
11

= T
22

= T
33

=
-
p

If a liquid is
incompressible

G ≠ f(p)

≠ f(p)

Then only
t
the
extra

or
viscous

stresses
cause deformation

T

=
-
p
I

+
t

and only the normal stress
differences

cause deformation

T
11

-

T
22

=
t
11

-

t
22

≡ N
1

(shear)

1
-
14

2. Simple Shear

But to balance angular momentum

Stress tensor for simple shear

Only 3 components:

T
12

T
11

T
22

=
t
11

t
22

≡ N
1

T
22

T
33

=
t
22

t
33

≡ N
2

in general

T
21

T
21

T
12

T
12

Rheologists use very simple
T

Hooke

Young

Cauchy

Gibbs
Einstein

(1678)

(~1801)

(1830’s)

(1880,~1905)

Stress Tensor Summary

T
11

T
11

n

2. in general
T

= f( time or rate, strain)

3. simple
T

for rheologically complex materials:

-

extension and shear

4.

T

= pressure + extra stress
=
-
p
I

+

t
.

5.
τ

causes deformation

6. normal stress differences cause deformation,

t
11
-
t
22

=
T
11
-
T
22

7. symmetric
T

=
T
T

i.e.
T
12
=
T
21

1.

stress at point

on any
plane

1
-
16

a new tensor !

s = w

y

w = y + s

w

y

s

P

Q

s′ = w′
-

y′

w’ = y’ + s’

w′

y′

s′

P

Q

x =

displacement function

describes how material points move

s’
is a vector connecting
two very close points in the
material, P and Q

1
-
17

Apply
F

to Uniaxial Extension

Displacement functions describe how coordinates of P in
undeformed state, x
i

have been displaced to coordinates of P in
deformed state, x
i
.

1
-
18

Can we write Hooke’s Law as ?

Assume:

1)
constant volume V′ = V

1

axis

1
-
19

Can we write Hooke’s Law as ?

Solid Body Rotation

expect no stresses

For solid body rotation,
expect
F

=
I

t

=0

䉵B

F

I

F

F
T

Need to get rid of rotation
create a new tensor!

1
-
20

B
ij

gives relative local change in area within the sample.

Finger Tensor

Solid Body Rotation

1
-
21

1. Uniaxial Extension

since
T
22

= 0

Neo
-
Hookean Solid

1
-
22

2. Simple Shear

agrees with experiment

Silicone rubber G = 160 kPa

1.

area change around a point

on any
plane

2. symmetric

3. eliminates rotation

4. gives Hooke’s Law in 3D

fits rubber data fairly well

predicts N
1
, shear normal stresses

Finger Deformation Tensor Summary

Course Goal: Understand Principles of Rheology:
(constitutive equations)

stress = f (deformation, time)

Simplest constitutive relations:

Newton’s Law:

Hooke’s Law:

shear thinning (thickening)

time dependent modulus G(t)

normal stresses in shear N
1

extensional > shear stress

u

>

Key Rheological Phenomena

VISCOUS LIQUID

2

The resistance which arises

From the lack of slipperiness

Originating in a fluid, other

Things being equal, is

Proportional to the velocity

by which the parts of the

fluids are being separated

from each other.

Isaac S. Newton (1687)

measured

in shear

1856

capillary (Poiseuille)

1880’s concentric cylinders

(Perry, Mallock,

Couette, Schwedoff)

Newton, 1687

Stokes
-
Navier, 1845

Bernoulli

Familiar materials have a wide range in viscosity

Barnes et al.
(1989).

measured

in shear

1856

capillary (Poiseuille)

1880’s concentric cylinders

(Perry, Mallock,

Couette, Schwedoff)

measured in extension

1906
Trouton

u

= 3

Newton, 1687

Stokes
-
Navier, 1845

Bernoulli

To hold his viscous pitch samples, Trouton forced a
thickened end into a small metal box. A hook was
attached to the box from which weights were hung.

“A variety of pitch which gave by the traction method
l

= 4.3 x 10
10

(poise) was found by the torsion
method to have a viscosity
m

= 1.4 x 10
10

(poise).”
F.T. Trouton (1906)

polystyrene 160
°
C

Münstedt (1980)

Goal

1.
Put Newton’s Law in 3 dimensions

rate of strain tensor 2
D

show

u

= 3

Separation and displacement

of point Q from P

s = w
-

y

s′ = w′
-

y′

w′

y′

s′

P

Q

w

y

s

P

Q

Alternate notation
:

Viscosity is “proportional to the
velocity

by which the parts of the
fluids are being
separated

from each other.”

Newton

Can we write Newton’s Law for viscosity as
t

=

L
?

solid body rotation

Rate of Deformation Tensor
D

t
2

t
2

Other notation:

Vorticity Tensor
W

Example 2.2.4 Rate of Deformation Tensor is a Time Derivative of
B
.

Show that 2D = 0 for solid body rotation

Here planes of fluid slide over each other like cards in a deck.

Newtonian Liquid

t

=

2
D

or

T

=
-
p
I

+

2
D

Time derivatives of the displacement functions

for simple, shear

Newtonian Liquid

Apply to Uniaxial Extension

t

=

2D

From definition of extensional viscosity

Newton’s Law in 3 Dimensions

predicts

0

low shear rate

predicts

u0

= 3

0

but many materials show large deviation

Newtonian Liquid

T
11

T
11

n

1.

stress

at point on
plane

Summary of Fundamentals

simple
T

-

extension and shear

T

= pressure + extra stress
=
-
p
I

+

t
.

symmetric
T

=
T
T

i.e.
T
12
=
T
21

2.
area change

around a point on plane

symmetric, eliminates rotation

gives Hooke’s Law in 3D, E=3G

3.

rate of separation

of particles

symmetric, eliminates rotation

gives Newton’s Law in 3D,

Course Goal: Understand Principles of Rheology:

stress = f (deformation, time)

NeoHookean: Newtonian:

shear thinning (thickening)

time dependent modulus G(t)

normal stresses in shear N
1

extensional > shear stress

u

>

Key Rheological Phenomena

t

=

2
D