# Lecture 7: Non-Newtonian Fluids

Mechanics

Nov 14, 2013 (4 years and 6 months ago)

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Lecture 7: Non
-
Newtonian Fluids

Classification of Non
-
Newtonian Fluids

Laminar Flow of a Non
-
Newtonian fluid in
Circular Pipes

Recommended text
-
book: W.F. Hughes, J.A.
Brighton,
Schaum's

outline of theory and
problems of fluid dynamics, New York:
McGraw Hill,

1999

1

Examples

Water and simple liquids; air and simple gases… are Newtonian
fluids.

Fluids in food industry, gels, polymers, slurries, drilling muds,
blood… are Non
-
Newtonian fluids.

The Non
-
Newtonian behaviour is frequently associated with
complex internal structure: fluid has large complex molecules
(like a polymer) or fluid is a heterogeneous solution (like a
suspension)...

1: Coal slurries having consistency of over 80% by volume of
powdered or crushed coal in water can be pumped long distances
with much less power requirements for pumping than pure
water.

2

Examples

2:
In the fracturing treatment of oil wells, materials have been
developed which when added to water make a fluid so thick that it
suspends sand, glass or metal pellets. Yet the same fluid can be
pumped down a well at enormous rates with less than half the friction
loss of water.

Such materials are used for fracturing of an oil reservoir. Fracturing is
used to increase the production of the well. First, a crack is initiated in
the producing zone. Then, fluid pumped down the well under high
pressure greatly extends this crack. The sand, or pellets act as propping
agents to hold the fracture open after the treatment. Fluid must be
delivered at a rapid rate to overcome the loss by diffusion of fluid
within the pores of the fractured rock.

3

Definition: Newtonian/Non
-
Newtonian
fluids

The viscous stress tensor for incompressible Newtonian fluid is

ik
ik

--

rate of shear strain

i
k
k
i
ik
x
v
x
v

Newtonian fluids
: linear proportionality between the shearing tensor
and the shearing rate.

Non
-
Newtonian fluids
: any different relation between the shearing
stress and the shearing rate.

4

Classification

Time Independent Fluids
(the relation between shearing stress and rate is
unique but non
-
linear)

Bingham plastics

Pseudoplastic fluids

Dilatant plastics

Time Dependent Fluids
(the shear rate depends on the shearing time or
on the previous shear rate history)

Thixotropic fluids

Reopectic fluids

Viscoelastic fluids
(the shear stress is determined by the shear strain and
the rate of shear strain)

5

Time Independent Fluids

ik
ik
f

τ

slope
η

0

2

3

1

0

Newtonian fluid

1

Bingham plastics

2

pseudoplastic fluids

3

dilatant fluids

1.
Bingham plastics

y
Examples
: slurries, plastics, emulsions such as paints, and
suspensions of finely solids in a liquid (e.g. drilling muds, which consist
primarily of clays suspended in water).

6

A progressively decreasing slope of shear stress vs. shear rate.

The slope can be defined as apparent viscosity:

At very high rates of shear in real fluids the apparent viscosity becomes
constant.

Examples
: paper pulp in water, latex paint, blood, syrup, molasses,
ketchup, whipped cream, nail polish

The simplest empirical model is the power law due to Ostwald:

The power law model is popular due to its simplicity.

For the cases where power law model does not give an adequate
representation, it might be practical to use the actual measured properties
of the fluid.

a

1
1

n
k
n
a
,

2. Pseudoplastic (shear thinning)
fluids

7

3. Dilatant (shear thickening) fluids

The apparent viscosity increases with increasing shear rate.

Can be represented by the power law model with
n
>1.

Less common (suspensions of corn starch or sand in water).

Applications:

Traction control
:
some all
-
wheel drive systems use a viscous coupling unit full of
dilatant fluid to provide power transfer between front and rear wheels. On high
traction surfacing, the relative motion between primary and secondary drive
wheels is the same, so the shear is low and little power is transferred. When the
primary drive wheels start to slip, the shear increases, causing the fluid to thicken.
As the fluid thickens, the torque transferred to the secondary drive wheels
increases, until the maximum amount of power possible in the fully thickened state
is transferred.

Body armour
:
application of shear thickening fluids for use as body armour,
allowing the wearer flexibility for a normal range of movement, yet providing
rigidity to resist piercing by bullets, stabbing knife blows, and similar attacks.

8

Time Dependent Fluids

Thixotropic fluids
(the shear stress
decreases

with time as the fluid is sheared)

As the fluid is sheared from the state of rest, it breaks down (on molecular
scale), but then the structural reformation will increase with time. An
equilibrium situation is eventually reached where the breakdown rate is equal
to build
-
up rate. If allowed to rest, the fluid builds up slowly and eventually
regains its original consistency.

Examples: many gels or colloids

Rheopectic

fluids
(the shear stress
increases

with time as the fluid is sheared).

Molecular structure is formed by shear and behaviour is opposite to that of
thixotropy
.

Example: beating and thickening of egg whites, inks.

9

Viscoelastic Fluids

A viscoelastic material exhibits both elastic and viscous properties.

The simplest type is one which is Newtonian in viscosity and obeys Hooke’s law
for the elastic part:

λ

is a rigidity modulus.

Simplest and popular model
--

Maxwell liquids:

Under steady flow, . If the motion is stopped the stress relaxes as

Movies:

-
1o3Lzs

t
exp

Examples: polymers, metals at
temperature close to their melting
point

10

Laminar flow of Bingham plastics

R

z

r
p

We consider steady plane parallel flow. The
governing equations are reduced to

rz
r
r
r
z
p
z

1
0
:
projection
(*)

By denoting and integrating (*) we obtain

A
z
p

r
c
Ar
rz
1
2

c
1
= 0 as the stress tensor must be bounded at
r
= 0,
i.e.

2
Ar
rz

11

τ
rz

r

R

r
p

τ
y

y
z
y
rz
y
z
τ
τ
r
v
rA
τ
τ
r
v

for
for

2
1
0
(**)

,
:
,
:

y
rz
y
y

0
For Bingham plastics, the rate of stress tensor is related to the shearing
stress as

or

A
r
r
v
r
r
y
p
z
p

2
0

:
at
12

Integration of (**) gives

R
r
r
r
R
r
R
A
dr
rA
v
p
y
r
R
y
z

,

2
2
4
2
1
1
Setting gives

A
r
r
y
p

2

0
1
2
2

r
r
r
R
A
v
p
p
p
,

The volumetric flow rate

4
4
2
3
1
2
3
4
1
8
RA
RA
A
R
Q
y
y

At
τ
y

we will have the formulae earlier obtained for the Poiseuille flow
.

13

Eugene Cook Bingham
, born 8 December
1878, died 6 November 1945. Bingham made
many contributions to rheology.

14

Lecture 8: Flow fields with negligible
inertia forces

Flow in slowly
-
varying channel

Lubrication theory

15

Flow in slowly
-
varying channels

For
Poiseuille

flow:

0

t
v

0

v
v

0

v

Plane
-
parallel flow

slowly varying channel

0

v
v

but small

We can always make the ratio

by choosing a sufficiently slow rate of variation of the cross
-
section.

1


v
v
v

16

Consider a steady flow along a circular tube with
R(x)
, with

x
dx
dR

and

x
A
dx
dp

In such a flow, . And

z
r
v
v

~
,
~
2
R
V
u

R
V
v
v
2
~


1
~


RV
v
v
v

Hence, , i.e. inertia force is negligible.

This approximation is useful in many different circumstances
e.g. the flow of a fluid squeezed out radially by pressing close
together two plane disks.

2
2
4
,
r
R
A
r
x
v
z

8
4
A
R
Q

Thus, the flow profile and the volumetric flow flux are

17

Lubrication theory

It is a matter of common experience that two solid bodies can
slide over one another easily when there is a thin layer of fluid
between them and that under certain conditions a high positive
pressure is set up in the fluid layer. This is used as a means of
substituting fluid
-
solid friction for the much larger friction
between two solid bodies in contact. In some case the fluid layer
is used to support a useful load, and is then called a
lubrication
bearing
.

α

h
2

h

h
1

Reynolds’
theory, 1886

U

1


RU
w
hich is usually satisfied under
practical conditions of
lubrication

18

h
y
h
U
y
h
y
A
u

2
Flow profile:

Poiseuille

flow

Couette

flow

Volumetric
flow flux:

Uh
Ah
y
u
Q
h
2
1
12
d
3
0

From here, the pressure gradient is

3
2
2
6
h
Q
h
U
A
x
p

Q

must be independent of
x

x
h
h

1
(1)

Integration of (1) gives

2
1
2
1
0
1
1
1
1
6
h
h
Q
h
h
U
p
p

19

Suppose the sliding block is completely immersed in the
fluid, so
p
=
p
0

when
h
=
h
2
, which enables us to determine
Q
,

2
1
2
1
h
h
h
h
U
Q

and

2
1
2
2
1
0
6
h
h
h
h
h
h
h
U
p
p

1
2
0
if
0
h
h
p
p

A lubrication layer will be able to support
a load normal to the layer only when the
layer is so arranged that the relative
motion of the two surfaces tends to drag
fluid from the wider to the narrow end.

2
1
max
h
LU
p

p
max

can be high if
h
1

is small
(
L

is a layer length)

20

p
-
p
0

x

The total normal force exerted on either of the two boundaries
by the fluid layer is

2
1
2
1
2
1
2
0
0
2
ln
6
h
h
h
h
h
h
U
dx
p
p
L

The total tangential force exerted by the fluid on the lower
plate

2
1
2
1
2
1
0
0
ln
2
3
2
h
h
h
h
h
h
U
dx
y
u
L
y

The tangential force on the upper boundary is

2
1
2
1
2
1
0
ln
3
2
h
h
h
h
h
h
U
dx
y
u
L
h
y

21

L
h
h
h
f
1
2
1
~
block

on the

force

normal
block

on the

force

tangential

This ratio is independent of the viscosity, and can be made
indefinitely small by reduction of
h
1

with
h
1
/
h
2

held constant.

α

is regarded as a given quantity, although in any case in
which the sliding block is free to move,
α

may be a variable,
but consideration of this is beyond of our scope.

22