Lecture 7-8

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.
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Definition: Newtonian/Non-Newtonian
fluids
The viscous stress tensor for incompressible Newtonian fluid is

ik
  
ik
 v
v
  


x
x

i
k
k
i
ik

 -- rate of shear strain

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.
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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)
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Time Independent Fluids

ik
 f 
ik

τ
0 – Newtonian fluid
1 – Bingham plastics
2 – pseudoplastic fluids
3 – dilatant fluids
1
2
0
slope η
3

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).
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2. Pseudoplastic (shear thinning)
fluids
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.
a
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:
  k 
a
 n 1 
,
n 1
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.
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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.
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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.
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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 exp   t  


Movies:
Examples: polymers, metals at
temperature close to their melting
point
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Laminar flow of Bingham plastics
R
rp
We consider steady plane parallel flow. The
governing equations are reduced to
z
z  projection
By denoting 
p
z
 A
:
0  
p
z

1 
r r
r  rz  (*)
and integrating (*) we obtain
 rz  
Ar
2

c1
r
c1 = 0 as the stress tensor must be bounded at r = 0,
i.e.
Ar
τrz
 rz  
rp R
r
2
τy
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For Bingham plastics, the rate of stress tensor is related to the shearing
stress as    y :   0 ,
   y :  
 rz   y

,
or
v
r

(**)
at r  r :
p
rz
z
0

1
2
v
r
z
v
rA    
0 
z
r
y
r 
p
2
for τ  τ
y
for τ  τ
y
y
A
12
Integration of (**) gives
1 
1
A

R  r
v      rA dr 
 
2
4

r
2
z
y
R
Setting r  r 
p

2
2


y

R  r , r  r  R
p
gives
y
A
2
R


v 
 1 ,


A  r

2

p
r r 0
p
p
The volumetric flow rate
R A 
4  2  1  2  
Q 
 
 
1  
8 
3  RA  3  RA  
4
4
y
y
At τy 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

v
For Poiseuille flow:
t
0
For steady flow in a
slowly varying channel


v   v  0

v  0
Plane-parallel flow


v   v  0 but small


 v   v
We can always make the ratio
 1

v
by choosing a sufficiently slow rate of variation of the crosssection.
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Consider a steady flow along a circular tube with R(x), with
dR
dx
  x 
dp
and
dx
  A x 
In such a flow, v r ~  v z . And   u ~  V ,
2
R
Hence,
V


 v   v ~ 
R
2


 v   v
 RV
~
 1 , i.e. inertia force is negligible.

v

Thus, the flow profile and the volumetric flow flux are
v z x ,r  
A
4
R
2
r
2

R A
4
Q 
8
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.
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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.
Reynolds’
theory, 1886
 RU

 1
α

h1
h

U
h2
which is usually satisfied under
practical conditions of
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lubrication
Flow profile:
A
u 
2
y h  y   U
Poiseuille flow
Volumetric
flow flux:
h
Q 
u dy 
0
Ah
h y
h
Couette flow
3
12 

1
Uh
2
From here, the pressure gradient is
p
2Q 
U
  A  6  2  3 
x
h 
h
(1)
Q must be independent of x. In addition, h  h 1   x
Integration of (1) gives
6 
p  p0 
U
 
1
 1
1 
1 
 
  Q  2  2 
h1 
h h1 
h
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Suppose the sliding block is completely immersed in the
fluid, so p=p0 when h=h2, which enables us to determine Q,
Q U
h 1h 2
h1  h 2
and
p  p0 
p  p0  0
if
6 U h 1  h h  h 2 

h 2  h1
p-p0
x
p max 
 LU
h
2
1
h
2
h 1  h 2 
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.
pmax can be high if h1 is small
(L is a layer length)
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The total normal force exerted on either of the two boundaries
by the fluid layer is
L
6 U  h 1
h1  h 2 
  p  p 0 dx   2  ln h  2 h  h 
2
1
2
0
The total tangential force exerted by the fluid on the lower
plate
L
 u 
2 U  h 1  h 2
h1 
    y  dx    3 h  h  2 ln h 
1
2
2
0
y 0
The tangential force on the upper boundary is
L
 u 
2 U  h 1  h 2
h1 
  
 ln
 dx 
3


y

h

h
h

 y h
 1
2
2 
0
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 tangential force on the block 

normal force on the block 
h  h
f 1~ 1
h2  L
This ratio is independent of the viscosity, and can be made
indefinitely small by reduction of h1 with h1/h2 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.
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