Rheology
Strain and Strain (Shear) Rate
• Strain
– a dimensionless quantity representing the
relative deformation of a material
Normal Strain
Shear Strain
Shear Stress is the intensity of force per unit area, acting tang
Simple Shear Flow
Fluid Viscosity
• Newtonian fluids
– viscosity is constant (Newtonian viscosity, )
  
• Non-Newtonian fluids
– shear-dependent viscosity (apparent
viscosity, )

  f ( ) 

Viscosity: Introduction
The viscosity is measure of the “fluidity” of the
fluid which is not captured simply by density or
specific weight. A fluid can not resist a shear
and under shear begins to flow. The shearing
stress and shearing strain can be related with a
relationship of the following form for common
fluids such as water, air, oil, and gasoline:
Viscosity: Introduction
du
 
dy
 is the absolute viscosity or dynamics viscosity of the
fluid, u is the velocity of the fluid and y is the vertical
coordinate as shown in the schematic below:
Viscosity
Viscosity is a property of fluids that indicates resistance to flow. When a force is
applied to a volume of material then a displacement (deformation) occurs. If two
plates (area, A), separated by fluid distance apart, are moved (at velocity V by a
force, F) relative to each other,
dv
  
dy
y=0
  
y = y-max
dv
dy

= Coefficient Viscosity ( Pa s)
Newton's law states that the shear stress (the force divided by area parallel to the force,
F/A) is proportional to the shear strain rate . The proportionality constant is
known as the (dynamic) viscosity
Shear stress
dv
  
dy
The unit of viscosity in the SI system of units
is pascal-second (Pa s)

In cgs unit , the unit of viscosity is
expressed as poise
1 poise = 0.1 Pa s
Shear rate
1 cP = 1 m Pa s
Newtonian fluid
  
 r (P2  P1 )  (2 rL)
2
P x  r 2
v(r ) 
  (2 rL)
P R 2
4 L
  r 2 
1    
  R  
dv
dy
2
( r ) Pr

 = P
2 rL
2L
4
=
PR
0
8 LV
Newtonian
From
and
(r 2 )
Pr
du
dv
P=


or 
2rL
2L
dy
dr
PR4
8LV
=
P =

8LV
R 4
  = 8v/D
Non-Newtonian Fluids
Flow Characteristic of Non-Newtonian Fluid
• Fluids in which shear stress is not
directly proportional to deformation rate
are non-Newtonian flow: toothpaste and
Lucite paint
(Casson Plastic)
(Bingham Plastic)
Viscosity changes with shear rate. Apparent viscosity (a
or ) is always defined by the relationship between shear
stress and shear rate.
Model Fitting - Shear Stress
Shear Rate
vs.
Summary of Viscosity Models
   

Newtonian
Pseudoplastic


K 
Dilatant


n

K
Bingham



Casson

Herschel-Bulkley

1
2


( n < 1)


+
y
0

n
1
2
( n > 1)
 

+

n

 1
1
2
2
c

y + K 
n
 or = shear stress, º = shear rate, a or  = apparent viscosity
m or K or K'= consistency index, n or n'= flow behavior index
Herschel-Bulkley model
(Herschel and Bulkley , 1926)
n
 du 
  m  +  0
 dy 
Values of coefficients in Herschel-Bulkley fluid model
Fluid
m
n
0
Herschel-Bulkley
>0
0<n<
>0
Minced fish paste, raisin paste
Newtonian
>0
1
0
Water,fruit juice, honey, milk,
vegetable oil
Shear-thinning
(pseudoplastic)
>0
0<n<1
0
Applesauce, banana puree, orange
juice concentrate
Shear-thickening
>0
1<n<
0
Some types of honey, 40 % raw
corn starch solution
Bingham Plastic
>0
1
>0
Toothpaste, tomato paste
Typical examples
Non-Newtonian Fluid Behaviour
The flow curve (shear stress vs. shear rate) is either
non-linear, or does pass through the origin, or both.
Three classes can be distinguished.
(1) time-independent fluids
(2) time-dependent fluids
(3) visco-elastic fluids
Time-Independent Fluid Behaviour
Fluids for which the rate of shear at any point is
determined only by the value of the shear stress at that
point at that instant; these fluids are variously known
as “time independent”, “purely viscous”, “inelastic”, or
“Generalised Newtonian Fluids” (GNF).
1. Shear thinning or pseudoplastic fluids
2. Viscoplastic Fluid Behaviour
3. Shear-thickening or Dilatant Fluid Behaviour
1. Shear thinning or pseudoplastic fluids
Viscosity decrease with shear stress. Over a limited range
of shear-rate (or stress) log (t) vs. log (g) is approximately a
straight line of negative slope. Hence
 yx = m(yx)n
where
(*)
m = fluid consistency coefficient
n = flow behaviour index
Eq. (*) is applicable with 0<n<1
Re-arrange Eq. (*) to obtain an expression for apparent
viscosity
a (= yx/yx)
Pseudoplastics
Flow of pseudoplastics is consistent with
the random coil model of polymer solutions
and melts. At low stress, flow occurs by
random coils moving past each other
without coil deformation. At moderate
stress, the coils are deformed and slip past
each other more easily. At high stress, the
coils are distorted as much as possible and
offer low resistance to flow. Entanglements
between chains and the reptation model also
are consistent with the observed viscosity
changes.
Why Shear Thinning occurs
Unsheared
Sheared
Anisotropic Particles align
with the Flow Streamlines
Random coil
Polymers
elongate and
break
Aggregates
break up
Courtesy: TA Instruments
Shear Thinning Behavior
 Shear thinning behavior is often a result of:
 Orientation of non-spherical particles in the direction of
flow. An example of this phenomenon is the pumping of
fiber slurries.
 Orientation of polymer chains in the direction of flow and
breaking of polymer chains during flow. An example is
polymer melt extrusion
 Deformation of spherical droplets to elliptical droplets in
an emulsion. An industrial application where this
phenomenon can occur is in the production of low fat
margarine.
 Breaking of particle aggregates in suspensions. An
example would be stirring paint.
Courtesy: TA Instruments
2. Viscoplastic Fluid Behaviour
Viscoplastic fluids behave as if they have a yield stress (0).
Until  0is exceeded they do not appear to flow. A Bingham
plastic fluid has a constant plastic viscosity
 yx   0B + B yx
 yx  0
for
 yx >  0B
for
 yx <  0B
Often the two model parameters 0B and B are treated as curve
fitting constants, even when there is no true yield stress.
3. Shear-thickening
or Dilatant Fluid Behaviour
yx = m(yx)n (*)
where m = fluid consistency coefficient
n = flow behaviour index
Eq. (*) is applicable with n>1.
Viscosity increases with shear stress.
Dilatant: shear thickening fluids that contain suspended
solids. Solids can become close packed under shear.
Time-dependent Fluid Behaviour
More complex fluids for which the relation between shear
stress and shear rate depends, in addition, on the duration
of shearing and their kinematic history; they are called
“time-dependent fluids”.
The response time of the material may be longer than
response time of the measurement system, so the
viscosity will change with time. Apparent viscosity
depends not only on the rate of shear but on the “time for
which fluid has been subject to shearing”.
Thixotropic : Material structure breaks down as shearing action
continues : e.g. gelatin, cream, shortening, salad dressing.
Rheopectic : Structure build up as shearing continues (not common
in food : e.g. highly concentrated starch solution over long periods
of time
Shear stress
Thixotropic
Rheopectic
Shear rate
Time independent
Time dependent
_
+
A
B
C
D
_
+
E
F
G
Non - newtonian
Rheological curves of Time - Independent and Time – Dependent Liquids
Visco-elastic Fluid Behaviour
Substances exhibiting characteristics of both ideal fluids
and elastic solids and showing partial elastic recovery,
after deformation; these are characterised as “viscoelastic” fluids.
A true visco-elastic fluid gives time dependent
behaviour.
Pseoudoplastic
Dilatant
Shear stress
Shear stress
Newtonian
Shear rate
Shear rate
Shear rate
Viscosity
Viscosity
Viscosity
Shear rate
Shear rate
Common flow behaviours
Shear rate
Examples
Newtonian flow occurs for simple fluids, such as water, petrol, and
vegetable oil.
The Non-Newtonian flow behaviour of many microstructured
products can offer real advantages. For example, paint should be
easy to spread, so it should have a low apparent viscosity at the
high shear caused by the paintbrush. At the same time, the paint
should stick to the wall after its brushed on, so it should have a high
apparent viscosity after it is applied. Many cleaning fluids and
furniture waxes should have similar properties.
Examples
The causes of Non-Newtonian flow depend on the colloid
chemistry of the particular product. In the case of water-based
latex paint, the shear-thinning is the result of the breakage of
hydrogen bonds between the surfactants used to stabilise the
latex. For many cleaners, the shear thinning behaviour results
from disruptions of liquid crystals formed within the products. It
is the forces produced by these chemistries that are responsible
for the unusual and attractive properties of these microstructured
products.
Measurement of Viscosity
Viscosity of a liquid can be measurement
Capillary Tube Viscosity
Rotational Viscometer
Viscosity: Measurements
A Capillary Tube Viscosimeter is one method of measuring
the viscosity of the fluid.
Viscosity Varies from Fluid to Fluid and is dependent on
temperature, thus temperature is measured as well.
Units of Viscosity are N·s/m2 or lb·s/ft2
Capillary Tube Viscometer
Newtonian
 r (P2  P1 )  (2 rL)
2
P x  r 2
v(r ) 
  (2 rL)
P R
4 L
2
  
dv
dy
 r
1   
  R 
2



2
( r ) Pr

 = P
2 rL
2L
=
PR
0
8 LV
4

PR
4
0
8 LV
m gt

2
8L
2
gR t

8L
  t
Capillary tube viscometer
• Reference sample with known density and
time will be used (at specified temperature).
1 1t1

 2  2t 2
• Time and density of sample will be evaluated
and then the viscosity can be determined.
Falling sphere method
• This method can be used for
assumption that falling is in Stoke
region.
• When the sphere is fell through the
fluid, distance and time of falling will
be measured (velocity).
Falling sphere method
D (s   f ) g
2
for v 
18
1 ( s   f 1 )t1

2 ( s   f 2 )t2
Example
• In a capilary flow experiment using buret,
30 ml of water was emptied in 10 s. A
40% sugar solution (specific gravity
1.179) having similar volume took 52.8 s
for emptying. What was the viscosity of
sugar solution?
Viscometers
In order to get meaningful (universal) values for the
viscosity, we need to use geometries that give the
viscosity as a scalar invariant of the shear stress or
shear rate. Generalized Newtonian models are good for
these steady flows: tubular, axial annular, tangential
annular, helical annular, parallel plates, rotating disks and
cone-and-plate flows. Capillary, Couette and cone-andplate viscometers are common.
Rotational Viscometer
Non-newtonian fluid
• from

= 2r2L

=
dω
r
dr
n
Ω  - K  rdω 
 dr 
2
2Lr


Integrate from r = Ro Ri and  = 0i
Non-newtonian fluid
 or a
º
• obtain


 4N 
- K

n


 4N 
K

 n 
n
 4N 
 -K 

n


n 1
n 1 4N 

 n
K
ln   ln n -1 + (n -1) ln 4N 
n
Linear : y = y-intercept + slope (x)


K and n
4
y = -0.7466x + 3.053
3.5
2
R = 0.9985
3
n = 0.25
ln Ua
2.5
2
1.5
K = 59.02
1
0.5
0
-1
-0.5
0
0.5
ln 4TTN
1
1.5
2
(shear thinning)
Example
• The shear rate for a power law fluid without
yield stress when using a single spindle
viscometer is given by (4N/n). Determine
the flow behavior index and consistency
coefficient based on the following data:
Apparent
viscosity (Pa.s)
5.0
RPM
1.5
60
6