RheologyRheology

Strain and Strain (Shear) RateStrain and Strain (Shear) Rate

• Strain– a dimensionless quantity representing the

relative deformation of a material

Normal Strain Shear Strain

• 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 FlowSimple Shear Flow

Fluid ViscosityFluid Viscosity• Newtonian fluids

– viscosity is constant (Newtonian viscosity, )

• Non-Newtonian fluids– shear-dependent viscosity (apparent

viscosity, )

• Newtonian fluids– viscosity is constant (Newtonian viscosity, )

• Non-Newtonian fluids– shear-dependent viscosity (apparent

viscosity, )

)(f

Viscosity: IntroductionViscosity: 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: IntroductionViscosity: Introduction

dy

du

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:

ViscosityViscosity ViscosityViscosity

dy

dv y = y-max

y = 0

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,

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

dy

dv

= Coefficient Viscosity ( Pa s)

Shear rate

She

ar s

tres

s

dy

dv

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

1 cP = 1 m Pa s

dy

dv

Newtonian fluidNewtonian fluid

22

14

)(R

r

L

RPrv

)2()( 122 rLPPr

)2(2 rLrxP = 2L

Pr

rL2)r(

P2

= 0

4

LV8

PR

NewtonianNewtonian

From

and

From

and

= r d

vd

y d

u d

2L

Pr

2

)( 2

orrL

rP

= VL

PR8

4 P = 4

8

R

VL

= 8v/D

Non-Newtonian FluidsNon-Newtonian Fluids

Flow Characteristic of Non-Newtonian FluidFlow 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

• Fluids in which shear stress is not directly proportional to deformation rate are non-

Newtonian flow: toothpaste and Lucite paint

(Bingham Plastic)

(Casson 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 vs. Shear Rate

K

K

K

n

n

y

n

c

y

n

n

n

( )

( )

1

1

12

0

Newtonian

Pseudoplastic

Dilatant

Bingham

Casson

Herschel-Bulkley

Summary of Viscosity Models

12

12

12

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)

Herschel-Bulkley model (Herschel and Bulkley , 1926)

0dy

dum

n

Values of coefficients in Herschel-Bulkley fluid model

Fluid m n 0 Typical examples

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

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 fluids2. Viscoplastic Fluid Behaviour3. 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<1Re-arrange Eq. (*) to obtain an expression for apparent viscosity a (=yx/yx)

PseudoplasticsPseudoplasticsFlow 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 occursWhy Shear Thinning occurs

Unsheared Sheared

Aggregatesbreak up

Random coilPolymers elongate and break

Anisotropic Particles alignwith the Flow Streamlines

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

yxBB

yx 0 forB

yx 0

0yx forB

yx 0

Often the two model parameters 0B and are treated as curve

fitting constants, even when there is no true yield stress.

2. Viscoplastic Fluid Behaviour

Viscoplastic fluids behave as if they have a yield stress (0). Until is exceeded they do not appear to flow. A Bingham plastic fluid has a constant plastic viscosity

3. Shear-thickening or Dilatant Fluid Behaviour

yx = m(yx)n (*)

where m = fluid consistency coefficientn = 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

Thixotropic

Rheopectic

Shear stress

Shear rate

Non - newtonian

Time independent Time dependent

A EC D F G B

_ _

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 “visco-elastic” fluids.

A true visco-elastic fluid gives time dependent behaviour.

Common flow behaviours

Newtonian Pseoudoplastic DilatantS

hea

r st

ress

Sh

ear

stre

ss

Sh

ear

stre

ss

Shear rate Shear rate Shear rate V

isco

sity

Vis

cosi

ty

Vis

cosi

ty

Shear rate Shear rate Shear rate

Newtonian flow occurs for simple fluids, such as water, petrol, andvegetable 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.

ExamplesExamples

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.

Examples

Measurement of ViscosityMeasurement of Viscosity

Capillary Tube ViscosityRotational Viscometer

Viscosity of a liquid can be measurement

Viscosity: MeasurementsViscosity: 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 ViscometerCapillary Tube Viscometer

NewtonianNewtonian

dydv

22

14

)(R

r

L

RPrv

)2()( 122 rLPPr

)2(2 rLrxP = 2L

Pr

rL2)r(

P2

= 0

4

LV8

PR

L

tgR

L

mgtLV

PR

8

8

8

2

2

0

4

t

Capillary tube viscometerCapillary tube viscometer

• Reference sample with known density and time will be used (at specified temperature).

• Time and density of sample will be evaluated and then the viscosity can be determined.

• Reference sample with known density and time will be used (at specified temperature).

• Time and density of sample will be evaluated and then the viscosity can be determined.

22

11

2

1

t

t

Falling sphere methodFalling 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).

• 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 methodFalling sphere method

22

11

2

1

)(

)(

t

t

fs

fs

18

)(2 gDvfor fs

ExampleExample

• 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?

• 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?

ViscometersViscometers

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-and-plate

viscometers are common.

Rotational ViscometerRotational Viscometer

Non-newtonian fluidNon-newtonian fluid

• from• from

drdωr

n

drrd- 2Lr2

ωKΩ

= 2r2L =

Integrate from r = Ro Ri and = 0i

Non-newtonian fluidNon-newtonian fluid

• obtain• obtain

nn

NnNnN 414K4K -n

-

nn

NnNnN 414K4K -n

-

º

or a

14K

nNn

14K

nNn

N 4ln 1)-(n 1-nnKln ln

Linear : y = y-intercept + slope (x)

K and n

y = -0.7466x + 3.053

R2 = 0.9985

0

0.5

1

1.5

2

2.5

3

3.5

4

-1 -0.5 0 0.5 1 1.5 2

ln 4TTN

ln U

a n = 0.25

K = 5902.

(shear thinning)

ExampleExample

• The shear rate for a power law fluid without yield stress when using a single spindle viscometer is given by (4N/n). Determine the flow behavior index and consistency coefficient based on the following data:

• The shear rate for a power law fluid without yield stress when using a single spindle viscometer is given by (4N/n). Determine the flow behavior index and consistency coefficient based on the following data:

Apparent viscosity (Pa.s)

RPM

5.0 6

1.5 60