Rheology Chapter

1.  Definition::
Rheology is the science of flow and deformation of matter.
 There are basically two types of fluids, defined by the relationship
between shear stress and shear rate.
These are:
 Newtonian
 Non-Newtonian

2.  Newtonian and Non-Newtonian Fluid
If the gradient m is constant, the fluid is termed as Newtonian fluid. Otherwise, it is known as non-
Newtonian fluid. Fig. 1.5 shows several Newtonian and non-Newtonian fluids.
Newtonian and Non-Newtonian Fluids (habib)
A Newtonian fluid is a fluid whose viscosity remains constant when the shear rate changes. The
viscosity will still change with a change in temperature or pressure, but a pushing it faster, the viscosity
stays constant.
Example:
Water and simple liquids such as ethyl alcohol; air and simple … are Newtonian fluids.
A non-Newtonian fluid is a fluid whose viscosity changes (drops) when the shear rate changes i.e.
increase.
Exmple :
Fluids in food industry, gels, polymers, slurries, drilling muds, blood
… are Non-Newtonian fluid.

3. Figure: Classification of fluids with shear stress as a function of shear rate.
Examples of Viscoelastic Fluid
 Paint
 Crude oil
 Asphalt
 Cosmetics
 Biological fluids
Blood
Protein solutions
 Pulp and coal slurries
 Toothpaste
 Grease
Foodstuffs
Ketchup
Dough
Salad dressing
 Plastics
Polymer melts
Rubbers &Polymer solutions
Classification of fluids
 Time Independent Fluids (the relation between shearing stress and rate is unique but
non-linear)
Bingham plastics
Pseudoplastic fluids
Dilatant plastics

4.  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)
Time-Independent Fluids
Bingham plastics :
depends on a critical/yield shear stress ( ) and then becomes constant.
Ex. sludge
paint
blood
Time-Independent Fluids
Bingham plastics :
Time-Independent Fluids
Power law fluids
Pseudoplastic fluids : decreases as the shear rate increases (shear rate thinning)
Ex. polymer melts
paper pulp in water
clay solutions
molasses
whipped cream

5.  Definition of Newtonian Fluid:
Newtonian fluid is any fluid that exhibits a viscosity that remains constant regardless of any external
stress that is placed upon it, such as mixing or a sudden application of force.
 Rheological Properties
 Stress
Shear stress
Normal stress
Normal Stress differences
 Viscosity
Steady-state (i.e. shear) ( Most commonly sought rheological quantity).
Extensional
Complex
 Viscoelastic Modulus
G’ – storage modulus
G” – loss modulus
 Creep, Compliance, Decay
 Relaxation times
 Tensile or Compressive Stress - Normal Stress
 Three kinds of differential stress occur::
 Tensional stress (or extensional stress), which stretches rock.
 Compressional stress, which squeezes rock; and
 Shear stress, which result in slippage and translation.

6. Robert Hooke
Robert Hooke (28 July 1635-3 March 1703) was an English natural philosopher. Robert
Hooke was born in 1635 in Freshwater on the Isle of Wight to John Hooke and Cecily Gyles.
Dashpot

7.  Dashpot: A model for Newtonian fluids consisting of a piston and
cylinder containing a viscous liquid.
Normal and Shear Stress ::
traction’s on the planes that intersect at the origin of figure can be subdivided into perpendicular and
parallel components to each plane. The component perpendicular to each plane is termed normal stress
( ) and the component parallel to each plane is termed shear stress ( ). Figure illustrates the
relationship between the traction (s) and the normal ( ) and shear stress ( ) components acting on a
single plane whose trace in two dimensions the line segment AB.

8. Shear Stress::
Stress parallel to the plane is usually denoted ‘shear stress’ and can be expressed as
τ = Fp /A, where
τ = shear stress ((Pa) N/m2
, psi)
Fp = parallel component force (N, lbf)
A = area (m2
, in2
)

9.  Ideal (elastic) Solid
Hooks Law
Response is independent of time and
the deformation is dependent on the spring constant.

10. Maxwell Model
James Clerk Maxwell (1831–1879)Edinburgh, Scotland
Died 5 November 1879
(1879-11-05) (aged 48)
In series connection of Hook’s Spring and Dash pot

11.  Maxwell Model
Figure: Stress- time plot for stress relaxation in the Maxwell model
 Maxwell model
 In series
 Viscous strain remains after load removal.

12. The rate of strain d/dt is equal to zero under
conditions of constant stress (s)
Then
Thus , according to above equation for the Maxwell model or
element, under conditions of constant strain, the stress will decrease
exponentially with time and at the relaxation time t=, s will be equal to
1/e=1/2.7 or 0.37 of its original values (so)
 Lord Kelvin
26 June 1824
17 December 1907
William Thomson is popularly known as 1st Baron Kelvin

13. He was the first British scientist to be elevated to the House of Lords. The title refers to the
River Kelvin, which flows close by his laboratory at the University of Glassgow. His home was
the imposing red sandstone mansion Netherhall, in Largs.
The Kelvin-Voigt model, also called the Voigt model, can be represented by a purely viscous
damper and purely elastic spring connected in parallel as shown in the picture.
A Kelvin–Voigt material, also called a Voigt material, is a viscoelastic material having the
properties both of elasticity and viscosity. It is named after the British physicist and engineer
Lord Kelvin and after German physicist Woldemar Voigt
 The spring and dash pot are parallel in the Voigt-Kelvin
model

14. If G is much larger than , the retardation time (/G) or  is small, and  is large if
 is much larger than G
While polymers melts and elastomers flow readily when stress is applied,
structural plastics must resist irreversible deformation and behave as elastic solids
when relatively small stresses are applied. These plastics are called ideal or
Bingham plastics as described
 Kelvin or Voigt model
 In parallel
 Nonlinear increase in strain with time
 Strain decreases with time after load removal because of the action
of the spring (and dashpot).

17. Descriptive term for a liquid having both viscous and elastic properties.
A viscoelastic liquid will deform and flow under the influence of an applied shear
stress, but when the stress is removed the liquid will slowly recover from some of
the deformation.
Viscoelastic fluids have molecules in which the load-deformation relationship is
time dependant.
The Newtonian model has no value in predicting the behaviour of a drilling fluid,
as the majority of drilling fluids do not conform to the govering Newtonian fluids.

18.  The Bingham Plastic model establishes a distinct relationship between
shear stress, yield point, plastic viscosity and shear rate.
 Shear stress:
The force required to overcome a fluid’s resistance to flow, divided by the area
that force is acting upon.
 Shear rate:
The relative velocity of the fluid layers divided by their normal separation
distance.

19.  Viscosity is the resistance a material has to change in form. This property
can be thought of as an internal friction.