Rheology and materials 1

Solid

Material deforms (a little) when a force is applied, then recovers when force is removed. (obeys Hook’s law)

Liquid

Material can keep any shape (of a container) without the application of a force.

Force is only required during the shape change

Hook’s Law

F=kx

Stress

F/A

Strain

Extension/initial length

Youngs Modulus

Stress/Strain

Shear stress

F/A - force is parallel to fluid

Shear strain

the ratio of the horizontal displacement to the height of the block = dx/h

Shear modulus

G=tau/gamma

Yield stress

the stress required to cause a solid to deform plastically.

Viscoelastic Fluids

remember what has been done to them and try to recover, shear rate depends on shear history

Newtonian fluid 

A linear relationship between shear stress and shear strain (constant viscosity)

Thixtropic fluid

(time thinning) – viscosity decreases with time and shearing. Very common.

Rheopectic fluid

(time thickening) – viscosity increases with time (and shear rate).  Less common.

Bingham Fluid

A viscous fluid that possesses a yield strength which must be exceeded before the fluid will flow.

Weissenburg number

Wi=shear rate*relaxation time, tells us about the amount of orientation generated by the deformation

Deborah number

De=relaxation time/observation time, tells us about the rate at which elastic energy is stored/released during the deformation.

Wi>1, De>1

elastic solid

Wi<1, De<1

Liquid

Relaxation time

Period of time when viscoelastic fluids change their molecular conformation, and return to equilibrium state from deformed state.

How can die swell be controlled

decreasing the De number by changing observation time/length

What properties affect rheolgy?

Temperature

composition

pressure

rate of temperature change

Ideal Fluid

Incompressible and will flow continuously when stress is applied

Ideal Solid

has an ideal crystalline structure and is noncompressable and nondeformable.

Why do we need constitutive equations?

To predict material responses:

•How materials will process?

•What forces will be required to convey/mix material?

•How big does our equipment need to be?

•How much energy will be required?

•How much time will it take?

•What will be the properties of the end-product?

Apparent Viscosity

is the shear stress applied to a fluid divided by the shear rate ( ). For a Newtonian fluid, the apparent viscosity is constant, and equal to the Newtonian viscosity of the fluid, but for non-Newtonian fluids, the apparent viscosity depends on the shear rate.

n>1

shear thickening

n<1

shear thinning

Limitations of power law

Most fluids only show power law over a specific range of shear rates - Carreau model can be used to overcome this

Herschel Bulkley model

Combination of Bingham plastic and power law

Constitutive model

a relationship between the forces (or stresses) acting on a material and the deformations and rates of deformation (or strains and strain rates).

Carreau Model Fluid

Newtonian at low shear rates and that display power-law fluid behaviour at high rates.

Fluid Models

Newtonian

Power law

Carreau

Carreau Yasuda

Bingham plastic

Herschel Bulkley

Maxwell (viscoelastic)

Casson

Rheometry

measurement of rheological properties usually done by obtaining relationship between shear stress and shear rate

Transient flows

Flows where the shear strain/rate is continuously increasing (or decreasing) (good for understanding flow)

Oscillatory flows

Flows where the shear strain is changing in a cyclic manner, typically varying sinusoidally (good for understanding structure)

What parameters can be measured to determine shear stress/rate?

Force/torque

velocity/angular velocity

distance

flowrate

geometry of fluid

What’s the difference between rheometer and viscometer?

Generally, rheometers are more
flexible and precise instruments
that allow for more complex fluids

Viscometers are simpler instruments
that often just give a single value
measurement
(suitable for Newtonian fluids)

Types of rheometer

rotational

shearing - parallel plates, cone and plate, couette

capillary

extensional

Shear rate range for rotational shearing rheometers

10e-2 - 10e3 

Shear rate range for capillary rheometers

10e1 - 10e5

Capillary rheometry

Examines flows through tubes/pipes driven by pressure at one end

Examples of capillary flows

fluids in pipes

blood in veins

injection moulding

How do capillary rheometers work?

They generate a pressure in a fluid and measure the mass flowrate

Melt flow indexer

simple capillary rheometer which [places weight on resevoir and measures mass flow in set time

High melt flow index (MFI)

low viscosity - more fluid came through

No slip condition

no slip between fluid and wall so velocity is equal to that of the wall

What changes with wall slip

the relationship between volumetric flowrate and shear rate

Corrected shear rate

4/R(vavg - vslip)

shear thinning flow in pipe

plug flow, poor mixing as without shearing force there is no flow/very high viscosity and shear is only present at walls

why do coatings not drip?

yield stress fluids used, when stress is below yield stress there is no flow. the coating is very thin and is spread easily due to being shear thinning