2010 fundamentals

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Rheology Fundamentals

Lecture Notes 2



Continuum Approach

• What is a continuum?

• What is continuum approach?

• Eulerian method of description• Lagrangian method of description



Stress

• Stress=force/unit area

• Tensile

• Compressive• Shear



Stress

•Body and surface forces

• Simple and isolated stresses

• Tensile

• Compression• Shear

• Stress vector

Shear and normal components

Fluid element

The stress vector at thepoint on the surface isgiven by

0

limS

f

S

f

S

y

x

z

n̂



Stress Tensor

• Stress tensor Numbers of unknowns

Sign convention

•Stress in static fluidsHydrostatic pressure

• Stress in dynamic fluidsPressure and viscous stresses

• Invariant of stress tensor

• Principal stresses

ˆ (3.58) f dSn

Total stress tensor

(3.83) pI

Extra stress tensor or

Viscous stress tensor or

Dissipative stress tensor



Viscous Stress Tensor for a

Newtonian Incompressible Fluid

( ) jiij

j i

vv

x x

Viscosity



Deformation

When you apply an external force to abody, you either cause a rigid bodymovement and/or you cause a change in

shape. In rheology, we are interested inthe description of the change in shape.

Change in shape is the change of distances between points within acontinuum and is a geometrical concept.



Strain

• Strain is the absolute amount of distortion

which occurs, expressed as a fraction of

an original dimension of the unstressed

sample. Strain is dimensionless.

• Tensile strain

• Compressive strain

• Shear strain

• Rate of strain



Relation Between Force and

Deformation

• Relation for ideal solids

• Relation for ideal fluids

• Relation for viscoelastic materials

• Constitutive equations or rheological

equations of state



Important Deformations in

Rheology

• Shear

•Elongation



Ideal Solids

Hook’s Law 1676

http://www.upscale.utoronto.ca/GeneralInterest/Harrison/Flash/ClassMe

chanics/HookesLaw/HookesLaw.html

http://www.upscale.utoronto.ca/GeneralInterest/Harrison/Flash/ClassMechanics/HookesLaw/HookesLaw.html




http://upload.wikimedia.org/wikipedia/commons/2/2a/Ressort_de_compression.jpg



Ideal Liquids

Newton’s Law The resistance of which arises from

the lack of slipperiness originating

In a fluid, other things being equal, is

proportional to the velocity by which

the parts o f the fluid are being separated

from each other. Isaac N. Newton (1687)From Rheology Principles, Measurements, and Applications

By C. W. Macosko, 1994, VCH Publishers, page 65

http://upload.wikimedia.org/wikipedia/en/a/a4/Laminar_shear.png



Visco-elastic Behavior

• Elasticoviscous fluids

• Viscoelastic solids

• Creep• Relaxation

• Material modelsLudwig Boltzmann



Voigt (Kelvin-Voigt) Model forViscoelastic Behavior

SpringElastic Modulus

Force)

(Position)

DashpotViscosity

Force (Speed)

Woldemar Voigt(1850-1919)

Lord Kelvin(1824-1907)

http://en.wikipedia.org/wiki/Image:Lord_Kelvin_photograph.jpg



http://en.wikipedia.org/wiki/Image:Kelvin_Voigt_diagram.svg



Maxwell Model for

Viscoelastic Behavior

James Clerk Maxwell1831-1879

http://en.wikipedia.org/wiki/Image:Maxwell_diagram.svg

http://en.wikipedia.org/wiki/Image:James_Clerk_Maxwell.png



Non-Newtonian Behavior

• Bingham body

• Yield stress

• Psudoplastic behavior

• Dilatancy• St. Venant body

• Apparent viscosity

• The power law or Ostwald model (Ostwald de Walle)

• Time-dependent behavior • Thixotropic

• Rheotropic or anti-thixotropic



References

• Chapter 1 of your text book

• Rheology Study of Flow by George Smoluk, SPEJournal, Vol 27, 1971.

• Chapter 5 of Introduction to Fluid Mechanics by Fox,McDonald, and Pritchard, 6th edition, Wiley

• Rheology Study of Flow, G.R. Smoluk, SPE Journal, Vol.

27, December 1971, pages 19-30

• http://www.rheologyschool.com/rheology_glossary.html

http://www.rheologyschool.com/rheology_glossary.html

http://www.rheologyschool.com/rheology_glossary.html

2010 fundamentals

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