Rheology
•
Study of deformation and flow of matter
•
Classical fluids quickly shape themselves into a container and classical solids maintain their shape indefinitely
– Intuitively, a fluid flows, and a solid does not!
–
Newtonian fluids have constant viscosity
–
Stress depends linearly on the rate of strain
•
Complex fluids may maintain their shape for some time, but eventually flow
–
Viscosity depends on applied strain
– Stress is nonlinear function of rate of strain
–
Properties may include
•
Shear thinning / thickening (e.g. paint / cornstarch in water)
•
Normal stresses – (leads to rod climbing for example)
• “Elastic turbulence” - low Reynolds number flows

Examples of Complex Fluids
•
Foods
–
Emulsions (mayonnaise, ice cream)
–
Foams (ice cream, whipped cream)
– Suspensions (mustard, chocolate)
–
Gels (cheese)
•
Biofluids
–
Suspension (blood)
– Gel (mucin)
–
Solutions (spittle)
•
Personal Care Products
–
Suspensions (nail polish, face scrubs)
– Solutions/Gels (shampoos, conditioners)
–
Foams (shaving cream)
•
Electronic and Optical Materials
–
Liquid Crystals (Monitor displays)
– Melts (soldering paste)
•
Pharmaceuticals
–
Gels (creams, particle precursors)
–
Emulsions (creams)
– Aerosols (nasal sprays)
•
Polymers

A goal of Rheology
Establishing the relationship between applied forces and geometrical effects induced by these forces at a point (in a fluid).
–
The mathematical form of this relationship is called the rheological equation of state, or the constitutive equation.
–
The constitutive equations are used to solve macroscopic problems related to continuum mechanics of these materials.
–
Equations attempt to model physical reality.

Different theories are appropriate for different problems
•
Continuum theories
–
Cornerstone of traditional fluid mechanics
•
Material is treated as a continuum, consider objects such as velocity, acceleration, stress at a point
•
So-called constitutive models give continuum description of stress
•
Stress may have many degrees of freedom depending on material composition
•
Limitations in model
•
Useful for straightforward solutions (relatively speaking – numerical, analytical…)
•
Multi-scale
–
Can be more flexible
•
Material may have small scale fluctuations which can be modeled directly
•
Need to communicate between levels
•
Computationally challenging

•
Stress
–
Shear stress
–
Normal stress
–
Normal Stress differences
•
Viscosity
–
Steady-state (i.e. shear)
–
Extensional
–
Complex
•
Viscoelastic Modulus
–
G
’ – storage modulus
–
G
” – loss modulus
•
Creep, Compliance, Decay
•
Relaxation times
• and many more …

Common Non-Newtonian Behavior
• shear thinning
• shear thickening
• yield stress
• viscoelastic effects
–
Weissenberg effect
–
Fluid memory
–
Die Swell

Shear Thinning and Shear Thickening
• shear thinning – tendency of some materials to decrease in viscosity when driven to flow at high shear rates , such as by higher pressure drops
Increasing shear rate

• shear thickening
– tendency of some materials to increase in viscosity when driven to flow at high shear rates

•
Tendency of a material to flow only when stresses are above a threshold stress
•
Eg. Ketchup or Mustard

•
Weissenberg Effect (Rod Climbing Effect)
– does not flow outward when stirred at high speeds

•
Fluid Memory
–
Conserve shape over time periods or seconds or minutes
–
Elastic like rubber
–
Can bounce or partially retract
–
Example: clay (plasticina)

•
Viscoelastic fluids subjected to a stress deform
– when the stress is removed, it does not instantly vanish
– internal structure of material can sustain stress for some time
– this time is known as the relaxation time, varies with materials
– due to the internal stress, the fluid will deform on its own, even when external stresses are removed
– important for processing of polymer melts, casting, etc..

Elastic and Viscoelastic Effects
Die Swell
– as a polymer exits a die, the diameter of liquid stream increases by up to an order of magnitude
– caused by relaxation of extended polymer coils, as stress is reduced from high flow producing stresses present within the die to low stresses, associated with the extruded stream moving through ambient air

Viscoelastic fluid – Elastic “turbulence” - Efficient mixing
(Low Re, “High” Wi) Groisman & Steinberg
Rotating plates
Mixing in micro channels
Arratia and Gollub et al., PRL 2006
Elastic fluid instabilities near hyperbolic points

Basic continuum and multi-scale models
•
Conservation of mass
   t

(
 u )

0
•
Conservation of momentum

( t u u u )
  g
•
Cauchy stress tensor : 
( , )
 
F t x
 u t x
( , )
Deformation
( , ( , ))

 x

X
Deformation gradient

•
Basic continuum and multi-scale models
Viscoelastic Fluid – dilute solution of polymer chains in a Newtonian solvent spring
End to end vector
R
Polymer moves via Brownian motion in fluid bead Smoluchowski equation gives evolution of probability density in phase space
¶ t y + Ñ
R
×
( y
R )
=
0 m i
Force on ith bead: s i
= x i
( s i
u i
)
kT
¶
¶ s i ln y +
F i

Stress:
  s
  p
Solvent Stress Polymer Stress
Polymer stress:

P
 
F R

C RR
T
Assume linear
Hooke ’ s law for bead forces
 s
  p I+2
 s
E
Incompressible fluid
  p

(
 p

G I)=0
Relaxation time
Evolution of polymer stress
Thermodynamic constant
  t
 u

( u
  u
T
)
Upper convected derivative


( t u u u ) p
 s u
 p

  p

(
 p

G I)

0 f
0
Scale of nonlinear terms to relaxation term is given by the dimensionless parameter

( / )
Weissenberg number

Some references:
• Dynamics of Polymeric Liquids, Vol. I and II , Bird, Armstrong,
Hassager, Wiley, 1987
• The Structure and Rheology of Complex Fluids , R. Larson,
Oxford U. Press, 1999
• Computational Rheology , R. G. Owens and T. N. Phillips,
Imperial College of London Press, 2002
• Mathematical Problems in Viscoelasticity , M. Renardy, W.
Hrusa, J. Nohel, Pitman Monographs and Surveys in Pure and
Applied Mathematics 35, Longman 1987
• An Introduction to Continuum Mechanics, M. E. Gurtin, volume
158 of Mathematics of Science and Engineering, Academic
Press, 1981