### Introduction to Rheology

```Complex Fluids with
Applications to Biology
2011/2012 VIGRE RFG
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
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Foods
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Examples of Complex Fluids
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)
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Pharmaceuticals
– Gels (creams, particle precursors)
– Emulsions (creams)
– Aerosols (nasal sprays)
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Polymers
Granular Flows
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
Rheological Properties
• Stress
– Shear stress
– Normal stress
– Normal Stress differences
• Viscosity
– Extensional
– Complex
• Viscoelastic Modulus
– G’ – storage modulus
– G” – loss modulus
• Creep, Compliance, Decay
• Relaxation times
• and many more …
Common Non-Newtonian Behavior
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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
• shear thickening – tendency of some materials to increase
in viscosity when driven to flow at high shear rates
Yield Stress
• Tendency of a material to flow only when stresses are above a
threshold stress
• Eg. Ketchup or Mustard
Elastic and Viscoelastic Effects
• Weissenberg Effect (Rod Climbing Effect)
– does not flow outward when stirred at high speeds
Elastic and Viscoelastic Effects
• Fluid Memory
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Conserve shape over time periods or seconds or minutes
Elastic like rubber
Can bounce or partially retract
Example: clay (plasticina)
Elastic and Viscoelastic Effects
• Viscoelastic fluids subjected to a stress deform
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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 :  (t , x)   ( F (t , x), u(t , x))
x(t , X )
Deformation
x
F (t , x(t , X )) 
X
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
Smoluchowski equation gives evolution of
probability density in phase space
¶ty + ÑR ×(y R) = 0
¶
mi si = -xi (si - ui ) - kT
lny + Fi
¶si
Stress:
  s  p
Solvent Stress
Polymer stress:
Polymer Stress
 P  FR  C RR
 s   pI+2s E
Assume linear
Hooke’s law for
Incompressible fluid
 p  ( p  G I)=0
Relaxation time
 
T
Evolution of polymer stress
Thermodynamic constant
   t  u   (u  uT )
Upper convected
derivative
Oldroyd-B equations
 ( t u  u u )  p  s u     p  f
 p  ( p  G I)  0
 u  0
Scale of nonlinear terms to relaxation term is given by the
dimensionless parameter
 (U / L)
Weissenberg number
Complex Fluids, an overview
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
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