1. Complex Fluids with Applications to Biology 2011/2012 VIGRE RFG

2. 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

3. 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

4. Granular Flows

5. 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.

6. 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

7. Rheological Properties
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 …

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

9. 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

10. Shear Thickening
shear thickening – tendency of some materials to increase in viscosity when driven to flow at high shear rates

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

12. Elastic and Viscoelastic Effects
Weissenberg Effect (Rod Climbing Effect)
does not flow outward when stirred at high speeds

13. Elastic and Viscoelastic Effects
Fluid Memory
Conserve shape over time periods or seconds or minutes
Elastic like rubber
Can bounce or partially retract
Example: clay (plasticina)

14. Elastic and Viscoelastic Effects
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..

15. 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

16. Mixing in micro channels
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
Wi – 13,

17. Basic continuum and multi-scale models
Conservation of mass
Conservation of momentum
Cauchy stress tensor :
Deformation
Deformation gradient

18. Basic continuum and multi-scale models
Viscoelastic Fluid – dilute solution of polymer chains in a Newtonian solvent
spring
bead
End to end vector
Polymer moves via Brownian motion in fluid
Smoluchowski equation gives evolution of probability density in phase space
Force on ith bead:

19. Stress:
Solvent Stress Polymer Stress
Assume linear Hooke’s law for bead forces
Polymer stress:
Incompressible fluid
Evolution of polymer stress
Relaxation time
Thermodynamic constant
Upper convected
derivative

20. Oldroyd-B equations
Scale of nonlinear terms to relaxation term is given by the dimensionless parameter
Weissenberg number

21. 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