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

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

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 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 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 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 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

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,

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

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:

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

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