Nonlinear Dynamics of Fluid Motion

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003

“Nonlinear Dynamics” has deepened our understanding of fluid dynamics, and taught us many lessons about nonlinear phenomena.

No movies in this version; some may be found at www.haverford.edu/physics-astro/Gollub/lab.html

Thanks to Collaborators

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003

Postdocs and graduate students: G. Voth, JC Tsai, W. Losert, A. Kudrolli, D. Vallette, B. Gluckman, J. Liu, O. Mesquita, T. Solomon, F. Simonelli, A. Dougherty, M. Heutmaker, M. Lowe, S. Ciliberto.Undergraduate collaborators: E. Henry, B. Pier, Antony Bak, David G.W. Cooper, David Rothstein, David Schalk, Mark Buckley, and Ben Bigger and many previous students.

NSF - DMR-0072203 (Haverford) and DMR-0072203 (Penn)

Central Ideas from Nonlinear Dynamics

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003

Using phase space to describe fluid systems: mode amplitudes and their evolution.Geometrical thinking: attracting and repelling fixed points; multiple attractors; limit cycle attractors; chaotic attractors representing nonperiodic flows.Understanding instabilities as bifurcations or qualitative changes in the phase space of a system, e.g. birth or death of fixed points.

Central ideas - II

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003

Common features shared by disparate systems:Supercritical vs. subcritical transitionsBifurcation scenarios The central role of symmetryPatterns and spatiotemporal chaos described by generic amplitude equations in large systems

A Few Examples

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003

The role of stretching in fluid mixingClustering of particles suspended in a liquid.Nonlinear waves on fluid interfacesMultiple stability in granular flow

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003

1. Stretching and Fluid Mixing

Measuring fluid stretching can illuminate mixing and illustrate nonlinear dynamics.

Greg Voth, George Haller, Greg Dobler, Tim Saint.

Types of Fluid Mixing

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003

Turbulent mixing: Random structures produced by fluid instability at high Reynolds number Re stretch and fold fluid elements. Chaotic mixing: Some laminar flows at modest Re can produce complex distributions of material. Fundamental process: Non-reversible stretching and folding of fluid elements, with diffusion at small scales. Chaos in real space.

Background on Stretching in Chaotic Mixing

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003

Chaotic mixing has been well studied, and the importance of stretching was emphasized and investigated by J. Ottino and colleagues Khakhar, Swanson, and Muzzio.Others (Rom-Kedar, Leonard, Wiggins) have illuminated the connection between nonlinear dynamics and mixing theoretically.What is new in our work is the experimental measurement of space and time resolved stretching fields.

Apparatus: 2D Magnetically Driven Fluid Layer

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003

Magnet Array

Electrodes

ft)sin(2 I(t) 0 πI=

Glycerol and water a few mm thick, containing fluorescent dye.

Periodic forcing:Top View:

Precise Particle Tracking

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003

~ 800 fluorescent particles tracked simultaneously.

Velocity Fields

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003

0.9 cm/sec

0

(p=5, Re=56)

Why Do Periodic Flows Mix? Breaking Time Reversal Symmetry

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003

To mix, the flow can be time periodic, but must not be time reversible:

relative to any starting time.Finite Reynolds numberRe=VL/ν can break time reversal symmetry:

( ) ( )t t≠ − −V V

Velocity at equal intervals before and after the moment of minimum velocity.

Breaking of Time Reversal Symmetry

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003

10

8

6

4

2

0

< ((

v(t)

+ v(

-t))2

>1/2

(mm

/s)

200150100500

Reynolds Number

Vx

Vy

Structures in the Poincaré Map of the Flow

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003

Displacements in one period, color coded.Hyperbolic Fixed Points

A small part (6%) of the flow.

Elliptic Fixed Points

0 cm 1.7 cm

Manifolds of Hyperbolic Fixed Points

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003

UnstableManifold

StableManifold

Typical of Hamiltonian Chaos

Definition of Stretching

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003

Stretching = lim (L/L0)

L0

LL0 0

Past Stretching Field: Stretching that a fluid element has experienced during the last ∆t. (Large near unstable man.)

Future Stretching Field: Stretching that a fluid element will experience in the next ∆t. (Large near stable man.)

Past Stretching Field

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003

Stretching is organized in sharp lines.

Re=45, p=1, ∆t=3

Future and Past Stretching Fields

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003

Future Stretching Field (Blue) marks the stable manifold.

Past Stretching Field (Red) marks the unstable manifold.

Circles mark hyperbolic points. A “heteroclinic tangle”.

Understanding the Dye Concentration Field

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003

Unstable manifold (past stretching field) and the dye concentration field

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003

Lines of large past stretching (unstable manifold) are aligned with the contours of the concentration field.

This is true at every time (phase).

Stretching is Inhomogeneous: PDF

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003

λ / < λ >Pr

obab

ility

Stretching over one periodLog(stretching)(Finite Time Lyap. Exp.)

Solid: Re=45, p=1, <λ>=1.9 per-1

Dotted: Re=100, p=5, <λ>=6.4 per-1(Re=100,p=5)

Decay of the Dye Concentration Field

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003

-1.5

-1.0

-0.5

0.0

Log

of S

tand

ard

Dev

iatio

n of

Dye

Inte

nsity

50403020100Time (periods)

Re=25 Re=55 Re=85 Re=100 Re=115 Re=145 Re=170

Log

Contrast

Can Mixing Rates Be Predicted?

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003

Antonsen et al. predict the long time decay of concentration variance based on the prob. dist. P(h,t) of finite time Lyapunov exponents (stretching)Using our measured P(h,t) and their theory, we predict mixing (variance decay) rates a factor of 10 larger than is observed.

Predicting Mixing Rates - (cont.)

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003

Why is mixing so slow? Large system Transport over long distances is important. Enhanced or “eddy” diffusion: By watching particles diffuse, we can succesfully predict the rate of decay of a scalar field.

Future

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003

The onset of non-periodic flow (2D turbulence) does not dramatically change mixing rates, to our surprise. Are the main features of non-periodic mixing captured by the periodic case discussed here? What will be the topological properties of stretching fields for non-periodic flows?Papers: G.A. Voth et al, Phys. Rev. Lett. (2002) and Phys. Fluids (2003)

2. Structure and Dynamics of HydrodynamicallyMediated Particle Clusters

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003

Fluid mediated interactions between particles lead to forces between them.

A variety of tunable patterns are produced, including chaotic states.

Greg Voth, Charles Thomas, Ben Bigger, Mark Buckley, Michael Brenner, Howard Stone, and JPG.

Observations of Clustering

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003

f=50 HzΓ=4.5

Related Phenomena

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003

Sedimentation – when particles fall in a fluid, they influence each other.Fluidized beds, widely used for catalysis.

Here: An example of physical self-assembly of ordered particulate structuresChaos in a system of interacting particles

Rayleigh Streaming

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003

Streaming flow for a particle in a fluid - no boundary From Van Dyke, An Album of Fluid Motion

Origin of Long Range Attraction

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003

Rayleigh: steady streaming flow away from poles and toward the equator generated outside the viscous boundary layer (0.2 mm) surrounding the particle. Predicted inflow velocity

(via collaborators M. Brenner and H. Stone).Particles follow this flow as they approach each other.However, see Voth and Otto, KC004, Tuesday 08:39

2 3( ) 0.53 /V r Aa rων= −

Approach Curves for 2 Particles

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003

F=20 Hz; Γ=2.9; A=0.4 mm (periodic)

F=50 Hz; Γ=4.6; A=0.15 mm (chaotic)

•Dashed: theory; Solid:experiments;

t = 0 at end of approach.

•At higher excitation amplitude, the particles do not touch. Repulsive force also.

-2 s 0.0 stime

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003

-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.5

0.0

0.5

1.0

y po

sitio

n (m

m)

x position (mm)

Particle TwoParticle One

Particle Three

Particle Four

N=4, Γ=2.97.(a) trajectories; (b) spacings vs.

time. Note transitions.

0 200 400 600 800 1000 12000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

pair

sepa

ratio

ns (m

m)

time (s)

Particles 2 & 1 Particles 3 & 1 Particles 4 & 2 Particles 4 & 3

Trapezoidal Cluster - Chaotic Motion

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003

(a) Trajectories (Γ=2.96) (b) separations vs. time. Nonperiodicoscillations.

0 200 400 600 800 1000 12000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Particles 2 & 1 Particles 3 & 1 Particles 3 & 2 Particles 4 & 1 Particles 4 & 3 Particles 5 & 2 Particles 5 & 3 Particles 5 & 4

pair

sepa

ratio

n (m

m)

time (s)

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Particle Five

Particle Four

Particle Three

Particle Two

y po

sitio

n (m

m)

x position (mm)

Particle One

Fluctuating Five Particle Cluster

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003

-6 -4 -2 0 2 4 6-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 6-6

-4

-2

0

2

4

6

Γ=3.44

x position (mm)

y po

sitio

n (m

m)

x position (mm)

y po

sitio

n (m

m)

Γ=5.53

Large cluster trajectories, for several accelerations.

Large Cluster Trajectories

Main Conclusions - Interacting Particles

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003

Many different structures, ordered, disordered, chaoticObservations suggest an effective interaction potential for two particles. But pairwiseinteractions do not suffice for many particles.Large and small clusters behave differently. More: Voth et al. PRL 88, 234301 (2002); C.C. Thomas & JPG, in preparation.

Complex wave patterns

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003Kudrolli, Pier, Edwards, JPG

Quasicrystalline wave pattern

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003

Sheared Granular Flow

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003

Shearing induces order; order modifies

Conclusions

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003

Ideas from nonlinear dynamics contribute to understanding fluid phenomena.Equally, fluids illuminate nonlinear dynamics and can be used to teach it.Fluids ought to play a larger role in physics teaching. I make a case for this in Physics Today, December 2003: “One of the oddities of contemporary physics education is the nearly complete absence of continuum mechanics…”Movies: www.haverford.edu/physics-astro/Gollub/lab.html

END

Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003