Nonlinear Dynamics of Fluid Motion
„
“Nonlinear Dynamics” has deepened our
understanding of fluid dynamics, and taught us
many lessons about nonlinear phenomena.
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No movies in this version; some may be found at
www.haverford.edu/physics-astro/Gollub/lab.html
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Thanks to Collaborators
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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)
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Central Ideas from Nonlinear Dynamics
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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.
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Central ideas - II
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Common features shared by disparate systems:
„ Supercritical vs. subcritical transitions
„ Bifurcation scenarios
„ The central role of symmetry
„ Patterns and spatiotemporal chaos described by
generic amplitude equations in large systems
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
A Few Examples
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The role of stretching in fluid mixing
Clustering of particles suspended in a liquid.
Nonlinear waves on fluid interfaces
Multiple 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.
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Types of Fluid Mixing
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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.
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Background on Stretching in Chaotic Mixing
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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.
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Apparatus:
2D Magnetically Driven Fluid Layer
Glycerol and water a few mm
thick, containing fluorescent dye.
Electrodes
Magnet Array
Top View:
Periodic forcing:
I(t) = I 0 sin(2πft)
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Precise Particle Tracking
z
~ 800 fluorescent
particles tracked
simultaneously.
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Velocity Fields
0.9 cm/sec
0
(p=5, Re=56)
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Why Do Periodic Flows Mix?
Breaking Time Reversal Symmetry
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To mix, the flow can be
time periodic, but must not
be time reversible:
V (t ) ≠ − V (−t )
relative to any starting time.
„ Finite Reynolds number
Re=VL/ν can break time
Velocity at equal intervals
reversal symmetry:
before and after the moment
of minimum velocity.
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
10
Vx
Vy
8
2
< ((v(t) + v(-t)) >
1/2
(mm/s)
Breaking of Time Reversal Symmetry
6
4
2
0
0
50
100
150
200
Reynolds Number
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Structures in the Poincaré Map of the Flow
Displacements in one
period, color coded.
Hyperbolic Fixed Points
A small part (6%) of
the flow.
Elliptic Fixed Points
0 cm
1.7 cm
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Manifolds of Hyperbolic Fixed Points
Unstable
Manifold
Stable
Manifold
Typical of
Hamiltonian Chaos
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Definition of Stretching
L
Stretching = lim (L/L0)
L0
0
Past Stretching Field: Stretching that a
fluid element has experienced during
the last ∆t. (Large near unstable man.)
L0
Future Stretching Field: Stretching that
a fluid element will experience in the
next ∆t. (Large near stable man.)
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Past Stretching Field
ƒ Stretching
is
organized in sharp
lines.
Re=45, p=1, ∆t=3
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Future and Past Stretching Fields
ƒFuture Stretching
Field (Blue) marks the
stable manifold.
ƒ Past Stretching Field
(Red) marks the
unstable manifold.
ƒCircles mark
hyperbolic points. A
“heteroclinic tangle”.
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Understanding the Dye Concentration Field
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Unstable manifold (past stretching field) and
the dye concentration field
ƒLines of large past
stretching (unstable
manifold) are
aligned with the
contours of the
concentration field.
ƒThis is true at
every time (phase).
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Stretching is Inhomogeneous: PDF
Probability
Log(stretching)
Stretching over one period (Finite Time Lyap. Exp.)
λ/<λ>
(Re=100,p=5)
Solid: Re=45, p=1, <λ>=1.9 per-1
Dotted: Re=100, p=5, <λ>=6.4 per-1
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Decay of the Dye Concentration Field
Re=25
Re=55
Re=85
Re=100
Re=115
Re=145
Re=170
Log
Contrast
Log of Standard Deviation of Dye Intensity
0.0
-0.5
-1.0
-1.5
0
10
20
30
40
50
Time (periods)
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Can Mixing Rates Be Predicted?
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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.
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Predicting Mixing Rates - (cont.)
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Why is mixing so slow? Large system Æ
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.
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Future
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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)
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
2. Structure and Dynamics of Hydrodynamically
Mediated Particle Clusters
Fluid mediated interactions
between particles lead to forces
between them.
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zA
variety of tunable patterns
are produced, including chaotic
states.
zGreg
Voth, Charles Thomas,
Ben Bigger, Mark Buckley,
Michael Brenner, Howard
Stone, and JPG.
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Observations of Clustering
f=50 Hz
Γ=4.5
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Related Phenomena
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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 structures
Chaos in a system of interacting particles
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Rayleigh Streaming
Streaming flow for a particle in a fluid - no boundary
From Van Dyke, An Album of Fluid Motion
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Origin of Long Range Attraction
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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
V (r ) = −0.53 Aa
2
ων / r
3
(via collaborators M. Brenner and H. Stone).
„ Particles follow this flow as they approach each
other.
„ However, see Voth and Otto, KC004, Tuesday
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
08:39
Approach Curves for 2 Particles
F=20 Hz; Γ=2.9;
A=0.4 mm (periodic)
time
0.0 s
•Dashed: theory; Solid:experiments;
-2 s
t = 0 at
F=50 Hz; Γ=4.6;
endA=0.15
of approach.
mm (chaotic)
•At higher excitation amplitude, the particles
do not touch. Æ Repulsive force also.
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Trapezoidal Cluster - Chaotic Motion
1.0
Particle Three
N=4, Γ=2.97.
(a) trajectories;
(b) spacings vs.
time.
Note transitions.
y position (mm)
0.5
Particle Four
0.0
-0.5
Particle Two
-1.0
Particle One
-1.0
-0.5
0.0
0.5
1.0
x position (mm)
1.8
pair separations (mm)
1.6
1.4
1.2
1.0
0.8
0.6
Particles 2 & 1
Particles 3 & 1
Particles 4 & 2
Particles 4 & 3
0.4
0.2
0.0
0
200
400
600
Fluid
time (s)
Dynamics Prize800
Lecture – Copyright1000
Jerry Gollub, 2003
1200
Fluctuating Five Particle Cluster
1.5
1.0
(a) Trajectories
(Γ=2.96)
(b) separations
vs. time.
Nonperiodic
oscillations.
Particle Five
y position (mm)
0.5
Particle Four
Particle Three
0.0
-0.5
-1.0
-1.5
-1.5
Particle Two
-1.0
Particle One
-0.5
0.0
0.5
1.0
1.5
x position (mm)
1.8
1.6
pair separation (mm)
1.4
1.2
1.0
0.8
Particles 2 & 1
Particles 3 & 1
Particles 3 & 2
Particles 4 & 1
Particles 4 & 3
Particles 5 & 2
Particles 5 & 3
Particles 5 & 4
0.6
0.4
0.2
0.0
0
200
400
600
time (s)
Fluid Dynamics800Prize Lecture – Copyright
Jerry Gollub, 2003
1000
1200
Large Cluster Trajectories
6
6
Γ=5.53
4
4
2
2
y position (mm)
y position (mm)
Γ=3.44
0
0
-2
-2
-4
-4
-6
-6
-6
-4
-2
0
x position (mm)
2
4
6
-6
-4
-2
0
2
4
6
x position (mm)
Large cluster trajectories, for several accelerations.
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Main Conclusions - Interacting Particles
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Many different structures, ordered, disordered,
chaotic
Observations suggest an effective interaction
potential for two particles. But pairwise
interactions 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.
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Complex wave patterns
Kudrolli, Pier, Edwards, JPG
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Quasicrystalline wave pattern
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Sheared Granular Flow
Shearing induces order; order modifies
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
Conclusions
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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/physicsastro/Gollub/lab.html
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003
END
Fluid Dynamics Prize Lecture – Copyright Jerry Gollub, 2003