Singularities and Topological Transitions: Breaking Away, Selective Withdrawal, Islets in the Stream

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Singularities and Topological Transitions
Breaking Away, Selective Withdrawal, Islets in
the Stream. Fermi National Accelerator
Laboratory - Jan. 17, 2007

Dynamic singularities infinitesmally small,
very short time
Example How does drop fall, break
apart? Topology changes - real
transition Neck radius ? 0 (curvature
???) Pressure ??? Cannot do simulation to
get to other side of snapoff
?
Is there understanding for these transitions like
that for thermodynamic phase transitions?
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Similar behavior Star formation
PILLARS OF CREATION IN A STAR-FORMING REGION Gas
Pillars in M1 - Eagle Nebula Hubble Space
Telescope 4/1/95
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Similar behavior Breakup of bacteria colonies
Elena Budrene - Harvard
Dynamic singularities appear everywhere in
physics - celestial ? microscopic ? nuclear
fission
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Who Did It Experiment
Theory/Simulation Sarah Case
Osman Basaran Itai Cohen
Michael Brenner Nathan Keim Pankaj Doshi
Xiangdong Shi Jens Eggers Jason Wyman
Wendy Zhang Lei Xu Applications
Milan Mrksich Seda Kizilel Dr. Marc
Garfinkel, Dr. Horacio Rilo Funding
NSF


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Rayleigh-Plateau Instability
R
L
Surface area decreases Unstable to perturbations
of wavelength ?? ?? 2? R. Pressure greatest at
minimum thickness ? liquid squeezed out.
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Bolas Spider
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Water falling in air
NOT like cartoon! 2 snapoffs One at top
One at bottom
Xiangdong Shi Michael Brenner
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Water Drops
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Look at point of snapoff - cone pointing into
sphere
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Look at point of snapoff - cone pointing into
sphere
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Water into Air
Singularity is same even though gravity points in
opposite direction
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How to Think About Shapes Scale
invariance(borrowed from statistical mechanics)
Breakup ? radius smaller than any other
length. Dynamics insensitive to all other
lengths. Flow depends only on shrinking radius.

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How to Think About Shapes Scale
invariance(borrowed from statistical mechanics)
Breakup ? radius smaller than any other
length. Dynamics insensitive to all other
lengths. Flow depends only on shrinking radius.
But Radius depends on flow (which depends
on radius (which depends on flow (which
depends on radius (which depends on
flow (which depends on radius (which depends on
flow (which depends on radius (which depends on
flow (which depends on radius (which depends on
flow . . . Self-similar structure Blow up any
part ? regain original. Universal shapes
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Similarity Solution
h(z,t) f(t) H(z-zo)/f(t)ß Similarity
solutions same AT DIFFERENT TIMES with different
magnifications along h and z.
Scaling determined by force balance at
singularity. PDE ? ODE
Keller and Miksis (83) Eggers, RMP (97)
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Different Regimes
Explore different asymptotic regimes by tuning
parameters. Depends on viscosity of inner
fluid, ? viscosity ratio of fluids, ? (air is a
fluid) density difference, ?? surface tension,
? nozzle diameter, D
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Scaling Profiles - Glycerol into Oil Stretching
axes by different amounts at different times
produces master curve
Itai Cohen, Michael Brenner Jens Eggers, Wendy
Zhang
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Singularities tame non-linearity of Navier-Stokes
Eqs. Role of scale invariance - borrowed from
critical phenomena Near singularity, dynamics
insensitive to all other lengths Emphasize what
is Universal But . . .
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Water into OilNot so simple
Persistence of memory. No similarity
solution. No universality! Separation of
scales but also of axial and radial length
scales.
I. Cohen, W. Zhang P. Doshi, O. Basaran P.
Howell, M. Siegel
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Water into OilContinued
Viscosity of water begins to matter ? creates
very fine thread.
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Remember the water drop in air?
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Remember the water drop in air?
What about air drop in water (i.e., a bubble)?
N. Keim, W. Zhang
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Remember the water drop in air?
What about air drop in water (i.e., a bubble)?
Singularity sensitive to small perturbations -
remembers axial asymmetry
N. Keim, W. Zhang
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The drop falls ? splashes
Is splash interesting?Break-up localizes energy
from the kinetic energy into singular points as
surface ruptures. How?Coronal splash
Lei Xu, Wendy Zhang
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Drop splashes
Drop of alcohol hitting smooth, dry slide
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Drop splashes
atmospheric pressure
1/3 atmospheric pressure (Mt. Everest)
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Impact Velocity vs. Threshold Pressure
V0 (m/s)
Splash
No Splash
Non-monotonic
PT (kPa)
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Selective Withdrawal
Light fluid (oil) Heavy fluid (water)
Itai Cohen Sarah Case, Jason Wyman Wendy Zhang
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Hump to Spout Transition vs. Flow Rate
I. Cohen
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Why care about singularity?
Possible Medical Application Type 1 Diabetes
Islets of Langerhans damaged. ? Transplant
(106) islet cells. Immune system attacks
foreign cells. ? Coat Islets with thin shell
not recognized by antibodies but permeable to
insulin, glucose . . . Islets irregular size,
100300 ?, shape. - must coat each cell
accordingly. ?Shrinkwrap using spout.
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Coating TechnologySelective withdrawal using
convergent flow profile
Illumination to harden polymer coat
Physics Itai Cohen Jason Wyman Chemistry Milan
Mrksich Hospital Dr. Horacio Rilo Dr. Marc
Garfinkel Seda Kizilel
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Coating Process
Bigger particles initiate breakup in thicker
region of spout
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Coating of Islet
Jason Wyman
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Big molecules excluded
Blue Capsule Maroon Islet (stained) Green
Lechtin-FITC (140 k Dalton)
For comparison Insulin 5.8 k
Da Immunoglobulin G 160 - 180 k Da
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Tuning Coat Thickness
Singularity important for thin coat!
I. Cohen, M. Mrksich
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Singularities and Scaling from thermodynamic to
topological transitions
A great idea is like a phantom ocean beating
upon the shores of human life in successive waves
of specialization. A. N. Whitehead
Simplified dynamics More subtle than critical
phenomena Small dimensions Applications -
Coating (medical), fabrication Surprises await
us even in the most familiar of phenomena!
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Navier-Stokes Equation
??v/?t (v . ?)v -?P ??2v F inertial
terms internal pressure viscous force body
force Incompressibility Equation Laplace
pressure equation for surface
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Glycerol/water into Air
100 x viscosity of water
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Glycerol/water into Air
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Glycerol/water into Air
Xiangdong Shi, Michael Brenner
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Scaling for different ?
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Atmospheric pressure (100kPa)
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Reduced pressure (17kPa)
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Dynamic singularities appear everywhere in
physics - from celestial to microscopic to
nuclear fission
I am an old man now and when I die and go to
Heaven there are two matters on which I hope for
enlightenment. One is quantum electrodynamics,
and the other is turbulent motion of fluids. And
about the former I am really rather optimistic.
Sir Horace
Lamb (1932)