Controlling Faraday waves with multifrequency forcing

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Controlling Faraday waves with multi-frequency
forcing

Mary SilberEngineering Sciences Applied
Mathematics Northwestern University http//www.es
am.northwestern.edu/silber Work with Jeff
Porter (Univ. Comp. Madrid) Chad Topaz (UCLA),
Cristián Huepe, Yu Ding Paul Umbanhowar
(Northwestern), and Anne Catllá (Duke)
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FARADAY CRISPATIONS
M. Faraday, Phil. Trans. R. Soc. Lond. (1831)
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FARADAY CRISPATIONS
M. Faraday, Phil. Trans. R. Soc. Lond. (1831)
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Edwards and Fauve, JFM (1994)
12-fold quasipattern
Bordeaux to Geneva 5cm, depth 3mm
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Kudrolli, Pier and Gollub, Physica D (1998)
Superlattice pattern
Birfurcation theoretic investigations of
superlattice patterns Dionne and Golubitsky,
ZAMP (1992) Dionne, Silber and Skeldon,
Nonlinearity (1997) Silber and Proctor, PRL (1998)
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Arbell Fineberg, PRE 2002
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FARADAY CRISPATIONS
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LINEAR STABILITY ANALYSIS
Benjamin and Ursell, Proc. Roy. Soc. Lond. A
(1954)
Considered inviscid potential flow
in modulated gravity
with free surface given by
Find satisfy the Mathieu equation
gravity-capillary wave dispersion relation
with
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MATHIEU EQUATION
Subharmonic resonance
From Jordan Smith
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• Huge, easily accessible control parameter space
• Multiple length scales compete (or cooperate)

(Naïve) Schematic of Neutral Stability Curve
overall forcing strength
m/2
n/2
p/2
q/2
wave number k
cf. Huepe, Ding, Umbanhowar, Silber (2005)
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• Huge, easily accessible control parameter space
• Multiple length scales compete (or cooperate)

Goal Determine how forcing function parameters
enhance (or inhibit) weakly nonlinear wave
interactions. Benefits Helps interpret existing
experimental results. Leads to design strategy
how to choose a forcing function that favors
particular patterns in lab experiments.
Approach equivariant bifurcation theory.
Exploit spatio-temporal symmetries (and remnants
of Hamiltonian structure) present in the
weak-damping/weak-driving limit. Focus on
(weakly nonlinear) three-wave interactions as
building blocks of spatially-extended patterns.
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• Lowest order nonlinear interactions
• Building blocks of more complex patterns

k1 k2 k3
k2
k3
qres
k1
Resonant triads Faraday waves Müller, Edwards
Fauve, Zhang Viñals,
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• Role in pattern selection a simple example

k2
k3
critical modes
damped mode
qres
k1
spatial translation, reflection, rotation by p
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• Role in pattern selection a simple example

k2
k3
critical modes
damped mode (eliminate)
qres
k1
center manifold reduction
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• Role in pattern selection a simple example

k2
suppressing, competitive
enhancing, cooperative
qres
k1
rhombic equations
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(No Transcript)
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consider free energy
nonlinear coupling coefficient
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forcing
Expanded TW eqns.
Hamiltonian structure
SW eqns.
time translation, time reversal symmetries
damping
Porter Silber, PRL (2002) Physica D (2004)
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• Parameter (broken temporal) symmetries

um denotes dominant driving frequency
time translation symmetry
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• Parameter (broken temporal) symmetries

time reversal symmetry
Hamiltonian structure (for )
(See, e.g., Miles, JFM (1984))
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• Enforce symmetries ? Travelling wave amplitude
  equations

damping
parametric forcing
damping
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• Enforce symmetries ? Travelling wave amplitude
  equations

Example (m,n) forcing, ?m-n
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• Enforce symmetries ? Travelling wave amplitude
  equations

Porter, Topaz and Silber, PRL PRE 2004
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• Strongest interaction is for W m
• Parametrically forcing damped mode can strengthen
  interaction
• Phases fu may tune interaction strength
• Only W n m is always enhancing (Hamiltonian
  argument)

ex. (m,n, p 2n 2m) forcing, W n mgt0
gt 0
Pp(F) gt 0
bres gt 0 for this case(can get signs for some
other cases)
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Zhang Viñals, J. Fluid Mech 1997
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k2
q
k1
Solvability condition at
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• Strongest interaction is for W m
• Parametrically forcing damped mode can strengthen
  interaction
• Phases fu may tune interaction strength
• Only W n m is always enhancing (bres gt 0)

ex. (6,7,2) forcing, b(?)computed from
Zhang-Viñals equations
W n m 1
W m 6
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Kudrolli, Pier and Gollub, Physica D (1998)
Topaz Silber, Physica D (2002)
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Epstein and Fineberg, 2005 preprint.
6/7/2 forcing frequencies
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53
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Epstein and Fineberg, 2005 preprint.
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43
53
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Arbell Fineberg, PRE, 2002

(3,2,4)forcing

(3,2)forcing
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(See J. Bechhoefer B. Johnson, Am. J. Phys.
1996)
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(Catllá, Porter and Silber, PRE, in press)
One-dim. waves Weakly nonlinear analysis from Z-V
equations.
C
sinusoidal
Capillarity parameter
Prediction based on 2-term truncated Fourier
series
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Huepe, Ding, Umbanhowar, Silber, 2005 preprint
Forcing function Neutral Curve
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Linear analysis, aimed at finding envelope of
neutral curves ( following Cerda Tirapegui,
JFM 1998) Lubrication approximation shallow,
viscous layer, low-frequency forcing Transform
to time-independent Schrödinger eqn.,1-d periodic
potential
WKB approximation
Matching across regions gives transition
matrices Periodicity requirement determines
stability boundary
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Exact numerical
WKB approximation
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• Determined how which parameters in periodic
  forcing function influence weakly nonlinear
  3-wave interactions.
• Weak-damping/weak-forcing limit leads to scaling
  laws and phase dependence of coefficients in
  bifurcation equations.
• Hamiltonian structure can force certain
  interactions to be cooperative, while others
  are competitive.
• Results suggest how to control pattern selection
  by choice of forcing function frequency content.
  ( cf. experiments by Finebergs group).
• Symmetry-based approach yields model-independent
  results arbitrary number of (commensurate)
  frequency components. (even infinite -- impulsive
  forcing)
• Shallow, viscous layers present new challenges