Controlling Faraday waves with multi-frequency forcing
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Controlling Faraday waves with multi-frequency forcing
Mary SilberEngineering Sciences & Applied Mathematics
Northwestern University
http://www.esam.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 patternBirfurcation 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
in modulated gravity
Benjamin and Ursell, Proc. Roy. Soc. Lond. A (1954)
Considered inviscid potential flow:
with free surface given by:
Find satisfy the Mathieu equation:
with gravity-capillary wave dispersion relation
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MATHIEU EQUATION
Subharmonic resonance
From: Jordan & Smith
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Unique capabilities of the Faraday system
• Huge, easily accessible control parameter space
• Multiple length scales compete (or cooperate) o
vera
ll fo
rcin
g s
tre
ngth
wave number k
(Naïve) Schematic of Neutral Stability Curve:
m/2 n/2 p/2 q/2
cf. Huepe, Ding, Umbanhowar, Silber (2005)
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Unique capabilities of the Faraday system• 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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Resonant triads• Lowest order nonlinear interactions
• Building blocks of more complex patterns
k1
k2 k3
k1 + k2 = k3
res
Resonant triads & Faraday waves: Müller, Edwards & Fauve, Zhang & Viñals,…
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Resonant triads
• Role in pattern selection: a simple example
k1
k2 k3
spatial translation, reflection, rotation by
res
critical modes damped mode
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Resonant triads
• Role in pattern selection: a simple example
k1
k2 k3
center manifold reduction
res
critical modes damped mode (eliminate)
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Resonant triads
• Role in pattern selection: a simple example
k1
k2
rhombic equations:
res
“enhancing”,
“cooperative”
“suppressing”,
“competitive”
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nonlinear coupling coefficient:
consider free energy:
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Organizing Centerforcing
dampingtime translation, time reversal symmetries
Hamiltonian structure
Expanded TW eqns.
SW eqns.
Porter & Silber, PRL (2002); Physica D (2004)
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Travelling Wave eqns.
• Parameter (broken temporal) symmetries
time translation symmetry:
u=m denotes dominant driving frequency
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Travelling Wave eqns.
• Parameter (broken temporal) symmetries
time reversal symmetry:
Hamiltonian structure (for ):
(See, e.g., Miles, JFM (1984))
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Travelling Wave eqns.
• Enforce symmetries Travelling wave amplitude equations
damping
parametric forcing
damping
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Travelling Wave eqns.
• Enforce symmetries Travelling wave amplitude equations
Time translation invariants:
Example: (m,n) forcing, =m-n
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Travelling wave eqns.
• Enforce symmetries Travelling wave amplitude equations
Focus on
Possible only for
At most 5 relevant forcing frequencies for fixed
Perform center manifold reduction to SW eqns.
Porter, Topaz and Silber, PRL & PRE 2004
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Key results
• Strongest interaction is for = m
• Parametrically forcing damped mode can strengthen interaction
• Phases u may tune interaction strength
• Only = n – m is always enhancing (Hamiltonian argument)
ex. (m,n, p = 2n – 2m) forcing, = n – m>0
Pp() > 0> 0
bres > 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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Direct Reduction to Standing Wave eqns
k1
k2
Solvability condition at :
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Demonstration of key results
• Strongest interaction is for = m
• Parametrically forcing damped mode can strengthen interaction
• Phases u may tune interaction strength
• Only = n – m is always enhancing (bres > 0)
= m = 6
= n – m = 1
ex. (6,7,2) forcing, b()computed from Zhang-Viñals equations:
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Example: Experimental superlattice pattern
Topaz & Silber, Physica D (2002)
Kudrolli, Pier and Gollub, Physica D (1998)
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Example: Experimental superlattice patternEpstein and Fineberg, 2005 preprint.
3:2
5:3
6/7/2 forcing frequencies:
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Example: Experimental superlattice patternEpstein and Fineberg, 2005 preprint.
3:2
4:3
5:3
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Example: Experimental quasipattern
Arbell & Fineberg, PRE, 2002
(3,2,4)forcing
(3,2)forcing
{}
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Example: Impulsive-Forcing(See J. Bechhoefer & B. Johnson, Am. J. Phys. 1996)
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Example: Impulsive-Forcing(Catllá, Porter and Silber, PRE, in press)One-dim. waves
Weakly nonlinear analysis from Z-V equations.
Capillarity parameter
C
Prediction based on 2-term truncated Fourier series:
sinusoidal
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Linear Theory: Shallow and Viscous CaseForcing function Neutral Curve
Huepe, Ding, Umbanhowar, Silber, 2005 preprint
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Linear Theory: Shallow and Viscous CaseLinear 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
Matching across regions gives transition matrices
Periodicity requirement determines stability boundary
WKB approximation:
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Linear Theory: Shallow and Viscous Case
Exact numerical:
WKB approximation:
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Conclusions
• 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 Fineberg’s group).
• Symmetry-based approach yields model-independent results; arbitrary number of (commensurate) frequency components. (even infinite -- impulsive forcing)
• Shallow, viscous layers present new challenges…