Controlling Faraday waves with
multi-frequency forcing
Mary Silber
Engineering 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)
FARADAY CRISPATIONS
– M. Faraday, Phil. Trans. R. Soc. Lond. (1831)
FARADAY CRISPATIONS
– M. Faraday, Phil. Trans. R. Soc. Lond. (1831)
Edwards and Fauve, JFM (1994)
12-fold quasipattern
Bordeaux to Geneva: 5cm, depth: 3mm
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)
Arbell & Fineberg, PRE 2002
FARADAY CRISPATIONS
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
with
satisfy the Mathieu equation:
gravity-capillary wave
dispersion relation
MATHIEU EQUATION
Subharmonic
resonance
From: Jordan & Smith
Unique capabilities of the Faraday system
•
Huge, easily accessible control parameter space
•
Multiple length scales compete (or cooperate)
overall forcing strength
(Naïve) Schematic of Neutral Stability Curve:
m/2
n/2
p/2
q/2
wave number k
cf. Huepe, Ding, Umbanhowar, Silber (2005)
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.
Resonant triads
•
Lowest order nonlinear interactions
•
Building blocks of more complex patterns
k1 + k2 = k3
k2
qres
k3
k1
Resonant triads & Faraday waves:
Müller, Edwards & Fauve, Zhang & Viñals,…
Resonant triads
•
Role in pattern selection: a simple example
k2
k3
critical modes
damped mode
qres
k1
spatial translation, reflection, rotation by p
Resonant triads
•
Role in pattern selection: a simple example
k2
k3
critical modes
damped mode
(eliminate)
qres
k1
center manifold reduction
Resonant triads
•
Role in pattern selection: a simple example
k2
“suppressing”,
“competitive”
“enhancing”,
“cooperative”
qres
k1
rhombic equations:
consider free energy:
nonlinear coupling coefficient:
Organizing Center
forcing
Hamiltonian
structure
Expanded
TW eqns.
SW
eqns.
time translation,
time reversal
symmetries
damping
Porter & Silber, PRL (2002);
Physica D (2004)
Travelling Wave eqns.
•
Parameter (broken temporal) symmetries
u=m denotes dominant
driving frequency
time translation symmetry:
Travelling Wave eqns.
•
Parameter (broken temporal) symmetries
time reversal symmetry:
Hamiltonian structure (for
):
(See, e.g., Miles, JFM (1984))
Travelling Wave eqns.
•
Enforce symmetries  Travelling wave amplitude equations
damping
parametric forcing
damping
Travelling Wave eqns.
•
Enforce symmetries  Travelling wave amplitude equations
Time translation invariants:
Example: (m,n) forcing, =m-n
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
Key results
•
Strongest interaction is for  = m
•
Parametrically forcing damped mode can strengthen interaction
•
Phases fu may tune interaction strength
•
Only  = n – m is always enhancing (Hamiltonian argument)
ex. (m,n, p = 2n – 2m) forcing,  = n – m>0
>0
Pp(F) > 0
bres > 0 for this case
(can get signs for some other cases)
Zhang & Viñals, J. Fluid Mech 1997
Direct Reduction to Standing Wave eqns
k2
q
k1
Solvability condition at
:
Demonstration of key results
•
Strongest interaction is for  = m
•
Parametrically forcing damped mode can strengthen interaction
•
Phases fu may tune interaction strength
•
Only  = n – m is always enhancing (bres > 0)
ex. (6,7,2) forcing, b()computed from Zhang-Viñals equations:
=n–m=1
=m=6
Example: Experimental superlattice pattern
Kudrolli, Pier and Gollub, Physica D (1998)
Topaz & Silber, Physica D (2002)
Example: Experimental superlattice pattern
6/7/2 forcing frequencies:
Epstein and Fineberg, 2005 preprint.
3:2
5:3
Example: Experimental superlattice pattern
Epstein and Fineberg, 2005 preprint.
3:2
4:3
5:3
Example: Experimental quasipattern
Arbell & Fineberg, PRE, 2002
(3,2,4)
forcing
{
}
q = 45
(3,2)
forcing
Example: Impulsive-Forcing
(See J. Bechhoefer & B. Johnson, Am. J. Phys. 1996)
Example: Impulsive-Forcing
One-dim. waves
Weakly nonlinear analysis
from Z-V equations.
(Catllá, Porter and Silber, PRE, in press)
C
sinusoidal
Capillarity parameter
Prediction based on 2-term
truncated Fourier series:
Linear Theory: Shallow and Viscous Case
Forcing function
Neutral Curve
Huepe, Ding, Umbanhowar, Silber, 2005 preprint
Linear Theory: Shallow and Viscous Case
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
Linear Theory: Shallow and Viscous Case
Exact numerical:
WKB approximation:
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…