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)

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 patternBirfurcation 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

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

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) 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)

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

k1 + k2 = k3

res

Resonant triads & Faraday waves: Müller, Edwards & Fauve, Zhang & Viñals,…

Resonant triads

• Role in pattern selection: a simple example

k1

k2 k3

spatial translation, reflection, rotation by

res

critical modes damped mode

Resonant triads

• Role in pattern selection: a simple example

k1

k2 k3

center manifold reduction

res

critical modes damped mode (eliminate)

Resonant triads

• Role in pattern selection: a simple example

k1

k2

rhombic equations:

res

“enhancing”,

“cooperative”

“suppressing”,

“competitive”

nonlinear coupling coefficient:

consider free energy:

Organizing Centerforcing

dampingtime translation, time reversal symmetries

Hamiltonian structure

Expanded TW eqns.

SW eqns.

Porter & Silber, PRL (2002); Physica D (2004)

Travelling Wave eqns.

• Parameter (broken temporal) symmetries

time translation symmetry:

u=m denotes dominant driving frequency

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 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)

Zhang & Viñals, J. Fluid Mech 1997

Direct Reduction to Standing Wave eqns

k1

k2

Solvability condition at :

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:

Example: Experimental superlattice pattern

Topaz & Silber, Physica D (2002)

Kudrolli, Pier and Gollub, Physica D (1998)

Example: Experimental superlattice patternEpstein and Fineberg, 2005 preprint.

3:2

5:3

6/7/2 forcing frequencies:

Example: Experimental superlattice patternEpstein and Fineberg, 2005 preprint.

3:2

4:3

5:3

Example: Experimental quasipattern

Arbell & Fineberg, PRE, 2002

(3,2,4)forcing

(3,2)forcing

{}

Example: Impulsive-Forcing(See J. Bechhoefer & B. Johnson, Am. J. Phys. 1996)

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

Linear Theory: Shallow and Viscous CaseForcing function Neutral Curve

Huepe, Ding, Umbanhowar, Silber, 2005 preprint

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:

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…