Controlling Faraday waves with multifrequency forcing Mary Silber
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: 5 cm, depth: 3 mm
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 k 1 + k 2 = k 3 k 2 qres k 3 k 1 Resonant triads & Faraday waves: Müller, Edwards & Fauve, Zhang & Viñals, …
Resonant triads • Role in pattern selection: a simple example k 2 k 3 critical modes damped mode qres k 1 spatial translation, reflection, rotation by p
Resonant triads • Role in pattern selection: a simple example k 2 k 3 critical modes damped mode (eliminate) qres k 1 center manifold reduction
Resonant triads • Role in pattern selection: a simple example k 2 “suppressing”, “competitive” “enhancing”, “cooperative” qres k 1 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 W Perform center manifold reduction to SW eqns. Porter, Topaz and Silber, PRL & PRE 2004
Key results • 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 = 2 n – 2 m) forcing, W = 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 k 2 q k 1 Solvability condition at :
Demonstration of key results • 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 > 0) ex. (6, 7, 2) forcing, b( )computed from Zhang-Viñals equations: W = n – m = 1 W = 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…
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