1 / 37

Controlling Faraday Waves with Multi-Frequency Forcing for Pattern Selection in Lab Experiments

Discover how different forcing function parameters influence weakly nonlinear wave interactions in the Faraday system, aiding in interpreting experimental results and designing optimal lab experiment strategies. By using equivariant bifurcation theory and focusing on resonant triads, explore the vast control parameter space and enhance pattern formation. Collaborating with experts, this study leverages Hamiltonian structure and spatial symmetries for a deeper understanding of spatio-temporal patterns.

hsimonson
Download Presentation

Controlling Faraday Waves with Multi-Frequency Forcing for Pattern Selection in Lab Experiments

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 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)

  2. FARADAY CRISPATIONS – M. Faraday, Phil. Trans. R. Soc. Lond. (1831)

  3. FARADAY CRISPATIONS – M. Faraday, Phil. Trans. R. Soc. Lond. (1831)

  4. Edwards and Fauve, JFM (1994) 12-fold quasipattern Bordeaux to Geneva: 5cm, depth: 3mm

  5. 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)

  6. Arbell & Fineberg, PRE 2002

  7. FARADAY CRISPATIONS

  8. LINEARSTABILITY 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

  9. MATHIEU EQUATION Subharmonic resonance From: Jordan & Smith

  10. Unique capabilities of the Faraday system • 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)

  11. 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.

  12. Resonant triads • 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,…

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

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

  15. Resonant triads • Role in pattern selection: a simple example k2 “suppressing”, “competitive” “enhancing”, “cooperative” qres k1 rhombic equations:

  16. consider free energy: nonlinear coupling coefficient:

  17. Organizing Center forcing Expanded TW eqns. Hamiltonian structure SW eqns. time translation, time reversal symmetries damping Porter & Silber, PRL (2002); Physica D (2004)

  18. Travelling Wave eqns. • Parameter (broken temporal) symmetries u=m denotes dominant driving frequency time translation symmetry:

  19. Travelling Wave eqns. • Parameter (broken temporal) symmetries time reversal symmetry: Hamiltonian structure (for ): (See, e.g., Miles,JFM (1984))

  20. Travelling Wave eqns. • Enforce symmetries  Travelling wave amplitude equations damping parametric forcing damping

  21. Travelling Wave eqns. Time translation invariants: • Enforce symmetries  Travelling wave amplitude equations Example: (m,n) forcing, =m-n

  22. Travelling wave eqns. Focus on Possible only for At most 5 relevant forcing frequencies for fixed W Perform center manifold reduction to SW eqns. • Enforce symmetries  Travelling wave amplitude equations Porter, Topaz and Silber, PRL & PRE 2004

  23. 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 = 2n – 2m) forcing, W = n – m>0 > 0 Pp(F) > 0 bres > 0 for this case(can get signs for some other cases)

  24. Zhang & Viñals, J. Fluid Mech 1997

  25. Direct Reduction to Standing Wave eqns k2 q k1 Solvability condition at :

  26. 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

  27. Example: Experimental superlattice pattern Kudrolli, Pier and Gollub, Physica D (1998) Topaz & Silber, Physica D (2002)

  28. Example: Experimental superlattice pattern 6/7/2 forcing frequencies: Epstein and Fineberg, 2005 preprint. 3:2 5:3

  29. Example: Experimental superlattice pattern Epstein and Fineberg, 2005 preprint. 3:2 4:3 5:3

  30. Example: Experimental quasipattern q = 45 Arbell & Fineberg, PRE, 2002 { (3,2,4)forcing } (3,2)forcing

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

  32. Example: Impulsive-Forcing (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:

  33. Linear Theory: Shallow and Viscous Case Huepe, Ding, Umbanhowar, Silber, 2005 preprint Forcing function Neutral Curve

  34. 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

  35. Linear Theory: Shallow and Viscous Case Exact numerical: WKB approximation:

  36. 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…

More Related