Controlling Faraday wave instabilities in viscous shallow fluids through the shape of the forcing function Cristián Huepe Unaffiliated NSF Grantee - Chicago, IL. USA Collaborators Yu Ding, Paul Umbanhowar and Mary Silber, Northwestern University Work supported by NASA Grant No NAG and NSF Grants No.DMS & DMS ________________________________________________________________

Outline Outline  Introduction and Motivation –Faraday waves in viscous shallow fluids –Shape of the linear neutral stability curves  Numerical and Experimental Analysis –Multi-frequency forcing function –Nontrivial bi-critical points  WKB Approximation in Lubrication Regime –Derivation –Envelope analysis

- Faraday waves can produce a rich variety of surface patterns. - Fluid parameters: - Patterns (& quasi-patterns) depend on the forcing function: Introduction and Motivation (Images from Jerry P. Gollub’s Haverford College web site.) z y x g -h 0

System is described by: Navier-Stokes equation Kinematic condition & force balance at surface Linear equations for and are found, where Navier-Stokes Faraday-Wave Solutions

 The linearized Navier-Stokes (N-S) equation for the z-dependence of the vertical component of the fluid velocity becomes:  With boundary conditions –At z=-h: –At z=0  This fully describes the dynamics of the system  We are interested in neutral stability curves

 Numerically, we expand and in a Floquet form {Kumar & Tuckerman [J. Fluid Mech. 279, 49 (1994)]}:  Marginal stability:  Harmonic & subharmonic responses: &  The system is reduced to: where is an algebraic expression independent of and is the n-th Fourier component of

Summarizing…  We find the linear neutral stability conditions using: –Standard linearized Navier-Stokes formulation –Free boundary conditions at surface –Idealized laterally infinite container –Finite depth  We find an eigenvalue expression for the critical forcing acceleration by extending the numerical linear stability analysis by Kumar & Tuckerman to arbitrary forcing functions  We compute neutral stability curves: –Critical acceleration at which each wavenumber becomes unstable

Motivation… Shallow & viscous (sinusoidal forcing) Shallow & viscous (multi-frequency forcing)  Study shallow & viscous case  Study multi-frequency (delta-like) forcing  “Tongue envelopes” appear [Bechhoefer and Johnson, American Journal of Physics, 1996]

 We define an “arbitrary” one-parameter family of forcing functions by:  As p grows, the forcing function changes as: Numerical & Experimental Study of Envelopes

Analysis of Envelopes for “our” forcing function Fixed parameters: (a) p=-2, (b) p=-0.3, (c) p=0.5, (d) p =1

Experimental Results  Close to p=1 we can predict a dramatic change in pattern for a small variation of the forcing. –From 1st subharmonic to 2nd harmonic tongue –For p=1.1: instability of 2 nd harmonic tongue, which is not a fundamental harmonic or subharmonic response to any of the three frequency components (top) p=0.9, (center) p=1.0, (bottom) p=1.1

My first experiment… p=0.9 p=1.1  Experimental limitations: –To excite higher tongues we need very low values of or –These are limited by experimental setup  For low h spurious effects may affect patterns  For low omega, the maximum oscillation amplitude (prop. to )  Larger patterns (lower unstable k) would require larger container Image sizes: 8.22cm x 8.22cm Fluid parameters: Same as in numerical calculations

 Can we understand analytically the origin of the “tongue envelopes” that cause these nontrivial instabilities?  Analytical approximation 1: Lubrication regime –Small ratio between and terms in Navier-Stokes equation –Ratio is of order, with: –Lubrication approximation valid for fluids that are shallow and viscous enough, with low oscillation frequency WKB Approximation in Lubrication Regime

The Lubrication Approximation  Approximate analytic description [Cerda & Tirapegui, Beyer & Friedrich] –Only involves: –Leads to damped Mathieu equation: with

The WKB Approximation (1 of 3)  We write the damped Mathieu equation as a Scrhödinger equation –Defining: Time becomes space (N.B.: Not a metaphysical statement) –We obtain: with  Neutral stability solutions of damped Mathieu equation = Eigenfunctions of Scrhödinger equation with boundary condition [Cerda & Tirapegui: J. Fluid Mech., 368, , 1998 ]

 The Wentzel-Kramers-Brillouin approximation is valid in lubrication regime since it is an expansion in the small quantity:  The solutions are: with The WKB Approximation (2 of 3) E<V(x) E>V(x)

The WKB Approximation (3 of 3)  The WKB matching conditions are given by: withand  The neutral stability condition becomes:

Envelope Analysis p = -2p = 1  p = -2: The instability tongue envelope can only have one minimum.  p = 1: The instability tongue envelope has multiple minima.

Fin  A WKB method relating the linear surface wave instabilities of a shallow viscous fluid and the shape of its forcing function was presented.  Conjecture: any forcing function with two extrema per cycle has neutral stability tongues with a single-minimum envelope.  Idea: Can we use piecewise-constant forcing to formulate the inverse problem of finding the forcing shape required for a given instability.  Paper: Forcing function control of Faraday wave instabilities in viscous shallow fluids Physical Review E 73, (2006)