Controlling Faraday wave instabilities Controlling Faraday wave instabilities in viscous shallow fluids through the in viscous shallow fluids through the shape of the forcing functionshape of the forcing function

CristiCristiáán Huepen Huepe Unaffiliated NSF Grantee - Chicago, IL. USAUnaffiliated NSF Grantee - Chicago, IL. USA

CollaboratorsCollaborators Yu Ding, Paul Umbanhowar and Mary Silber, Yu Ding, Paul Umbanhowar and Mary Silber,

Northwestern UniversityNorthwestern University

Work supported by NASA Grant No NAG3-2364 and NSF Grants No.DMS-0309667 & DMS-0507745.

________________________________________________________________

OutlineOutline

Introduction and MotivationIntroduction and Motivation– Faraday waves in viscous shallow fluidsFaraday waves in viscous shallow fluids– Shape of the linear neutral stability curvesShape of the linear neutral stability curves

Numerical and Experimental AnalysisNumerical and Experimental Analysis– Multi-frequency forcing functionMulti-frequency forcing function– Nontrivial bi-critical pointsNontrivial bi-critical points

WKB Approximation in Lubrication RegimeWKB Approximation in Lubrication Regime

– DerivationDerivation– Envelope analysisEnvelope analysis

Controlling faraday wave instabilities in viscous shallow fluids Controlling faraday wave instabilities in viscous shallow fluids CRM Montreal - 2007 CRM Montreal - 2007

- Faraday waves can produce a rich variety of surface patterns.

- Fluid parameters:

- Patterns (& quasi-patterns) depend on the forcing function:

Introduction and MotivationIntroduction and Motivation

(Images from Jerry P. Gollub’s Haverford College web site.)

h,,,

tf

z

yx

g-h

0

),,( tyx

,

0

),(),,( txptxv

tf

System is described by:Navier-Stokes equation

Kinematic condition & force balance at surface

Linear equations for and are found, where

Navier-Stokes Faraday-Wave SolutionsNavier-Stokes Faraday-Wave Solutions

vztfgpvvvt 2ˆ)(

1)(

yyxxtzz

vvv

yx

z RRpp

110

tzueu zykxki

zyx ,~

tzuz ,~ t~

te ykxki yx ~

The linearized Navier-Stokes (N-S) equation for the z-dependence The linearized Navier-Stokes (N-S) equation for the z-dependence of the vertical component of the fluid velocity becomes:of the vertical component of the fluid velocity becomes:

With boundary conditionsWith boundary conditions– At z=-h:At z=-h:

– At z=0At z=0

This fully describes the dynamics of the systemThis fully describes the dynamics of the system

We are interested in neutral stability curvesWe are interested in neutral stability curves

0kv 0 kz v

kkt v 02 kzz vk

kkzzzt kktfgvkk 2222 /)(12

022 kzzzzt vkk

kc

Numerically, we expand and in a Floquet form Numerically, we expand and in a Floquet form {Kumar & Tuckerman [J. Fluid Mech. {Kumar & Tuckerman [J. Fluid Mech. 279279, 49 (1994)]}:, 49 (1994)]}:

Marginal stability:Marginal stability: Harmonic & subharmonic responses: &Harmonic & subharmonic responses: &

The system is reduced to:The system is reduced to:

where is an algebraic expression independent of where is an algebraic expression independent of

and is the n-th Fourier component ofand is the n-th Fourier component of

..)( ccezwevtji

jj

tik

k

0

nnn fA

..)( cceetji

jj

tik

0 2/

tji

jjetf

nA

nf

tf

Summarizing…Summarizing…

We find the linear neutral stability conditions using:We find the linear neutral stability conditions using:

– Standard linearized Navier-Stokes formulationStandard linearized Navier-Stokes formulation

– Free boundary conditions at surfaceFree boundary conditions at surface

– Idealized laterally infinite containerIdealized laterally infinite container

– Finite depthFinite depth

We find an We find an eigenvalue expressioneigenvalue expression for the for the critical forcing accelerationcritical forcing acceleration by by extending the numerical linear stability analysis by Kumar & Tuckerman to extending the numerical linear stability analysis by Kumar & Tuckerman to arbitrary forcing functionsarbitrary forcing functions

We compute neutral stability curves:We compute neutral stability curves:

– Critical accelerationCritical acceleration at which each at which each wavenumber wavenumber becomes unstable becomes unstable

k

Motivation…Motivation…

Shallow & viscous(sinusoidal forcing)

Shallow & viscous(multi-frequency forcing)

Study shallow & viscous caseStudy shallow & viscous case Study multi-frequency (delta-like) forcingStudy multi-frequency (delta-like) forcing ““Tongue envelopes” appearTongue envelopes” appear

[Bechhoefer and Johnson, American Journal of Physics,

1996]

We define an “arbitrary” one-parameter family of forcing functions by:We define an “arbitrary” one-parameter family of forcing functions by:

As p grows, the forcing function changes as:As p grows, the forcing function changes as:

tttNtf ppp 5cos53cos3cos5.2

~

Numerical & Experimental Study of Envelopes Numerical & Experimental Study of Envelopes

Analysis of Envelopes for “our” forcing functionAnalysis of Envelopes for “our” forcing function

Fixed parameters:Fixed parameters:

)10(2

3.0

/20

46

/95.0 3

Hz

cmh

cmdyn

cS

cmg

(a) p=-2, (b) p=-0.3, (c) p=0.5, (d) p =1

Experimental ResultsExperimental Results

Close to p=1 we can predict a Close to p=1 we can predict a dramatic change in pattern for a dramatic change in pattern for a small variation of the forcing.small variation of the forcing.

– From 1st subharmonic to 2nd From 1st subharmonic to 2nd harmonic tongueharmonic tongue

– For p=1.1: instability of 2For p=1.1: instability of 2ndnd harmonic tongue, which is not a harmonic tongue, which is not a fundamental harmonic or fundamental harmonic or subharmonic response to any of subharmonic response to any of the three frequency components the three frequency components

(top) p=0.9, (center) p=1.0, (bottom) p=1.1

tttN

tfpp

p

5cos53cos3cos5.2~

My first experiment…My first experiment…

p=0.9 p=1.1

Experimental limitations:Experimental limitations:

– To excite higher tongues we need very low values of orTo excite higher tongues we need very low values of or

– These are limited by experimental setupThese are limited by experimental setup

For low h spurious effects may affect patternsFor low h spurious effects may affect patterns For low omega, the maximum oscillation amplitude (prop. to ) For low omega, the maximum oscillation amplitude (prop. to ) Larger patterns (lower unstable k) would require larger containerLarger patterns (lower unstable k) would require larger container

Image sizes: 8.22cm x 8.22cm

Fluid parameters: Same as in numerical calculations

h

2

Can we understand analytically the origin of the “tongue envelopes” Can we understand analytically the origin of the “tongue envelopes” that cause these nontrivial instabilities?that cause these nontrivial instabilities?

Analytical approximation 1: Analytical approximation 1: Lubrication regimeLubrication regime

– Small ratio between and terms in Navier-Stokes equationSmall ratio between and terms in Navier-Stokes equation

– Ratio is of order , with:Ratio is of order , with:

– Lubrication approximation valid for fluids that are shallow and viscous Lubrication approximation valid for fluids that are shallow and viscous enough, with low oscillation frequencyenough, with low oscillation frequency

WKB Approximation in Lubrication RegimeWKB Approximation in Lubrication Regime

vt v

2 2/l

/ toprop.

)1 (if or )1 if(/1~

layerboundary of size sticcharacteri

surface penetratesmotion fluid that distance~

hkhhkk

l

The Lubrication ApproximationThe Lubrication Approximation

Approximate analytic description Approximate analytic description

[Cerda & Tirapegui, Beyer & Friedrich][Cerda & Tirapegui, Beyer & Friedrich]

– Only involves:Only involves:

– Leads to damped Mathieu equation:Leads to damped Mathieu equation:

withwith hkBhkBkk 21

2

hkBkgkk 222 /

0)(12 2 kkkkkk tf

/2kg

gk

rkik etz

The WKB Approximation (1 of 3)The WKB Approximation (1 of 3)

We write the damped Mathieu equation as a ScrhWe write the damped Mathieu equation as a Scrhödinger equationödinger equation

– Defining:Defining:

Time becomes space Time becomes space (N.B.: Not a metaphysical statement)(N.B.: Not a metaphysical statement)

– We obtain:We obtain:

withwith

Neutral stability solutions of damped Mathieu equation = Neutral stability solutions of damped Mathieu equation = Eigenfunctions of ScrhEigenfunctions of Scrhödinger equation with boundary conditionödinger equation with boundary condition

[Cerda & Tirapegui: J. Fluid Mech., 368,195-228, 1998]

// x

kkexx

tx

22kkE

021 xxVEx

xfxV kk2

xex ki /22

The The Wentzel-Kramers-BrillouinWentzel-Kramers-Brillouin approximation is valid in lubrication regime approximation is valid in lubrication regime since it is an expansion in the small quantity:since it is an expansion in the small quantity:

The solutions are:The solutions are:

with with

The WKB Approximation (2 of 3)The WKB Approximation (2 of 3)

4

2

22

~~

l

xVE k

xdxPDxdxPCxP

x

xdxPBxdxPAxP

x

x

x

x

x

x

x

x

x

~~cos~~sin1

~~exp~~exp1

00

00

/xVExP

E<V(x)

E>V(x)

The WKB Approximation (3 of 3)The WKB Approximation (3 of 3)

The WKB matching conditions are given by:The WKB matching conditions are given by:

withwith

andand

The neutral stability condition becomes:The neutral stability condition becomes:

/2cosh2

Tr,

k

MkQ

j

jj

j

j

B

AM

B

A

1

1

2/)cos()sin(

)sin()cos(2

jj

jjj jj

jj

ee

eeM

dxxPj

j

b

aj ~ dxxPj

j

a

bj 1 ~

Envelope AnalysisEnvelope Analysis

/2cosh

cos2logcosh,

k

aa

a kQ

21

2121

,

,

CCHkQ

SSHCCHkQ

Cd

SCd

p = -2 p = 1

p = -2:p = -2:The instability tongue The instability tongue envelope can only envelope can only have one minimum.have one minimum.

p = 1: p = 1: The instability tongue The instability tongue envelope has multiple envelope has multiple minima.minima.

FinFin A WKB method relating the linear surface wave instabilities of a A WKB method relating the linear surface wave instabilities of a

shallow viscous fluid and the shape of its forcing function was shallow viscous fluid and the shape of its forcing function was presented.presented.

Conjecture: any forcing function with two extrema per cycle has neutral Conjecture: any forcing function with two extrema per cycle has neutral stability tongues with a single-minimum envelope.stability tongues with a single-minimum envelope.

Idea: Can we use piecewise-constant forcing to formulate the inverse Idea: Can we use piecewise-constant forcing to formulate the inverse problem of finding the forcing shape required for a given instability.problem of finding the forcing shape required for a given instability.

Paper:Paper:Forcing function control of Faraday wave instabilities in viscous shallow fluidsForcing function control of Faraday wave instabilities in viscous shallow fluidsPhysical Review EPhysical Review E 73, 016310 (2006) 73, 016310 (2006)