Spreading under the action of:

-gravity-centrifugal force (spin coating)

- surface tension gradients

Contact Line Instability in Driven Films

Jennifer RieserRoman GrigorievMichael Schatz

School of Physics andCenter for Nonlinear Science

Georgia Institute of Technology

Contact Line Instability:Experiments and Theory

Supported by NSF and NASA

Transients & Hydrodynamic Transition

Controversy in Contact Line Problem

Important in Turbulent Transition?

Quantitative connection between experiment and theory

Optically-Driven Microflow

TContact line

Fluid flow

FLUID

The boundary conditions at the tail are different:experiment - constant volume

theory (slip model) - constant flux

Initial State (experiment and theory)

Fluid flow

1 mm

Silicone oil (100cS) on horizontal glass substrate

Contact line instability

Disturbance Amplitude(Ambient Perturbations)

log(A)

time (s)

Undisturbed system allows measurement of only the most unstable wavelength and the corresponding growth rate.

Numerous Previous Experiments:(Cazabat, et al. (1990), Kataoka & Troian (1999))

Optical Perturbations

Temperaturegradient

ResultantContact

LineDistortion(fingers)

Top view

Wavelengthof perturbation,

PerturbationThickness, w

Finger Formation

Wavenumber (2.5 mm1)

Time (s)

Disturbance amplitude (experiment)

log(A)Contact

LineDistortion

One Mechanism: Induce Transverse Counterflow to Suppress Instability

Other (Streamwise) Counterflow Mechanisms

Feedback control

Film mobility reduced by:*heating the front of the capillary ridge*cooling front and heating back of ridge

Effect of Feedback Depends onSpatial Profile

The feedback is applied on the right side of the film. On the left the film evolves under the action of a constant uniform temperature gradient.

Feedback control (experiment)

Slip model of thermally driven spreading

Non-dimensional evolution equation for thickness:

vz

pz v2hp 2

vv zh

3

2 3 2( )t xh h h h h

The boundary conditions at the tail are different:experiment - constant volume

theory (slip model) - constant flux

Initial State (experiment and theory)

Fluid flow

Dynamics of small disturbances, :

Linear stability

iqyo gehh

24

12

0)(,)( LqLqLqLgqLgt

)()( '0 xhxg o

1)(0 xf

0L

†0L

( ) is non-normal (not self-adjoint)L q

ASYMPTOTICGROWTH

time (s)

Growth rate

β0(q)

ContactLine

Distortion

ln(A)

MeasuringEigenvalues

Wavenumber (2.5 mm1)

Growth rates measured for externally imposed monochromatic initial disturbances with different wavenumbers.

Linear stability analysis correctly predicts most unstable wavenumber, but overpredicts growth rate by

about 40%

Dispersion curve

Transient Growth

TRANSIENTGROWTH

ASYMPTOTICGROWTH

NONLINEAREFFECTS

time (s)

Wavenumber (25 cm1)

Growth rate

β0(q)

ContactLine

Distortion

ln(A)

Linear operator L(q) not self-adjoint: L+(q) ≠L(q) The eigenvectors are not orthogonal

Transient Growth & Non-normality

Normal (eigenvalues<0) Norm

TimeNon-normal (eigenvalues<0) Norm

Time

Transient Growth & Non-normality

Non-normal (one positive eigenvalue)

L2 Norm

Time

Time

ln(A)

Transient Growth in Contact Lines:

Transient amplification: ~1000Nonlinear (Finite Amplitude) Rivulet formation possible

Gravitationally-Driven Spreading (Theory)

Bertozzi & Brenner (1997)Kondic & Bertozzi (1999)Ye & Chang (1999)

Davis & Troian (2003)Transient amplification: < 10

Thermally-Driven Spreading (Theory)

Grigoriev (2003)

Davis & Troian (2003)

Gravitationally-Driven Spreading (Experiments)

de Bruyn (1992)

Rivulets observed for “stable” parameter values

Ellingson & Palm (1975), Landhal (1980), Farrell (1988), Trefethan et al. (1993), Reshotko (2001), White (2002, 2003)

Chapman (2002), Hof, Juel & Mullin (2003)

Transient Growth in Turbulent Transition

+ (Eigenvalue) Linear stability fails in shear flows

Predicted transient amplification: 103-104

+ Mechanism for Bypass Transition

Transient Growthof Disturbances

Finite amplitudenonlinear instability

+ Importance still subject of controversy

+ Shear Flows are highly nonnormal

0, ( ,0)

0

|| ( , ) ||( ) exp[ ( ) ]max

|| ( ,0) ||

|| exp[ ( ) ( )] ||

pp

t h x p

p

h x tq q t

h x

L q q t

Optimal Transient AmplificationTheory

Transient Amplification Measurements

γexp ≡ e-t = e-t

A

f (A (tf ))hf

hi hi

hi

e 0

0

|| ( , ) ||( ) exp[ ( ) ]

|| ( ,0) ||

( ) || exp[ ( ) ( )] ||

x

th

h x tq q t

h x

q L q q t

Transient AmplificationTheory and Experiment

EXPERIMENT

Wave number

Modeling ExperimentalDisturbances

h

(m)

0

1

0.4 1.0 1.4

X(cm)

Localized Disturbancein Model

X

Y

Transient Amplification(Quantitative Comparison)

Optimal Transient Amplification (p norm)

(0) occurs as p t † †

0 0exp( (0) ) exp( (0) )i i ii

L t g t f g f

0OPTIMAL DISTURBANCE .f const

0 0|| || | | system size (X)p

p pf f dx † 2

0 0 0 0 0 2 1-

0 0

1|| || || || || ||(0) =

|| || || ||Xp p

pp p

pg f f g f

f f

In the limit Transient Amplification is

Arbitrarily Large

X

0( ) || exp[ ( ) ( )] ||p pq L q q t

Optimal Disturbance

For finite, transient growth stops ( )

For 0, disturbance decays due to surface tension

X t X

q

h

A X h t u t h

X

Grigoriev (2005)

+ Transient Growth in Contact linesQuantitative connections between theory and

experiment appear possible.

+ Work in Progress

Transient growth vs q

Transient growth in gravitationallydriven films

Summary