Contact Line Instability in Driven Films

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Contact Line Instability in Driven Films

• Spreading under the action of
• gravity
• centrifugal force (spin coating)
• - surface tension gradients

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Contact Line InstabilityExperiments and Theory
Jennifer Rieser Roman Grigoriev Michael Schatz
School of Physics and Center for Nonlinear
Science Georgia Institute of Technology
Supported by NSF and NASA
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Transients Hydrodynamic Transition
Controversy in Contact Line Problem Important
in Turbulent Transition? Quantitative
connection between experiment and theory
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Optically-Driven Microflow
Fluid flow
FLUID
Contact line
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Initial State (experiment and theory)
Fluid flow
The boundary conditions at the tail are
different experiment - constant volume theory
(slip model) - constant flux
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Contact line instability
1 mm
Silicone oil (100cS) on horizontal glass
substrate
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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))
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Optical Perturbations
Temperature gradient
Top view
Resultant Contact Line Distortion (fingers)
Wavelength of perturbation, l
Perturbation Thickness, w
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Finger Formation
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Disturbance amplitude (experiment)
log(A)
Contact Line Distortion
Time (s)
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Feedback control
One Mechanism Induce Transverse Counterflow to
Suppress Instability
Other (Streamwise) Counterflow Mechanisms
Film mobility reduced by heating the front of
the capillary ridge cooling front and heating
back of ridge
Effect of Feedback Depends on Spatial Profile
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Feedback control (experiment)
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.
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(No Transcript)
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Slip model of thermally driven spreading
f
Non-dimensional evolution equation for thickness
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Initial State (experiment and theory)
Fluid flow
The boundary conditions at the tail are
different experiment - constant volume theory
(slip model) - constant flux
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Linear stability
Dynamics of small disturbances,

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MeasuringEigenvalues
ln(A)
Contact Line Distortion
ASYMPTOTIC GROWTH
Growth rate ß0(q)
time (s)
Wavenumber (2.5 mm-1)
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Dispersion curve

• 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

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Transient Growth
ln(A)
Contact Line Distortion
TRANSIENT GROWTH
ASYMPTOTIC GROWTH
Growth rate ß0(q)
time (s)
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Transient Growth Non-normality
? Linear operator L(q) not self-adjoint L(q)
?L(q) ? The eigenvectors are not orthogonal
Normal (eigenvalueslt0)
Norm
Time
Non-normal (eigenvalueslt0)
Norm
Time
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Transient Growth Non-normality
Non-normal (one positive eigenvalue)
L2 Norm
Time
ln(A)
Time
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Transient Growth in Contact Lines
Gravitationally-Driven Spreading (Experiments)
de Bruyn (1992)
Rivulets observed for stable parameter values
Gravitationally-Driven Spreading (Theory)
Bertozzi Brenner (1997) Kondic Bertozzi
(1999) Ye Chang (1999)
Transient amplification 1000 Nonlinear (Finite
Amplitude) Rivulet formation possible
Davis Troian (2003)
Transient amplification lt 10
Thermally-Driven Spreading (Theory)
Davis Troian (2003)
Grigoriev (2003)
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Transient Growth in Turbulent Transition
Ellingson Palm (1975), Landhal (1980), Farrell
(1988), Trefethan et al. (1993), Reshotko
(2001), White (2002, 2003)Chapman (2002), Hof,
Juel Mullin (2003)
(Eigenvalue) Linear stability fails in shear
flows
Shear Flows are highly nonnormal
Predicted transient amplification 103-104
Mechanism for Bypass Transition
Transient Growth of Disturbances
Finite amplitude nonlinear instability
Importance still subject of controversy
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Optimal Transient AmplificationTheory
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Transient Amplification Measurements
f (A (tf ))
dhf
?exp e-bt
e-bt
dhi
dhi
dhi
A
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Transient AmplificationTheory and Experiment
EXPERIMENT
Wave number
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Modeling ExperimentalDisturbances
1
? h (?m)
0
0.4
1.0
1.4
X(cm)
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Localized Disturbancein Model
Y
X
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Transient Amplification(Quantitative Comparison)

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Optimal Transient Amplification (p norm)
In the limit Transient Amplification
is Arbitrarily Large
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Optimal Disturbance
Grigoriev (2005)
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Summary
Transient Growth in Contact
lines Quantitative connections between theory
and experiment appear possible. Work in
Progress Transient growth vs q Transient
growth in gravitationally driven films