1. Contact Line Instability in Driven Films
Spreading under the action of:
gravity
centrifugal force (spin coating)
- surface tension gradients

2. Contact Line Instability: Experiments 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

3. & Hydrodynamic Transition
Transients
& Hydrodynamic Transition
Controversy in Contact Line Problem
Important in Turbulent Transition?
Quantitative connection
between experiment and theory

4. Optically-Driven Microflow
Fluid flow
FLUID
Contact line

5. Initial State (experiment and theory)
Fluid flow
The boundary conditions at the tail are different:
experiment - constant volume
theory (slip model) - constant flux

6. Contact line instability
1 mm
Silicone oil (100cS) on horizontal glass substrate

7. 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))

8. Optical Perturbations
Temperature
gradient
Top view
Resultant
Contact
Line
Distortion
(fingers)
Wavelength
of perturbation, l
Perturbation
Thickness, w

9. Finger Formation

10. Disturbance amplitude (experiment)
log(A)
Contact
Line
Distortion
Time (s)
Wavenumber (2.5 mm-1)

11. Feedback control Effect of Feedback Depends on Spatial Profile
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

12. 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.

14. Slip model of thermally driven spreading
Non-dimensional evolution equation for thickness:

15. Initial State (experiment and theory)
Fluid flow
The boundary conditions at the tail are different:
experiment - constant volume
theory (slip model) - constant flux

16. Linear stability
Dynamics of small disturbances, :

17. Measuring Eigenvalues
ln(A)
Contact
Line
Distortion
ASYMPTOTIC
GROWTH
Growth rate
β0(q)
time (s)
Wavenumber (2.5 mm-1)

18. 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%

19. Transient Growth ln(A) Contact Line Distortion time (s) β0(q)
NONLINEAR
EFFECTS
ASYMPTOTIC
GROWTH
Growth rate
β0(q)
time (s)
Wavenumber (25 cm-1)

20. Transient Growth & Non-normality
 Linear operator L(q) not self-adjoint: L+(q) ≠L(q)
 The eigenvectors are not orthogonal
Normal (eigenvalues<0)
Norm
Time
Non-normal (eigenvalues<0)
Norm
Time

21. Transient Growth & Non-normality
(one positive eigenvalue)
L2 Norm
Time
ln(A)
Time

22. 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: < 10
Thermally-Driven Spreading (Theory)
Davis & Troian (2003)
Grigoriev (2003)

23. 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:
+ Mechanism for Bypass Transition
Transient Growth
of Disturbances
Finite amplitude
nonlinear instability
+ Importance still subject of controversy

24. Optimal Transient Amplification Theory

25. Transient Amplification Measurements
dhf
f (A (tf ))
γexp ≡ e-bt = e-bt
dhi
dhi
dhi A

26. Transient Amplification Theory and Experiment
Wave number

27. Modeling Experimental Disturbances1
 h
(m)
0.4
1.0
1.4
X(cm)

28. Localized Disturbance in ModelY X

29. Transient Amplification (Quantitative Comparison)

30. Optimal Transient Amplification (p norm)
In the limit
Transient Amplification is
Arbitrarily Large

31. Optimal Disturbance Grigoriev (2005)
In this slide we describe the optimal disturbance with zero wavenumber as an example. A zero wavenumber perturbation corresponds to varying the film thickness uniformly throughout the film. The relaxation of such a disturbance results in driving the fluid front due to conservation of volume. In the upper figure the solid line describes the initially perturbed film. The volume added due to the perturbation, represented by A*delta-h equals the volume added by propagating the front as seen in the lower figure.
The transient amplification scales with the size of the disturbance, X.
Grigoriev (2005)

32. 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
In conclusion, we now have experimental techniques to allow for quantitative comparison with theoretical models for thin film flows. Indications of transient growth have been observed in experiments and found to be consistent with theoretical predictions underlining the importance of non-normal effects in such flows in regard to the contact line instability. The ability to impose optically induced disturbances aids in capturing the evolution of the system driven by tailored perturbations to the flow. Our current and future efforts are towards additional experimental measurements to compare with theoretical models and improve theoretical models to incorporate specific experimental effects such as optically induced perturbations.