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

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

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

Optically-Driven Microflow Fluid flow FLUID Contact line

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

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

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

Finger Formation

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

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

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.

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

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

Linear stability Dynamics of small disturbances, :

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

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%

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)

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

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

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)

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

Optimal Transient Amplification Theory

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

Transient Amplification Theory and Experiment Wave number

Modeling Experimental Disturbances 1  h (m) 0.4 1.0 1.4 X(cm)

Localized Disturbance in Model Y X

Transient Amplification (Quantitative Comparison)

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

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)

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.