(186af) Stability and Nonlinear Evolution of Coating Flows Over Locally Heated Surfaces
The dynamics and stability of a liquid film flowing over a locally-heated surface under the influence of a body force are analyzed using a long-wave analysis that accounts for inertial effects. The temperature gradient at the leading edge of the heater induces a gradient in surface tension, or Marangoni stress, that opposes the bulk flow and leads to the formation of a pronounced capillary ridge. The stability of the film to spanwise perturbations is analyzed for a range of Marangoni, Reynolds, Biot, and capillary numbers. A rivulet instability is predicted to develop above a critical Marangoni number (corresponding to a sufficiently large ridge) for a finite band of wavenumbers separated from zero. For films with appreciable heat loss from the free-surface (from convection or evaporation), an oscillatory, thermocapillary instability is also predicted. The competition between these two qualitatively different instabilities is investigated and found to lead to interesting nonlinear dynamics and pattern formation. Furthermore, for small inertia the thermocapillary instability leads to a bifurcation to a temporally periodic base state that is susceptible to a secondary instability, yielding wavy, oscillatory rivulets that can cause film rupture. The influence of the governing dimensionless groups on potentially chaotic, two-dimensional base states is studied, and the enhancement to heat transfer to the film for these flows is computed. Inertia is found to be destabilizing, and its influence on the nonlinear evolution of perturbations is explored.
Due to the non-normality in the governing linear operators for the stability analysis, a non-modal analysis is used to determine the transient evolution of perturbations to the film. This analysis predicts very large non-modal amplification of perturbations with a transverse wave number that corresponds to the intersection of the leading branches of the discrete and continuous spectra. Nonlinear simulations of the three-dimensional film evolution are performed for a linear combination of small perturbations of various wave numbers. These simulations reveal a finite amplitude instability for very small perturbations that leads to rivulets corresponding exactly to the wave number that experiences the largest non-modal amplification, which differs from the wave number corresponding to the largest eigenvalue. These results from the nonlinear simulations thus validate the findings of the non-modal analysis. As part of this analysis, the regions of the film that are most sensitive to perturbations are identified. The addition of topographical features to the substrate near these regions is explored as a mechanism of suppressing the instability through its influence on the capillary pressure gradient in the flowing film.