(108b) Linear Stability And Transient Dynamics Of Non-Inertial Coating Flows Over Locally Heated Surfaces
The dynamics and linear stability of a liquid film flowing over a locally-heated surface under the influence of gravity are analyzed using a long-wave lubrication analysis. The temperature gradient at the leading edge of the heater induces a gradient in surface tension (Marangoni stress) that opposes the gravitationally-driven flow and leads to the formation of a pronounced capillary ridge. The shape of the free-surface near the heater is computed, and the stability of the film to spanwise perturbations is analyzed for a range of Marangoni numbers, substrate inclination angles, and temperature profiles corresponding to both semi-infinite and finite-width heaters. The amplitude of the capillary ridge increases with the magnitude of the temperature gradient but asymptotes to a constant value for very steep temperature increases. 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, which is consistent with published experimental results and direct numerical simulation. This rivulet instability can ultimately lead to film rupture and dry-out, which can restrict the use of liquid films on heated surfaces. As the substrate inclination angle is decreased from vertical, the wavelength of the instability increases, and the film becomes stable beyond a threshold inclination angle due to the increased hydrostatic component of the pressure. This behavior is similar to that of a liquid film spreading down an inclined plane, and an energy analysis is used to gain further insight into the instability. Because the spatial non-uniformity of the base state gives rise to non-normal linearized operators that govern the evolution of perturbations, a non-modal analysis is used to determine the transient evolution of perturbations to the film. The structure of optimal perturbations of different wavenumber is computed to elucidate the most sensitive regions of the film. Results are contrasted to those for non-inertial coating flows over substrates with topographical features, which exhibit similar capillary ridges but are strongly stable to perturbations. The analysis is also extended to volatile films, which can undergo a thermocapillary instability that competes with the rivulet instability and leads to interesting dynamics, including film rupture that produces moving contact lines and subsequent film evolution.