Linear stability analysis of thin liquid film flow over a heterogeneously heated substrate

Abstract

The linear stability of a thin film of volatile liquid flowing over a surface with embedded, regularly spaced heaters is investigated. The temperature gradients at the upstream edges of the heaters induce gradients in surface tension that create a pronounced non-uniformity in the film profile due to the formation of capillary ridges. The Governing equations for the evolution of the film thickness are derived within the lubrication approximation, and three important parameters that affect the dynamics and stability of the film are identified. The computed two-dimensional, steady solutions for the local film thickness reveal that due to evaporation there is a slight change in the height of capillary ridge at subsequent heaters downstream. Using a linear stability analysis, it is shown that, as for a single heater, the film is susceptible to two types of instabilities. A rivulet instability leads to spanwise-periodic rivulets, and an oscillating thermocapillary instability leads to streamwise, time-periodic oscillations in the film thickness. The critical Marangoni number is calculated for both types of instability for a range of parameter values. The effect of the number of heaters, heater width, and gap between the heaters on the critical Marangoni number is computed and analyzed. For small evaporation rates and less volatile films, the presence of multiple heaters has almost no noticeable effect on the film stability. For larger evaporation rates and more volatile films, additional heaters decrease the Marangoni number at instability onset. The destabilizing effect of multiple heaters is sensitive to the heater geometry and spacing. Furthermore, the limitations of streamwise periodic boundary conditions for analyzing the stability of such flows are discussed. Computations on the transient and nonlinear growth of perturbations are also presented and indicate that the results of eigenanalysis are physically determinant.