The dynamics and linear stability of a liquid film flowing over a locally heated surface are studied using a long-wave lubrication analysis. The temperature gradient at the leading edge of the heater induces a gradient in surface tension that opposes the gravitationally driven flow and leads to the formation of a pronounced capillary ridge. The resulting free-surface shapes are computed, and their stability to spanwise perturbations is analyzed for a range of Marangoni numbers, substrate inclination angles, and temperature profiles. Instability is predicted above a critical Marangoni number for a finite band of wave numbers separated from zero, which is consistent with published results from experiment and direct numerical simulation. An energy analysis is used to gain insight into the effect of inclination angle on the instability. Because the spatial nonuniformity of the base state gives rise to nonnormal linearized operators that govern the evolution of perturbations, a nonmodal, transient analysis is used to determine the maximum amplification of small perturbations to the film. The structure of optimal perturbations of different wave numbers is computed to elucidate the regions of the film that are most sensitive to perturbations, which provides insight into ways to stabilize the flow. The results of this analysis are contrasted to those for noninertial coating flows over substrates with topographical features, which exhibit similar capillary ridges but are strongly stable to perturbations.

The dynamics and linear stability of a volatile liquid film flowing over a locally heated surface are investigated. The temperature gradient at the leading edge of the heater induces a gradient in surface tension that leads to the formation of a pronounced capillary ridge. Lubrication theory is used to develop a model for the film evolution that contains three key dimensionless groups: a Marangoni parameter ͑M͒, an evaporation number ͑E͒, and a measure of the vapor pressure driving force for evaporation ͑K͒, which behaves as an inverse Biot number. The two-dimensional, steady solutions for the local film thickness are computed as functions of these parameters. A linear stability analysis of these steady profiles with respect to perturbations in the spanwise direction reveals that the operator of the linearized system can have both a discrete and a continuous spectrum. The continuous spectrum exists for all values of the spanwise wave number and is always stable. The discrete spectrum, which corresponds to eigenfunctions localized around the ridge, appears for values of M larger than a critical value for a finite band of wave numbers separated from zero. Above a second, larger critical value of M, a portion of the discrete spectrum becomes unstable, corresponding to rivulet formation at the forward portion of the capillary ridge. For sufficiently large heat transfer at the free surface, due either to phase change or to convection, a second band of unstable discrete modes appears, which is associated with an oscillatory, thermocapillary instability above the heater. The critical Marangoni parameter above which instability develops, M crit ͑K , E͒, has a nonmonotonic dependence on the steepness of the temperature increase at the heater, in contrast to the monotonic decrease for a nonvolatile film at vanishing Biot number. An energy analysis reveals that the dominant instability mechanism resulting from perturbations to the film thickness is either streamwise capillary flow or gravity for weakly volatile fluids and thermocapillary flow due to spanwise temperature gradients for more volatile fluids. The stability results are rather sensitive to the steepness of the temperature increase and heater width due to the nonlinear coupling of gravity, capillary pressure gradients, thermocapillary flow, and evaporation through the base states.

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The stability of a thin, volatile liquid film falling under the influence of gravity over a locally heated, vertical plate is analyzed in the noninertial regime using a model based on long-wave theory. The model is formulated to account for evaporation that is either governed by thermodynamic considerations at the interface in the one-sided limit or limited by the rate of mass transfer of the vapor from the interface. The temperature gradient near the upstream edge of the heater induces a gradient in surface tension that opposes the gravity-driven flow, and a pronounced thermocapillary ridge develops in the streamwise direction. Recent theoretical analyses predict that the ridge becomes unstable above a critical value of the Marangoni parameter, leading to the experimentally observed rivulet structure that is periodic in the direction transverse to the bulk flow. An oscillatory, thermocapillary instability in the streamwise direction above the heater is also predicted for films with sufficiently large heat loss at the free surface due to either evaporation or strong convection in the adjoining gas. This present work extends the recent linear stability analysis of such flows by Tiwari and Davis ͓Phys. Fluids 21, 022105 ͑2009͔͒ to a nonmodal analysis of the governing non-self-adjoint operator and computations of the nonlinear dynamics. The nonmodal analysis identifies the most destabilizing perturbations to the film and their maximum amplification. Computations of the nonlinear dynamics reveal that small perturbations can be sufficient to destabilize a linearly stable film for a narrow band of wave numbers predicted by the nonmodal, linearized analysis. This destabilization is linked to the presence of stable, discrete modes that appear as the Marangoni parameter approaches the critical value at which the film becomes linearly unstable. Furthermore, the thermocapillary instability leads to a new, time-periodic base state. This transition corresponds to a Hopf bifurcation with increasing Marangoni parameter. A linear stability analysis of this time-periodic state reveals further instability to transverse perturbations, with the wave number of the most unstable mode about 50% smaller than for the rivulet instability of the steady base state and exponential growth rate about three times larger. The resulting film behavior is reminiscent of inertial waves on locally heated films, although the wave amplitude is larger in the present case near the heater and decays downstream where the Marangoni stress vanishes. The film's heat transfer coefficient is found to increase significantly upon the transition to the time-periodic flow.