Nonmodal and nonlinear dynamics of a volatile liquid film flowing over a locally heated surface

Abstract

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.

Stabilization of thin liquid films flowing over locally heated surfaces via substrate topography

“…A detailed investigation of liquid films flowing over locally heated flat surfaces based on Eqs. ͑3͒ and ͑6͒ has been the focus of previous studies, [24][25][26] including base profiles, linear stability analysis, and computations of the non-modal and nonlinear dynamics. Also discussed in these studies are comparisons of the base states and dispersion curves based on the model used here for flat surfaces to those computed in previous studies 20,21 that differ primarily in the choice of boundary conditions for the linear stability analysis, the temperature profile, and some parameter values.…”

“…Further details on the reduction of the NavierStokes and energy equations can be found in previous work. 24,47 For consistency with some previous studies of instabilities in thin liquid films flowing over locally heated, planar surfaces, [24][25][26] the temperature profile along the substrate is modeled as…”

“…[19][20][21][22][23][24][25][26] In particular, for films flowing under the influence of gravity over an inclined surface containing a rectangular heater, a thermocapillary ridge develops at the upstream edge of the heater where the Marangoni stress opposes the bulk flow. This ridge breaks up into an array of regular, spanwise rivulets above a critical value of the temperature increase at the heater 27,28 ͑characterized by a dimensionless Marangoni parameter that quantifies the gradient in surface tension͒.…”

Stability of a localized heated falling film with insoluble surfactants

“…Oscillatory unstable mode and rivulet unstable mode were discussed [10]. Furthermore, Tiwari and Davis [11] studied the non-modal stability [10]. In their paper [11], they identified the optimal perturbations and the maximum amplification so as to understand the energy transfer from the base state to disturbances.…”

“…Furthermore, Tiwari and Davis [11] studied the non-modal stability [10]. In their paper [11], they identified the optimal perturbations and the maximum amplification so as to understand the energy transfer from the base state to disturbances. They reported that the rupture region was between the thermocapillary ridge and rivulet due to evaporation.…”