Abstract
Accurate low-dimensional models for the dynamics of falling liquid films subject to localized or time-varying heating are essential for applications that involve patterning or control. However, existing modelling methodologies either fail to respect fundamental thermodynamic properties or else do not accurately capture the effects of advection and diffusion on the temperature profile. We argue that the best-performing long-wave models are those that give the surface temperature implicitly as the solution of an evolution equation in which the wall temperature alone (and none of its derivatives) appears as a source term. We show that, for both flat and non-uniform films, such a model can be rationally derived by expanding the temperature field about its free-surface values. We test this model in linear and nonlinear regimes, and show that its predictions are in remarkable quantitative agreement with full Navier–Stokes calculations regarding the surface temperature, the internal temperature field and the surface displacement that would result from temperature-induced Marangoni stresses.
Highlights
This study is motivated by the desire to utilize active feedback controls to manipulate the inertially driven instabilities of a falling liquid film
With all other parameters fixed, we find that increasing Ma increases the surface deformation h, accompanied by only small changes in the surface temperature profile S
We have explored in detail the formulation of long-wave models for predicting the effect of localized wall heating on surface temperature, and via variation of surface tension with temperature, on the dynamics of falling liquid films
Summary
This study is motivated by the desire to utilize active feedback controls to manipulate the inertially driven instabilities of a falling liquid film. Thompson et al (2016a) showed that observations of the film surface shape combined with actuation applied by same-fluid blowing and suction through the rigid wall supporting the flow could be used to apply feedback control to a falling film, allowing stabilization of a wide range of states, including a uniform film, unstable travelling waves and even arbitrary states which would not be solutions of the original. Practical implementation of feedback via blowing and suction through the supporting wall presents significant challenges in terms of delivering well-controlled, real-time inputs, and would likely require invasive modifications of any industrial process. A more appealing method of applying feedback is presented by the possibility of applying time-dependent and spatially varying heating of the solid wall, again in response to observations, using a system such as illustrated in figure 1. The applied heating would affect the hydrodynamic flow through the temperature dependence of physical properties such as surface tension
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