Oscillatory thermocapillary instability of a film heated by a thick substrate

Abstract

In this work we consider a new class of oscillatory instabilities that pertain to thermocapillary destabilization of a liquid film heated by a solid substrate. We assume the substrate thickness and substrate-film thermal conductivity ratio are large so that the effect of substrate thermal diffusion is retained at leading order in the long-wave approximation. As a result, system dynamics are described by a nonlinear partial differential equation for the film thickness that is nonlocally coupled to the full substrate heat equation. Perturbing about a steady quiescent state, we find that its stability is described by a non-self adjoint eigenvalue problem. We show that, under appropriate model parameters, the linearized eigenvalue problem admits complex eigenvalues that physically correspond to oscillatory (in time) instabilities of the thin film height. As the principal results of our work, we provide a complete picture of the susceptibility to oscillatory instabilities for different model parameters. Using this description, we conclude that oscillatory instabilities are more relevant experimentally for films heated by insulating substrates. Furthermore, we show that oscillatory instability where the fastest-growing (most unstable) wavenumber is complex, arises only for systems with sufficiently large substrate thicknesses.1 oscillatory instability can be a challenging task. Alternatively, the long wave approximation offers a convenient means to couple free surface deformation to other time-dependent physical processes of interest.Several authors have investigated oscillatory instabilities of thin liquid films in the context of the long-wave approximation. In many cases, e.g. Podolny et al. (2005) and Bestehorn and Borcia (2010), such instabilities originate from the coupling between the local thickness and bulk concentration of a film composed of a binary mixture. In addition to the bulk concentration dynamics, Morozov et al. (2014) investigated oscillatory instability with the added effect of absorption/desorption kinetics between interfacial and bulk film surfactant concentration. In other cases, oscillatory instabilities have been uncovered in multiple stacked layers of films, as described theoretically by coupled sets of film thickness evolution equations (Nepomnyashchy and Simanovskii (2007), Beerman and Brush (2007)). Multi-layer film configurations do not, however, guarantee oscillatory modes: for example, such instabilities were not obtained by Pototsky et al. (2005) who investigated the dewetting dynamics of isothermal, ultrathin bilayers. Of particular interest to the present work are oscillatory instabilities reported by Shklyaev et al. (2012) in a model of thin-film thermocapillary destabilization from below. While there are similarities between that work and the present, we point out one important difference: in Shklyaev et al. (2012), the instability is driven by imposing a heat flux at the film-substrate interface; instead, in the present work we consider the full time-dependent heat-transfer in the ...

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