We study the dynamics of a thin liquid film on a compliant substrate in the presence of thermocapillary effect. A set of long-wave equations are derived to investigate the effects of fluid gravity $(G)$, fluid inertia (Re), and Marangoni stresses (Ma) on the dynamics of the liquid film and the compliant substrate. By performing linear stability analysis and time-dependent computations of the long-wave equations, we examine two different cases: thin-film flows on a horizontally compliant substrate $(\ensuremath{\beta}=0$, where $\ensuremath{\beta}$ is the inclined angle) and down a vertically compliant substrate $(\ensuremath{\beta}=\ensuremath{\pi}/2)$, respectively. For $\ensuremath{\beta}=0$, we neglect fluid inertia and identify two different modes: (1) sinuous mode, where the deformations of liquid-air and liquid-substrate interfaces are in phase, which is induced by the fluid gravity, and (2) varicose mode, where the deformations of two interfaces are in phase opposition, which is induced by the Marangoni stresses. For $\ensuremath{\beta}=\ensuremath{\pi}/2$, we consider a weak fluid inertia and only observe the varicose mode driven by fluid inertia and Marangoni stresses. However, because the gravity direction is parallel to the substrate, the fluid gravity modifies the varicose mode, making the deformations of two interfaces out of phase. In particular, we also seek the nonlinear traveling-wave solutions in the case of $\ensuremath{\beta}=\ensuremath{\pi}/2$, revealing that fluid inertia and/or heating effect enhance the height and speed of the traveling waves. In both cases, the introduction of a strong wall heating gives rise to large deformations of both the thin liquid film and the compliant substrate.

Thermocapillary thin-film flows on a compliant substrate.

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Comment on ”low-dimensional models for vertically falling viscous films” C. Ruyer-Quil, Paul Manneville

Comment on: Low-dimensional models for vertically falling viscous films. Authors' reply

We investigate the stability of a thin film flow on a compliant substrate with broken time-reversal symmetry. The broken time-reversal symmetry induces odd viscosity in the liquid. Using coupled equations encompassing the film and substrate, we ascertain that inclusion of a compliant substrate induces instability within the system. Nonetheless, the introduction of odd viscosity notably enhances its stability. Furthermore, under conditions of weak nonlinearity, the incorporation of odd viscosity can effectively mitigate the occurrence of chaotic behavior arising from compliant substrates. Numerical simulations confirm that odd viscosity has a substantial impact on substrate deflection. 

Role of odd viscosity on falling films over compliant substrates

Despite the extensive literature on liquid falling films, the majority of them have focused on clean falling films on rigid walls. In this study, the falling film contaminated with insoluble surfactant on a flexible substrate is considered in the numerical simulations. The flow dynamics of film is studied by using the Navier–Stokes equations coupled with interface tracking approach. The surfactant concentration is calculated according to the ratio between surfactant mass and the surface area of film. The fully developed solitary-like wave accompanied with front capillary ripples is obtained. The results show that the presences of interfacial surfactant and flexible substrate have a considerable influence upon the dynamics of falling film. The surfactant, which acts to reduce the celerity of interfacial wave, can suppress the dispersion as well. The wall damping plays a role to deplete the deformation energy of short-wavelength deflection. As the wall damping coefficient increases, the proceeding capillary ripples of the interfacial humps can be suppressed, and the secondary circulation flow is intensified. The wall tension, which can inhibit the flexible wall fluctuation, tends to strengthen the proceeding capillary ripples and promote the celerity of interfacial solitary-like wave.

Numerical study of the surfactant-covered falling film flowing down a flexible wall

Microscopic mechanisms of wave effect on heat transfer enhancement in condensate films

Coherent wave structures on falling fluid films flowing down a flexible wall

Using an approximate method, families of non-linear steady-traveling periodic waves in a two-layer falling film have been found for the first time. Computed waves have qualitatively similar behavior as that of those found in homogeneous films but the quantitative characteristics of the waves strongly depend on additional similarity parameters in the two-layer films. In particular, the average location of the interface affects the bifurcation scheme of the waves.

/pdf/on-wave-families-in-a-two-layer-falling-film-41tvtod649.pdf

On wave families in a two-layer falling film

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Coherent wave structures on falling fluid films flowing down a flexible wall

On wave families in a two-layer falling film