Dynamics of falling liquid films.

Falling liquid films are examples of open flows which undergo a sequence of supercritical instabilities giving way to a spatio-temporal weak disorder organized around interfacial waves in interaction. The long-wave nature of the waves and the laminar state of the flow enable to derive reduced sets of equations or models that are amenable to thorough analytic investigations. This paper reviews the phenomenology of falling film flows and discusses recent low-dimensional modeling attempts. Some open questions and perspectives are also considered.

Wavy regimes of film flow down a fiber.

We consider axisymmetric traveling waves propagating on the gravity-driven flow of a liquid down a vertical fiber. Our starting point is the two-equation model for the flow derived in the study by Ruyer-Quil et al. [J. Fluid Mech. 603, 431 (2008)]. The speed, amplitude, and shape of the traveling waves are obtained for a wide range of parameters by using asymptotic analysis and elements from dynamical systems theory. Four different regimes are identified corresponding to the predominance of four different physical effects: advection by the flow, azimuthal curvature, inertia, and viscous dispersion. Construction of the traveling-wave branches of solutions reveals complex transitions from one regime to another. A phase diagram of the different regimes in the parameter space is constructed.

Interaction of three-dimensional hydrodynamic and thermocapillary instabilities in film flows.

We study three-dimensional wave patterns on the surface of a film flowing down a uniformly heated wall. Our starting point is a model of four evolution equations for the film thickness h , the interfacial temperature theta , and the streamwise and spanwise flow rates, q and p , respectively, obtained by combining a gradient expansion with a weighted residual projection. This model is shown to be robust and accurate in describing the competition between hydrodynamic waves and thermocapillary Marangoni effects for a wide range of parameters. For small Reynolds numbers, i.e., in the "drag-gravity regime," we observe regularly spaced rivulets aligned with the flow and preventing the development of hydrodynamic waves. The wavelength of the developed rivulet structures is found to closely match the one of the most amplified mode predicted by linear theory. For larger Reynolds numbers, i.e., in the "drag-inertia regime," the situation is similar to the isothermal case and no rivulets are observed. Between these two regimes we observe a complex behavior for the hydrodynamic and thermocapillary modes with the presence of rivulets channeling quasi-two-dimensional waves of larger amplitude and phase speed than those observed in isothermal conditions, leading possibly to solitarylike waves. Two subregions are identified depending on the topology of the rivulet structures that can be either "ridgelike" or "groovelike." A regime map is further proposed that highlights the influence of the Reynolds and the Marangoni numbers on the rivulet structures. Interestingly, this map is found to be related to the variations of amplitude and speed of the two-dimensional solitary-wave solutions of the model. Finally, the heat transfer enhancement due to the increase of interfacial area in the presence of rivulet structures is shown to be significant.