Surfaces with topography promote rivulet flow patterns, which are characterized by a high cumulative length of contact lines. This property is very advantageous for evaporators and cooling devices, since the local evaporation rate in the vicinity of contact lines (microregion evaporation) is extremely high. The liquid flow in rivulets is subject to different kinds of instabilities, including the long-wave falling film instability (or the kinematic-wave instability), the capillary instability, and the thermocapillary instability. These instabilities may lead to the development of wavy flow patterns and to the rivulet rupture. We develop a model describing the hydrodynamics and heat transfer in flowing rivulets on surfaces with topography under the action of gravity, surface tension, and thermocapillarity. The contact line behavior is modeled using the disjoining pressure concept. The perfectly wetting case is described using the usual h3 disjoining pressure. The partially wetting case is modeled using the integrated 6–12 Lennard-Jones potential. The developed model is used for investigating the effects of the surface topography, gravity, thermocapillarity, and the contact line behavior on the rivulet stability. We show that the long-wave thermocapillary instability may lead to splitting of the rivulet into droplets or into several rivulets, depending on the Marangoni number and on the rivulet geometry. The kinematic-wave instability may be completely suppressed in the case of the rivulet flow in a groove.

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