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Long-wave instabilities of non-uniformly heated falling films. (English) Zbl 1053.76024
Summary: We consider a thin liquid layer falling down an inclined plate and subjected to non-uniform heating. The plate temperature is assumed to be linearly distributed, and both directions of the temperature gradient with respect to the flow are investigated. The film flow is influenced not only by gravity and mean surface tension, but in addition by the thermocapillary force acting along the free surface. The coupling of thermocapillary instability and surface-wave instabilities is studied for two-dimensional disturbances. Applying the long-wave theory, we derive a nonlinear evolution equation. When the plate temperature is decreasing in the downstream direction, linear stability analysis exhibits a film stabilization, compared to a uniformly heated film. In contrast, for increasing temperature along the plate, the film becomes less stable. Numerical solution of the evolution equation indicates the existence of permanent finite-amplitude waves of different kinds. The shape of the waves depends mainly on the mean flow and mean surface tension, but their amplitudes and phase speeds are influenced by thermocapillarity.
MSC:
76E17Interfacial stability and instability (fluid dynamics)
76A20Thin fluid films (fluid mechanics)
76D45Capillarity (surface tension)
80A20Heat and mass transfer, heat flow