Journal of Fluid Mechanics

Long-wave instabilities of non-uniformly heated falling films

a1 Department of Fluid Mechanics, Institute of Mechanics, Sofia 1113, Bulgaria
a2 Institute of Physics, University of Liege, B 4000 Liege, Belgium
a3 Microgravity Research Center, Universite Libre de Bruxelles, B 1050 Brussels, Belgium


We consider the problem of a thin liquid layer falling down an inclined plate that is 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 not only influenced 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, a nonlinear evolution equation is derived. 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 the mean surface tension, but their amplitudes and phase speeds are influenced by thermocapillarity.

(Received July 24 2000)
(Revised July 18 2001)