Long-wave instabilities of non-uniformly heated falling films

Abstract

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.

Validity domain of the Benney equation including the Marangoni effect for closed and open flows

“…We believe, this ensures that the Benney equation will remain a helpful model to study thin film flows, especially to identify new phenomena but also in a limited parameter range for quantitative predictions. Actually, many other effects may be added to the Benney equation like evaporation (Joo et al 1991), Van der Waals force (Tan et al 1966), chemical reaction (Trevelyan et al 2002), topolographical effects (Kalliadasis et al 2000), non-uniform heating (Miladinova et al 2002;Kabov et al 2001;Scheid et al 2002;Skotheim et al 2002;Kalliadasis et al 2003b), etc. The validity of the Benney equation should in the future also be addressed including these additional effects.…”

Numerical Investigation of the Liquid Film Flows with Evaporation at Thermocapillary Interface

“…The influences of various effects on the flow character are investigated analytically, numerically and experimentally [1][2][3][4][5][6]. Evaporation at the interface should be taken into account in the modeling of such processes [1][2][3][7][8][9]. A number of works are devoted to the numerical study of flow of evaporating thin liquid layers [2][3][4].…”

“…Evaporation at the interface should be taken into account in the modeling of such processes [1][2][3][7][8][9]. A number of works are devoted to the numerical study of flow of evaporating thin liquid layers [2][3][4]. The boundary conditions should provide a fulfillment of the conservation laws on the interface [10][11][12].…”

The Liquid Film Flow with Evaporation: Numerical Modelling

“…A large number of both theoretical [1][2][3][4][5] and experimental investigations [6] are devoted to the study of viscous incompressible fluids with evaporation. The interest to such currents is caused, in particular, by the wide possibilities of their practical application in modern industrial processes.…”

“…Great attention is paid to the thin liquid films. The mathematical models of the flows in the thin layer approximation and their numerical study are presented in [1][2][3][4][5][7][8][9][10]. Special attention while modelling of the flows with interfaces should be given to the formulation of the boundary conditions [11][12][13].…”