Instability of a thin film flowing on a rotating horizontal or inclined plane

Abstract

In this paper the instability of a thin fluid film flowing under the effects of gravity, Coriolis, and centrifugal forces is investigated. It is supposed that the film flows far from the axis of rotation on a plane which may be horizontal or inclined with respect to the horizontal. In the former case, the flow is only driven by the centrifugal force while in the latter case, the flow is driven by the components of centrifugal force and gravity along the plane. This case may also be considered as the flow down a rotating cone but far from the apex. The stabilizing influence of rotation on the film flow increases with the rotation rate. Up to a certain critical rate of rotation, the film flowing down the rotating inclined plane (or cone) is more stable than the flow on the horizontal rotating plane while above this rate of rotation the situation is reversed. The instability above the critical rate is associated with a finite wave number in contrast to the vanishing wave number of the instability below the critical rate. The possibility of Ekman layer instabilities is also investigated. An equation describing the nonlinear evolution of surface waves is also obtained. Moreover, this equation is simplified for the case in which the amplitudes are very small. An equation including dissipation as well as dispersion is derived whose solutions may possess solitary waves, as in the case of similar equations considered in the literature. These solutions are likely to correspond to the solitary spiral waves observed in experiments.