Effect of an electric field on the stability of contaminated film flow down an inclined plane

The stability of a liquid film flowing down an inclined plane is considered when the film is contaminated by an insoluble surfactant and subjected to a uniform normal electric field. The liquid is treated as a perfect conductor and the air above the film is treated as a perfect dielectric. Previous studies have shown that, when acting in isolation, surfactant has a stabilizing influence on the flow while an electric field has a destabilizing influence. The competition between these two effects is the focus of the present study. The linear stability problem is formulated and solved at arbitrary parameter values. An extended form of Squire's theorem is presented to argue that attention may be confined to two-dimensional disturbances. The stability characteristics for Stokes flow are described exactly; the growth rates of the normal modes at finite Reynolds number are computed numerically. We plot the neutral curves dividing regions of stability and instability, and trace how the topology of the curves changes as the intensity of the electric field varies both for a clean and for a contaminated film. With a sufficiently strong electric field, the neutral curve for a clean film consists of a lower branch trapping an area of stable modes around the origin, and an upper branch above which the flow is stable. With surfactant present, a similar situation obtains, but with an additional island of stable modes disjoint from the upper and lower branches.