Instability of odd viscosity falling liquid films with insoluble surfactants

The linear and weakly nonlinear stability of odd viscosity film with insoluble surfactants flowing down an inclined plane under a normal electric field is investigated analytically. Using the long-wave expansion method, the nonlinear evolution equations for liquid film thickness and surfactant concentration are derived. Through the normal mode analysis, the effects of surface surfactant, odd viscosity, and electric field on the neutral stability curve and the temporal growth rates are calculated to explore the linear stability of the film. Two modes, i.e., Kapitza mode and surfactant mode, are identified. Linear results show that the presence of surfactants and odd viscosity has a stabilizing effect, while electric field has a destabilizing effect on flowing of thin film. Based on the Ginsburg-Landau equation, the primary bifurcations in the phase diagram for two types of modes are investigated. The results reveal the destabilizing nature with increasing Marangoni number and viscosity ratio for surfactant mode and the stabilizing nature for Kapitza mode.