Film drainage between two drops: Vortex formation in thin liquid films

in this study, a mathematical formalism that takes into account the surfactant effect on the drainage of the interfacial film between two drops is considered. the effects of thermal perturbations and van der waals forces are neglected. in the mathematical formalism the navier-stokes equations within the lubrication approximation are coupled to a diffusion-convection equation leading to an evolution equation for the interfacial film. this last equation is solved by using the numerical method of lines coupled with an implicit runge-kutta method for the integration with respect to time. as a result of the inclusion of interfacial tension gradients a non oscillating dimple arises, even beginning with an initial condition corresponding to a plane interfacial film.