Linear stability of a draining film squeezed between two approaching droplets

In the present paper we analyze the effect of infinitesimal non-axisymmetric perturbations in determining the critical gap thickness at which a draining, finite radius thin-film becomes unstable. The film is part of the suspending fluid trapped between two approaching deformable drops under the action of a flow field. We carry out a linear stability analysis in the context of a quasi-static approximation where the rate of growth of the disturbances is assumed to be much faster than the rate of film drainage. An analytical solution is derived for the model in the special case of a uniformly thick film, for two types of perturbation: fixed-end and free-end. It is shown, for this special case, when the hydrodynamic force pushing the drops together from the external flow is constant, that the four most unstable disturbances are of the free-end kind, associated with the lowest frequency modes of azimuthal variation in the film thickness. Higher modes are stabilized by surface tension. Our analysis also shows that adopting the unretarded form of the van der Waals disjoining pressure yields results similar to the analysis when electromagnetic retardation effects are included in the calculation. A second case is analyzed where the film is also of uniform thickness but its lateral extent and the gap thickness are both time-dependent. This case was included to extend the predictions to glancing drop-collisions where the external hydrodynamic force is time-dependent. We find that there is a maximum capillary number below which the film becomes unstable, and that there is range of angles in the trajectory where the film becomes unstable, but that outside this range the film is stable.