Discrete self-similarity in the formation of satellites for viscous cavity break up

The breakup of a jet of a viscous fluid with viscosity ¿1 immersed into another viscous fluid with viscosity ¿2 is considered in the limit when the viscosity ratio ¿=¿1/¿2 is close to zero. We show that, in this limit, a transition from ordinary continuous self-similarity to discrete self-similarity takes place as ¿ decreases. It follows that instead of a single point breakup, the rupture of the inner jet occurs through the appearance of an infinite sequence of filaments of decreasing size that will eventually produce infinite sequences of bubbles of the inner fluid inside the outer fluid. The transition can be understood as the result of a Hopf bifurcation in the system of equations modeling the physical problem.M.A. Fontelos is supported by the research Grant from the Spanish Ministry of Economy and Competitiveness MTM2017-89423-P