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

The breakup of a jet of a viscous fluid with viscosity $\mu _{1}$ immersed into another viscous fluid with viscosity $\mu _{2}$ is considered in the limit when the viscosity ratio $\lambda =\mu _{1}/\mu _{2}$ is close to zero. We show that, in this limit, a transition from ordinary continuous selfsimilarity to discrete selfsimilarity takes place as $\lambda $decreases. The result being 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 modelling the physical problem.