Stability of similarity solutions of viscous thread pinch-off

In this paper we compute the linear stability of similarity solutions of the breakup of viscous liquid threads, in which the viscosity and inertia of the liquid are in balance with the surface tension. The stability of the similarity solution is determined using numerical continuation to find the dominant eigenvalue of the correction problem. Stability of the first two solutions identified in Brenner et al.
[1] are considered. We find that the primary similarity solution, which is that seen in experiments and simulations, is linearly stable with a complex nontrivial eigenvalue, which could explain the phenomenon of break-up producing sequences of small satellite droplets of decreasing radius near a main pinch-off point. The second solution is seen to be linearly unstable. These linear stability
results compare favorably to numerical simulations for the stable similarity solution, while a profile starting near the unstable similarity solution is shown to very rapidly leave the linear regime.