A renormalization group analysis of bubble breakup.

The self-similarity has been discussed repeatedly for the singular dynamics such as breakup of a fluid drop and its resemblance to critical phenomena in thermodynamic transitions has also been pointed out. Although critical phenomena have been well understood by the renormalization group (RG) theory, the counterpart has not been developed for the breakup problem. Here, we apply an RG analysis developed in mathematics for partial differential equations (PDEs) without noise terms to the bubble breakup, or the formation of a fluid drop surrounded by a more viscous fluid. As a result, we show a wide class of nonlinear and complex PDEs shares the same self-similar solution with a simple PDE that describes the interfacial phenomena, forming the bubble-breakup universality class. We reveal that the experimentally observed self-similar dynamics appear as a stable fixed point of the RG. The framework clarifies that the physical origin of the emergence of the self-similar solution is the invariance of the governing equation under a scale transformation, where the invariance, if not initially exists, could be aquired after the repetition of RG. The present study elucidates that the self-similarity and universality in the hydrodynamic analog emerges as a result of the physics at small scales becoming so important, just as the universality in critical phenomena appears as a result of the physics at large scales becoming so important.