A renormalization group analysis of the axisymmetric bubble breakup inspired by recent non-axisymmetric studies

Recently, we have revealed in an original experimental setup, in which a disk entrains air into viscous liquid in a confined geometry, a number of distinct dynamical regimes for the air-liquid interface, which include two regimes for non-axisymmetric bubble breakup [1-4]. The regimes have been found to correspond to distinctly different self-similar dynamics, which suggests deep analogy with critical phenomena. To explore the analogy, we focus on a remoralization group (RG) theory developed by mathematicians for partial differential equations (PDEs) without stochastic noise [5], which, like the original RG by Wilson, consists of “two steps,” and apply it to the well-understood problem of axisymmetric bubble breakup [6] that occurs in a non-confined highly viscous liquid, where viscosity and capillarity compete with each other. As a result, we find that the self-similar solution appears as RG fixed points, from which a unique stable solution is selected by a stability analysis as the experimentally observed solution. Our analysis further shows that the solution is shared by a universality class, i.e., a wide class of PDEs, which includes very complex and nonlinear PDEs, and that the origin of universality is the scale invariance of the governing equation [7]. If the time allows, we further discuss [8] the combined use of "two-step" and "field-theoretic" RG theories for a non-linear diffusion equation, where "field-theoretic" one [9] is developed by physicists and uses a singular perturbation, whose divergence is removed by the renormalization technique in quantum field theory.



References

[1] Hana Nakazato, Yuki Yamagishi, and Ko Okumura, Phys. Rev. Fluids, 3:054004, 2018.

[2] Hana Nakazato and Ko Okumura, Phys. Rev. Research, 4(1):013150, 2022.

[3] Shoko Ii and Ko Okumura, under revision.

[4] Ikumi Yoshino and Ko Okumura, under revision.

[5] J. Bricmont, A. Kupiainen, and G. Lin, Commun. Pure Appl. Math, 47:893, 1994.

[6] J. Eggers and M. A. Fontelos, Singularities: formation, structure, and propagation, Cambridge University Press, 2015.

[7] Ko Okumura, Sci. Rep., 15:34507, 2025.

[8] Ko Okumura, in preparation.

[9] N. Goldenfeld. Lectures on Phase Transitions and the Renormalization Group. Addison-Wesley Pub., 1992.