Okumura Ko
Physics Department and Soft Matter Center, Ochanomizu University, 2-1-1 Ohtsuka, Bunkyo-ku, 112-8610, Tokyo, Japan.
Sci Rep. 2025 Oct 3;15(1):34507. doi: 10.1038/s41598-025-17667-x.
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.
对于诸如液滴破裂等奇异动力学过程中的自相似性已被反复讨论，并且其与热力学转变中的临界现象的相似性也已被指出。尽管临界现象已通过重整化群（RG）理论得到了很好的理解，但对于破裂问题的对应理论尚未得到发展。在此，我们将数学中针对无噪声项的偏微分方程（PDEs）所发展的RG分析应用于气泡破裂，即由更粘性流体包围的液滴的形成过程。结果，我们表明一类广泛的非线性和复杂PDEs与描述界面现象的简单PDE具有相同的自相似解，从而形成了气泡破裂普适类。我们揭示了实验观测到的自相似动力学表现为RG的一个稳定不动点。该框架阐明了自相似解出现的物理根源是控制方程在尺度变换下的不变性，其中这种不变性如果最初不存在，在重复进行RG之后可以获得。本研究阐明了流体动力学模拟中的自相似性和普适性是由于小尺度物理变得如此重要而产生的，正如临界现象中的普适性是由于大尺度物理变得如此重要而出现的一样。
