https://doi.org/10.1140/epje/s10189-023-00292-9
Regular Article – Flowing Matter
A framework for crossover of scaling law as a self-similar solution: dynamical impact of viscoelastic board
1
Deep-Sea Nanoscience Research Group, Research Center for Bioscience and Nanoscience, Research Institute for Marine Resource Utilization, Japan Agency for Marine-Earth Science and Technology (JAMSTEC), 2-15 Natsushima-cho, 237-0061, Yokosuka, Kanagawa, Japan
2
Advanced Statistical Dynamics, Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa Oiwakecho, 606-8502, Sakyo-ku, Kyoto, Japan
a hmaruoka1987@gmail.com, hirokazu.maruoka@yukawa.kyoto-u.ac.jp
Received:
26
July
2022
Accepted:
22
April
2023
Published online:
24
May
2023
In this paper, a new framework for crossover of scaling law is proposed: a crossover of scaling law can be described by a self-similar solution. A crossover emerges as a result of the interference from similarity parameters of the higher class of the self-similarity. This framework was verified for the dynamical impact of a solid sphere onto a viscoelastic board. All the physical factors including the size of spheres and the impact of velocity are successfully summarized using primal dimensionless numbers which construct a self-similar solution of the second kind, which represents the balance between dynamical elements involved in the problem. The self-similar solution gives two different scaling laws by the perturbation method describing the crossover. These theoretical predictions are compared with experimental results to show good agreement. It was suggested that a hierarchical structure of similarity plays a fundamental role in crossover, which offers a fundamental insight into self-similarity in general.
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