A framework for crossover of scaling law as a self-similar solution : dynamical impact of viscoelastic board

Abstract

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 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 on crossover, which offers a fundamental insight to self-similarity in general.

21 pages, 6 Figures and 9 Supplemental Figures

References in corpus (2)

• A dissipative force between colliding viscoelastic bodies: Rigorous approach
• Universal equation of state describes osmotic pressure throughout gelation process

Cited by in corpus (3)

• Data-driven discovery of self-similarity using neural networks
• Prigogine's temporalization of physics: two agnostic attitudes of physicists
• Early-stage impact dynamics in dense suspensions of millimeter-sized particles