Nagayama M, Monobe H, Sakakibara K, Nakamura KI, Kobayashi Y, Kitahata H. On the reaction-diffusion type modelling of the self-propelled object motion.
Sci Rep 2023;
13:12633. [PMID:
37537247 PMCID:
PMC10400585 DOI:
10.1038/s41598-023-39395-w]
[Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/16/2023] [Accepted: 07/25/2023] [Indexed: 08/05/2023] Open
Abstract
In this study, we propose a mathematical model of self-propelled objects based on the Allen-Cahn type phase-field equation. We combine it with the equation for the concentration of surfactant used in previous studies to construct a model that can handle self-propelled object motion with shape change. A distinctive feature of our mathematical model is that it can represent both deformable self-propelled objects, such as droplets, and solid objects, such as camphor disks, by controlling a single parameter. Furthermore, we demonstrate that, by taking the singular limit, this phase-field based model can be reduced to a free boundary model, which is equivalent to the [Formula: see text]-gradient flow model of self-propelled objects derived by the variational principle from the interfacial energy, which gives a physical interpretation to the phase-field model.
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