Nagayama Masaharu, Monobe Harunori, Sakakibara Koya, Nakamura Ken-Ichi, Kobayashi Yasuaki, Kitahata Hiroyuki
Research Center of Mathematics for Social Creativity, Research Institute for Electronic Science, Hokkaido University, Hokkaido, 060-0812, Japan.
Department of Mathematics, Graduate School of Science, Osaka Metropolitan University, Osaka, 599-8531, Japan.
Sci Rep. 2023 Aug 3;13(1):12633. doi: 10.1038/s41598-023-39395-w.
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.
在本研究中，我们基于艾伦 - 卡恩（Allen-Cahn）型相场方程提出了一个自推进物体的数学模型。我们将其与先前研究中使用的表面活性剂浓度方程相结合，构建了一个能够处理具有形状变化的自推进物体运动的模型。我们数学模型的一个显著特点是，通过控制单个参数，它既可以表示可变形的自推进物体，如液滴，也可以表示固体物体，如樟脑盘。此外，我们证明，通过取奇异极限，这个基于相场的模型可以简化为一个自由边界模型，该模型等同于通过变分原理从界面能导出的自推进物体的[公式：见原文]梯度流模型，这为相场模型提供了物理解释。
