On the reaction-diffusion type modelling of the self-propelled object motion

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 selfpropelled 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 L-2-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.

Automated summary

In this study, a mathematical model of self-propelled objects based on the Allen-Cahn type phase-field equation is proposed. By combining it with the equation for the concentration of surfactant, the model is able to handle both shape change and motion of self-propelled objects. The model can represent both deformable and solid objects by controlling a single parameter. Moreover, it is demonstrated that the phase-field based model can be reduced to a free boundary model by taking the singular limit, which gives a physical interpretation to the model.