![]() ![]() Therefore, we aimed to construct a mathematical model suitable for numerical computation at a low cost that can be connected to the natural model from the physics viewpoint. Thiele and coauthors reported the two-dimensional model for the self-propelled droplet 23, but no systematic study has not been demonstrated. The other approach for the droplet motion is to adopt the lubrication approximation. A mathematical model based on the membrane motion model has been proposed for the droplet motion however, it is not easy to handle as a model of droplet dynamics due to its substantial computational cost 22. ![]() On the other hand, as for mathematical modelling for the self-propelled system with shape deformation, the mathematical modelling approach has not yet been satisfactorily performed. Formulations of two-dimensional motion models for the former case have already been made, and some comparisons of experimental results with those of mathematical models and mathematical analyses have been performed 20, 21. Such surface-tension-driven systems are classified into two cases the systems without shape deformation, like camphor disks, and those with shape deformation, like pentanol droplets. In this study, we focus on the spatio-temporal behaviours in self-propelled systems, such as the motion of camphor disks and pentanol droplets, where the mechanism for the self-propelled motion is believed to be primarily governed by surface tension gradient. Mathematical models for the motility of non-living materials are also investigated by constructing the mathematical model, for example, camphor particles 15, 16, pentanol droplets 17, running oil droplets 18, and blebbing oil droplets 19. In recent years, various types of cell motility have been analysed through mathematical models, including keratocyte motility 10, cell population motility 11, 12, 13, and cell division 14. Motions of living organisms, such as birds, insects, bacteria, and cells 1, 2, 3, 4 and those in non-living materials, such as camphor disks, swimming droplets, running droplets, and Janus particles 5, 6, 7, 8, 9, are regarded as the motion in self-propelled systems. ![]() Such a particle or droplet can move by the internal mechanism some can move by deforming themselves, and others move by changing the characters in the neighbouring field. A self-propelled system is composed of a particle or droplet that exhibits self-propelled motion by consuming the free energy under non-equilibrium conditions. ![]()
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