Chapter 2: Mathematical Model and Analyses on Spontaneous Motion of Camphor Particle

When a camphor particle is placed on a water surface, it supplies camphor molecules to the water surface, which decrease the surface tension. Owing to the difference in surface tension originating from the difference in camphor concentration, the camphor particle starts to move. In this chapter, we introduce a mathematical model for the motion of a single camphor particle, and present the procedures to analyse the model. The original model is composed of a partial differential equation describing the time evolution of the concentration profile of camphor molecules and ordinary differential equations describing the time evolution of position and characteristic angle of the camphor particle. In the analysis, we derive the reduced ordinary differential equation regarding the dynamics of the camphor particle position and characteristic angle and discuss it considering the bifurcation theory of dynamical systems. We also discuss the effects of the particle shape based on the theoretical analysis.