Spontaneous motion of a camphor particle with a triangular modification from a circle

The spontaneous motion of a camphor particle with a slight modification from a circle is investigated. The effect of the shape on the motion is examined by the perturbation method. We introduce a slight $n$-mode modification from a circle, where the profile is described by $r = R(1 + \epsilon \cos n\theta)$ in polar coordinates. The results predict that a camphor particle with an $n=3$ mode modification from a circle, i.e., a triangular modification, moves in the direction of a corner for a small particle, while it moves in the direction of a side for a large particle. The numerical simulation results well reproduce the theoretical prediction. The present study will help understand the effect of the particle shape on spontaneous motion.