Theoretical study on the translation and rotation of an elliptic camphor particle

Abstract

The spontaneous motion of an elliptic camphor particle floating on water is studied theoretically and experimentally. Considering a mathematical model for the motion of an elliptic camphor particle in a two-dimensional space, we first investigate the asymptotic solutions with numerical computation. We then introduce a small parameter ε into the definition of the particle shape, which represents an elliptic deformation from a circular shape and, by means of perturbation theory, we analytically calculate the travelling solution to within O(ε). The results show that short-axis-directed travelling solutions primarily bifurcate from stationary solutions and that long-axis-directed ones are secondary which means that elliptic camphor particles are easier to move in the short-axis direction. Furthermore, we show that rotating solutions bifurcate from stationary solutions and that the bifurcation point changes with O(ε2), which suggests that elliptic camphor disks easily exhibit translational motion, rather than rotational, within the small deformation. Finally, our theoretical suggestions are confirmed by an experiment.