Rotational motion of a camphor disk in a circular region

Abstract

In a two-dimensional axisymmetric system, the system symmetry allows rotational or oscillatory motion as stable stationary motion for a symmetric self-propelled particle. In the present paper, we studied the motion of a camphor disk confined in a two-dimensional circular region. By reducing the mathematical model describing the dynamics of the motion of a camphor disk and the concentration field of camphor molecules on a water surface, we analyzed the reduced equations around a bifurcation point where the rest state at the center of the system becomes unstable. As a result, we found that rotational motion is stably realized through the double-Hopf bifurcation from the rest state. The theoretical results were confirmed by numerical calculation and well corresponded to the experimental results.

Self-propulsion of symmetric chemically active particles: Point-source model and experiments on camphor disks

“…There are several reports of isotropic swimmers such as camphor, benzoquinone, or phenanthroline disks [44][45][46][47][48][49]. Yet, despite much progress [6,[50][51][52], the theoretical description of propulsion is not yet complete. Two basic questions have remained unanswered.…”

Realization of Self‐Rotating Droplets Based on Liquid Metal

“…This is analogous to the rotation of the comma‐shaped camphor fragments under the action of an unbalanced Laplace pressure. [ 33–35 ] In Figure 5b, the case is similar. The torques, caused by the inward and outward forces, F 1 and F 2 , result in a continuously left‐handed rotation.…”

“…The rotation caused by a comma shape is well known from the phenomenon that comma‐shaped camphor fragments can spontaneously rotate in water. [ 33–35 ] The asymmetric shape creates additional torque on the camphor fragments and promotes continuous rotation. [ 33–35 ] In Figure 1a, the droplet rotation is analogous to the case of camphor fragments.…”

“…[ 33–35 ] The asymmetric shape creates additional torque on the camphor fragments and promotes continuous rotation. [ 33–35 ] In Figure 1a, the droplet rotation is analogous to the case of camphor fragments. The asymmetric shape causes the nonequilibrium Laplace pressure, which contributes to the droplet rotation.…”

“…In this case, the convex and concave parts of the camphor fragment, that is, the positive curvature and negative curvature part, produce torque on the camphor fragment and cause it to rotate. [ 33–35 ] In Figure 5a, the curvature of the comma‐shaped droplet varies at different points along the contact line. The different curvatures lead to unequal Laplace pressures, which are generally given by equation Δ p = γ (1/ r 1 + 1/ r 2 ), where γ is the surface tension of the liquid, and r 1 and r 2 are two orthogonal curvatures of the curved liquid surface, respectively.…”