Three-Dimensional Trajectories of Spheroidal Particles in Two-Dimensional Flow Fields

We investigate the motion of homogeneous, spheroidal particles immersed in an incompressible, viscous fluid. We assume the particles to be more dense than the surrounding fluid and small enough that inertia is negligible with respect to viscous forces. We give exact solutions for the motion of the particle’s center of mass for steady, linear flows, either irrotational or without strain. For a weakly strained, two-dimensional, rotational flow we give an asymptotic approximation to the solutions, and we compare it with numerical solutions. In the presence of vorticity we find that the spheroid moves along three-dimensional, non-planar paths. With pure strain the three-dimensionality of the paths is transient. If a two-dimensional rotational flow is perturbed by strain, then the generic path of a spheroid is an open curve, even if all the streamlines of the flow are closed. We conclude by speculating about the significance of these findings for the ecology of phytoplankton.