  In the Stokesian limit, the streamline topology around a single neutrally
buoyant sphere is identical to the topology of pair-sphere pathlines, both in
an ambient simple shear flow. In both cases there are fore-aft symmetric open
and closed trajectories spatially demarcated by an axisymmetric separatrix
surface. This topology has crucial implications for both scalar transport from
a single sphere, and for the rheology of a dilute suspension of spheres. We
show that the topology of the fluid pathlines around a neutrally buoyant freely
rotating spheroid, in simple shear flow, is profoundly different, and will have
a crucial bearing on transport from such particles in shearing flows. To the
extent that fluid pathlines in the single-spheroid problem and
pair-trajectories in the two-spheroid problem, are expected to bear a
qualitative resemblance to each other, the non-trivial trajectory topology
identified here will also have significant consequences for the rheology of
dilute suspensions of anisotropic particles.
