  A growing body of work on the dynamics of eukaryotic flagella has noted that
their oscillation frequencies are sufficiently high that the viscous
penetration depth of unsteady Stokes flow is comparable to the scales over
which flagella synchronize. Incorporating these effects into theories of
synchronization requires an understanding of the global unsteady flows around
oscillating bodies. Yet, there has been no precise experimental test on the
microscale of the most basic aspects of such unsteady Stokes flow: the orbits
of passive tracers and the position-dependent phase lag between the oscillating
response of the fluid at a distant point and that of the driving particle.
Here, we report the first such direct Lagrangian measurement of this unsteady
flow. The method uses an array of $30$ submicron tracer particles positioned by
a time-shared optical trap at a range of distances and angular positions with
respect to a larger, central particle, which is then driven by an oscillating
optical trap at frequencies up to $400$ Hz. In this microscale regime, the
tracer dynamics is considerably simplified by the smallness of both inertial
effects on particle motion and finite-frequency corrections to the Stokes drag
law. The tracers are found to display elliptical Lissajous figures whose
orientation and geometry are in agreement with a low-frequency expansion of the
underlying dynamics, and the experimental phase shift between motion parallel
and orthogonal to the oscillation axis exhibits a predicted scaling form in
distance and angle. Possible implications of these results for synchronization
dynamics are discussed.
