  Viscous streaming has emerged as an effective method to transport, trap, and
cluster inertial particles in a fluid. Previous work has shown that this
transport is well described by the Maxey-Riley equation augmented with a term
representing Saffman lift. However, in its straightforward application to
viscous streaming flows, the equation suffers from severe numerical stiffness
due to the wide disparity between the time scales of viscous response,
oscillation period, and slow mean transport, posing a severe challenge for
drawing physical insight on mean particle trajectories. In this work, we
develop equations that directly govern the mean transport of particles in
oscillatory viscous flows. The derivation of these equations relies on a
combination of three key techniques. In the first, we develop an inertial
particle velocity field via a small Stokes number expansion of the particle's
deviation from that of the fluid. This expansion clearly reveals the primary
importance of Fax\'en correction and Saffman lift in effecting the trapping of
particles in streaming cells. Then, we apply Generalized Lagrangian Mean theory
to unambiguously decompose the transport into fast and slow scales, and
ultimately, develop the Lagrangian mean velocity field to govern mean
transport. Finally, we carry out an expansion in small oscillation amplitude to
simplify the governing equations and to clarify the hierarchy of first- and
second-order influences, and particularly, the crucial role of Stokes drift in
the mean transport. We demonstrate the final set of equations on the transport
of both fluid and inertial particles in configurations involving one and two
weakly oscillating cylinders. Notably, the new equations allow numerical time
steps that are $O(10^3)$ larger than the existing approach with little
sacrifice in accuracy, allowing more efficient predictions of transport.
