Structure and particle transport in second-order stokes flow.

Second-order streaming in a thin fluid layer driven by one or two opposed, tangentially oscillating wavy walls is theoretically investigated. In contrast to the well-studied problem of oscillatory flow past a stationary boundary, the present problem is subject to a nonhomogeneous second-order boundary velocity condition. A combination of steady Reynolds stresses and boundary forcing thus drives the streaming flow; indeed, under most conditions, boundary-forced flow dominates Reynolds-stress-driven flow. The first part of the paper examines parametric effects on second-order flow structure. Under low-Reynolds-stress conditions and during single-boundary forcing, flow structure remains essentially independent of all parameters, including the Stokes layer thickness, the fluid layer thickness, and the forcing wave form. Three approximations to the full second-order solution, valid under low-Reynolds-stress conditions, are used to explain these results. In the case of dual-boundary forcing, no corresponding universal behaviors are observed; flow structure exhibits sensitivity to all problem parameters. The second part of the paper investigates particle transport during quasistatic second-order streaming. Here, slow, superposed, large-amplitude oscillations of one wall produce the time-dependent, quasisteady flows of interest. Collective particle motion in the direction of large-scale boundary displacement and filamentary motion in the opposite direction, features consistent with transport in traveling waves [E. Moses and V. Steinberg, Phys. Rev. Lett. 60, 2030 (1988)], characterize short-time transport. Long-time or asymptotic transport, in contrast, is characterized by particle attraction or repulsion to or from period-one elliptic points and attraction toward limit cycles on the Poincare map.