Quasisteady Particle Transport in Slowly Varying Periodic Streaming Flows

Abstract

When a cylindrical probe vibrates laterally in a fluid at a Reynolds number of order 10, a circulatory pattern of flow is established in the fluid near the probe. In a plane transverse to the probe, a symmetric arrangement of four streaming cells is maintained by the probe's vibration. A computational model has shown that inertial particles introduced to such a flow will be attracted to the centers of the streaming cells. We present preliminary experimental data supporting this prediction and outline a strategy whereby particles captured in this way can be transported from place to place as a result of periodic variations in the excitation of a fluid containing two independent probes. Specifically, we model the time-averaged flow around two probes by superposing two copies of a velocity field obtained analytically for the single-probe case and demonstrate that a cyclic change in the two-probe flow can engender acyclic variations in the positions and character of fixed points. A fixed point that's initially attractive to inertial particles can be moved from the vicinity of one probe to the vicinity of another and then annihilated or altered via bifurcation so that it will surrender the particles it carries. If parametric variations in such a flow are slow relative to the dynamics of particles migrating to attracting points, then the net displacement of inertial particles from one trapping point to another as a result of a cyclic change in the flow exhibits certain features of a geometric phase.