Streaming in a channel bounded by an ultrasonically oscillating beam and its cooling efficiency

In this article the oscillating and streaming flow fields in a channel composed of two long parallel beams, one of which is stationary and the other of which oscillates with an ultrasonic frequency in a standing wave form, are investigated. The perturbation technique is utilized under the assumption that the oscillation amplitude is much smaller than the channel width and that the Reynolds number, which is defined by the oscillating frequency and the standing wave number, is much greater than unity. A three-layer structure of both the oscillating and streaming flow fields, which is composed of two very thin boundary layers near the beams and the core region between the boundary layers, is found in the channel. The oscillating velocity fields in all three layers are obtained analytically. The streaming fields within both boundary layers are also obtained analytically based on the oscillating fields. It is found that the streaming velocities approach constant values at the edges of the boundary layers and thus provide slip velocities for the streaming field in the core region. The core-region streaming velocity field is then obtained numerically by solving the Navier-Stokes equations in the stream function-vorticity formulation. Based on the core-region streaming field, which dominates most of the channel, the temperature field is computed for two cases: both beams are kept at constant but different temperatures (case A); and the oscillating beam is kept at a constant temperature while the stationary beam is subjected to a uniform constant heat flux (case B). Cases of different channel widths are computed and a critical width is found. When the channel width is smaller than the critical one, for each half standing wavelength distance along the beams, two symmetric eddies are observed, which occupy almost the whole width of the channel. In this case, the Nusselt number increases with the increase of the channel width. After the critical width, two layers of asymmetric eddies are observed near the oscillating beam and the Nusselt number decreases and approaches unity with further increase of the channel width. The abrupt change of the streaming field and the Nusselt number as the channel width goes through its critical value may be due to a bifurcation caused by instability of the vortex structure in the fluid layer.