The balancing of thermocapillary flow in a floating zone by ripple-driven streaming

It is known that streaming can be generated in a liquid bridge by vibrating an end wall. This phenomenon has been used in an attempt to minimize thermocapillary flow during crystal growth in a floating zone, by inducing such a streaming running in the opposite direction to it [Grugel et al., J. Cryst. Growth 142, 209 (1994)]. In the present theoretical study, the nature of this streaming and its effects on the average flow and temperature fields in a floating zone are investigated. It is noticed that in the experiment, the applied frequencies were high enough such that the corresponding wavelengths of the capillary ripples were much smaller than the dimensions of the zone. It is believed that the ripples were a traveling wave that generated the streaming in the direction of the wave propagation as a result of Stokes drift. For such a wave to be traveling, it must be dissipated sufficiently by viscosity upon reaching the other wall to guarantee negligible reflection there. Accordingly, a model is formulated to study the reduction of thermocapillary flow in a floating zone by means of streaming. It is found that for a half zone, streaming can minimize the thermocapillary flow near the vibrating wall and make the temperature uniform across the zone there. For a full zone, streaming can minimize the flow but cannot make the temperature uniform near the wall. For a long full zone, streaming can similarly minimize the flow. But in addition, the temperature near each wall is uniform with or without streaming.