Compressibility effects on steady streaming from a noncompact rigid sphere

The problem of steady streaming around a rigid isolated sphere in a plane standing acoustic field is considered. Existing results in the literature have been generalized to allow for noncompactness of the sphere, and the influence of fluid compressibility on the streaming behavior has been included. It is found that in the high-frequency limit of interest for which the streaming is strongest, the effective steady slip velocity at the edge of the inner boundary layer region that is responsible for driving the steady streaming in the bulk of the fluid in the outer region, has a complex variation over the surface of the sphere that depends on (i) the sphere position (with respect to the node/antinode of the acoustic field), (ii) the extent of sphere compactness, and (iii) on a well-defined function (representing compressibility effects) of the fluid Prandtl number and its ratio of specific heats. Not surprisingly, the contribution from this function is negligible when the host fluid is a liquid. The steady streaming behavior around the sphere is demonstrated with the help of flow streamlines for various cases in the diffusive limit of weak outer flow for low streaming Reynolds numbers.