Acoustic/steady streaming from a motionless boundary and related phenomena: generalized treatment of the inner streaming and examples

As originally realized by Nyborg (J. Acoust. Soc. Am., vol. 30, 1958, P. 329), the problem of the inner acoustic/steady streaming can be analysed in quite general terms. The inner streaming is the one that develops in the high-frequency limit in a thin Stokes (shear-wave) layer at a boundary, in contrast to the outer streaming in the main bulk of the fluid. The analysis provides relevant inner-streaming characteristics through a given distribution of the acoustic amplitude along the boundary. Here such a generalized treatment is revisited for a motionless boundary. By working in terms of surface vectors, though in elementary notations, new compact and easy-to-use expressions are obtained. The most important ones are those for the effective (apparent) slip velocity at the boundary as seen from a perspective of the main bulk of the fluid, which is often the sole driving factor behind the outer streaming, and for the induced (acoustic) steady tangential stress on the boundary. As another novel development, non-adiabatic effects in the Stokes layer are taken into account, which become apparent through the fluctuating density and viscosity perturbations, and whose contribution into the streaming is often ignored in the literature. Some important particular cases, such as the axisymmetric case and the incompressible case, are emphasized. As far as the application of the derived general inner-streaming expressions is concerned, a few examples provided here involve a plane acoustic standing wave, which either grazes a wall parallel to its direction (convenient for the estimation of the non-adiabatic effects), or into which a small (compared to the acoustic wavelength) rigid sphere is placed. If there are simultaneously two such waves, out-of-phase and, say, in mutually orthogonal directions, a disk placed coplanarly with them will undergo a steady torque, which is calculated here as another example. Two further examples deal with translational high-frequency harmonic vibrations of particles relative to an incompressible fluid medium, viz, of a rigid oblate spheroid (along its axis) and of a sphere (arbitrary three dimensional). The latter can be a fixed rigid sphere, one free to rotate or even a (viscous) spherical drop, for which the outer streaming and the internal circulation are also considered.