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

Abstract

As originally realized by Nyborg (1958), the problem of the inner acoustic/steady streaming can be analyzed 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 with the outer streaming in the main bulk of the fluid. The analysis consists in expressing 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. Important particular cases, such as the axisymmetric case and the incompressible case, are dealt with. 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 highfrequency harmonic vibrations of particles relative to an incompressible fluid medium, viz. of a rigid oblate spheroid (along its axis) and of a sphere (arbitrary 3-d). 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 considered as well.

Numerical study of thermoviscous effects in ultrasound-induced acoustic streaming in microchannels

“…The streaming velocity has been calculated for different channel heights indicated by different colors and symbols. The full line is the analytical single-wall solution by Rednikov and Sadhal [15] for a standing acoustic wave parallel to a single planar wall. For all channel heights, the streaming velocity shows an almost linear dependence with positive slope on the equilibrium temperature.…”

“…It was later extended, among others, by Schlicting [10], Nyborg [11], Hamilton [12,13], and Muller et al [14]. Recently, Rednikov and Sadhal [15] have included the temperature dependence of the dynamic viscosity and shown that this can lead to a significant increase in the magnitude of the streaming velocity. In the present work, we present a numerical study of this and related thermoviscous effects.…”

“…In this paper, we extend our numerical model for a rectangular microchannel [20] to include the thermoviscous effects, which were treated analytically in the special case of a single planar infinite rigid wall by Rednikov and Sadhal [15]. The extension is done by including the dependence on the oscillatory first-order temperature and density fields in the dynamic shear viscosity, previously taken to be constant.…”

Ultrasound-induced acoustophoretic motion of microparticles in three dimensions

“…We use these models without any new contributions. However, so far theoretical treatments of the acoustic streaming generated by standing acoustic waves have not included an analysis of the effect of the vertical sidewall in rectangular microchannels, instead the focus has been on the idealized parallel-plate geometry [9][10][11][12][13][14] or single-wall systems [15][16][17][18]. Also, in most theoretical work either the radiation force or the streaming effects have been studied separately, but not combined with wall effects to obtain a complete description of microparticle * bruus@fysik.dtu.dk acoustophoresis.…”

Theoretical study of time-dependent, ultrasound-induced acoustic streaming in microchannels

“…The phenomenon of acoustic streaming was first described theoretically by Lord Rayleigh [4] in 1884, and has later been revisited, among others, by Schlicting [5], Nyborg [6], Hamilton [7,8], Rednikov and Sadhal [9], and Muller et al [10], to extend the fundamental treatment of the governing equations and to solve the equations for various open and closed geometries.…”