The behaviour of a particle in orthogonal acoustic fields

We are concerned with the response of an unconstrained particle, solid or liquid, placed in an acoustic field which consists of two orthogonal sound waves. These have the same amplitude and wavenumber, but differ in phase by π/2. The particle may be either a circular cylinder or a sphere. The effect of the superimposed waves is to create a time-averaged torque on the particle which causes it to rotate with uniform angular velocity. Throughout, a suitably defined Strouhal number is assumed to be large, with the solution developed in appropriate inverse powers of it. The particle size is assumed to be much smaller than the acoustic wavelength. At leading order it is shown that solid and liquid cylinders behave in a similar manner, in the sense that the liquid is in solid-body rotation. For a spherical liquid drop, the dominant time-averaged motion of it is also a solid-body rotation when the drop viscosity is large compared with that of the fluid environment; however, superposed on this is a time-averaged recirculatory flow within the droplet in the form of a pair of toroidal vortices.