Transverse (lateral) instantaneous force of an acoustical first-order Bessel vortex beam centered on a rigid sphere

In a recent report [F.G. Mitri, Z.E.A. Fellah, Ultrasonics 51 (2011) 719-724], it has been found that the instantaneous axial force (i.e. acting along the axis of wave propagation) of a Bessel acoustic beam centered on a sphere is only determined for the fundamental order (i.e. m=0) but vanishes when the beam is of vortex type (i.e. m>0, where m is the order (or helicity) of the beam). It has also been recognized that for circularly symmetric beams (such as Bessel beams of integer order), the transverse (lateral) instantaneous force should vanish as required by symmetry. Nevertheless, in this commentary, the present analysis unexpectedly reveals the existence of a transverse instantaneous force on a rigid sphere centered on the axis of a Bessel vortex beam of unit magnitude order (i.e. |m|=1) not reported in [F.G. Mitri, Z.E.A. Fellah, Ultrasonics 51 (2011) 719-724]. The presence of the transverse instantaneous force components of a first-order Bessel vortex beam results from mathematical anti-symmetry in the surface integrals, but vanishes for the fundamental (m=0) and higher-order Bessel (vortex) beams (i.e. |m|>1). Here, closed-form solutions for the instantaneous force components are obtained and examples for the transverse components for progressive waves are computed for a fixed and a movable rigid sphere. The results show that only the dipole (n=1) mode in the scattering contributes to the instantaneous force components, as well as how the transverse instantaneous force per unit cross-sectional surface varies versus the dimensionless frequency ka (k is the wave number in the fluid medium and a is the sphere's radius), and the half-cone angle β of the beam. Moreover, the velocity of the movable sphere is evaluated based on the concept of mechanical impedance. The proposed analysis may be of interest in the analysis of transverse instantaneous forces on spherical particles for particle manipulation and rotation in drug delivery and other biomedical or industrial applications.