Acoustic radiation force acting on a heavy particle in a standing wave can be dominated by the acoustic microstreaming

We numerically investigate the contribution of the microstreaming to the acoustic radiation force acting on a small elastic spherical particle placed into an ultrasonic standing wave. When an acoustic wave scatters on a particle the acoustic radiation force and the microstreaming appear as nonlinear time-averaged effects. The compressible Navier–Stokes equations are solved up to second order in terms of the small Mach number using a finite element method. We show that when the viscous boundary layer thickness to particle radius ratio is sufficiently large and the particle is sufficiently dense, the acoustic microstreaming dominates the acoustic radiation force. In this case, our theory predicts migration of the particle to the velocity node (pressure antinode).

Figure 1

The first-order COMSOL model; (a) the geometry and (b) the mesh of the model, which is symmetric with respect to the $z$ axis of the cylindrical coordinate system $(r,\theta ,z)$.

Figure 2

(a) Comparison of Eulerian streaming patterns around a rigid sphere by Rednikov and Sadhal [25], and (b) by our numerical model. The particle of $10\mu \text{m}$ radius is positioned at a velocity antinode of a standing wave with frequency of $0.6\text{MHz}$, in water. The flow direction is indicated by arrows. (c) The ARF on a polystyrene sphere, and (d) on a copper sphere in water, with respect to the frequency $f$. The particle of $1\mu \text{m}$ radius is positioned between the velocity node and the antinode.

Figure 3

The ARF on a copper sphere ($a=1\mu \text{m}$) in oil. (a) The force with respect to the relative thickness of the viscous boundary layer. The forces are normalized with respect to the force $F_{\mathrm{inv}}$ from the inviscid model [6]. The particle is positioned between the velocity node and the antinode. The boundary layer thickness changes due to the frequency sweep, indicated by the acoustic wavelength $\lambda$. (b) The ARF for different theories and our numerical model COMSOL. The force is analyzed with respect to the particle position in a standing wave with a frequency of $0.5\text{MHz}$. Our model agrees with the model by Doinikov [8] and predicts a different sign of the ARF compared to the models by Settnes and Bruus [10] and Yosioka and Kawasima [6].

Figure 4

The individual force contributions to the ARF, according to Eqs. (11, 12, 13), for a copper sphere in oil, positioned between the velocity node and the antinode. The forces are plotted with respect to the distance of the integration surface from the particle surface, i.e., $\xi =r-a$, normalized with the thickness of the viscous boundary layer $\delta$. Here, the thickness of the fluid domain was increased to $140\times a$. The ARF according to Doinikov [8] and Settnes and Bruus [10] is included for a reference.