The ability to trap a single cell or microparticle in three dimensions is important for biomedical and microfluidic applications. Single-beam acoustic tweezers based on focused waves offer a compact and biocompatible solution with their high spatial resolution and intensity gradient. However, the three-dimensional (3D) trapping remains limited because the weak axial restoring radiation forces cannot always surpass the pushing drag force from the acoustic bulk streaming (ABS), particularly at high frequencies. The combined effect of the acoustic radiation force (ARF, $F_{\mathrm{rad}}$) and the drag force ($F_{\mathrm{drag}}$) induced by the ABS on a microparticle has not yet been studied in the free space. It is known that the ARF depends on the square of the pressure amplitude at the focus as $p_{\text{foc}}^2$, whereas the drag force induced by the ABS remains an open question with respect to $p_{\text{foc}}$ at different flow conditions. In this study, we propose a unified theoretical-numerical framework to compare the contribution of the ARF and the drag force by the ABS, and we systematacially derive an explicit scaling law for the streaming velocity $U_0\sim p_{\text{foc}}^n$ with the hydrodynamic flow from the viscous to the inertial regimes. It is found that the exponential power $n$ of $p_{\text{foc}}$ is bounded by $n=2$ for the viscous limit (the flow Reynolds number ${\text{Re}}_{\lambda }\ll 1$) and by $n=4/3$ for the inertial limit (${\text{Re}}_{\lambda }\gg 1$), and $n\in (4/3,2)$ throughout the transition regime (${\text{Re}}_{\lambda }\sim 1$). In addition, the Schiller-Naumann model is introduced to facilitate a more accurate estimation of the drag force compared to the Stokes model. Based on the combined effect, we find that the trap ratio of the axial ARF over the drag force can vary nonmonotonically with $p_{\text{foc}}$, which contradicts the conventional expectation of the monotone increase. This work provides a theoretical foundation for the 3D trapping of a single cell with single-beam acoustical tweezers and guidance for the optimized parameters.

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• Acoustic forces
• Acoustofluidics