Competition between acoustic radiation force and streaming-induced drag force in focused beams for 3D cell trapping

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 provide a compact and biocompatible approach because of their high spatial resolution and strong intensity gradients. However, 3D trapping remains challenging, especially at high frequencies, because the weak axial restoring radiation force may not overcome the pushing drag force caused by acoustic bulk streaming in free space. The combined effect of acoustic radiation force and streaming-induced drag force on a microparticle has not been systematically studied. Although the radiation force scales with the square of the focal pressure amplitude p_foc, the scaling of streaming-induced drag force with p_foc under different flow conditions remains unclear. Here, we establish a unified theoretical and numerical framework to compare these two effects and derive an explicit scaling law, U0 ~ p_foc^n, for the streaming velocity from the viscous to the inertial regime. We show that n = 2 in the viscous limit (Re_lambda << 1), n = 4/3 in the inertial limit (Re_lambda >> 1), and n lies between 4/3 and 2 in the transition regime (Re_lambda ~ 1). We further introduce the Schiller-Naumann model to estimate the drag force more accurately than the Stokes model. On this basis, we find that the ratio of axial radiation force to drag can vary non-monotonically with p_foc, contrary to the conventional expectation of monotonic increase. This work provides a theoretical basis for optimizing single-beam acoustic tweezers for stable 3D trapping of single cells.