History force on coated microbubbles propelled by ultrasound

In this paper the unsteady translation of coated microbubbles propelled by acoustic radiation force is studied experimentally. A system of two pulsating microbubbles of the type used as contrast agent in ultrasound medical imaging is considered, which attract each other as a result of the secondary Bjerknes force. Optical tweezers are used to isolate the bubble pair from neighboring boundaries so that it can be regarded as if in an unbounded fluid and the hydrodynamic forces acting on the system can be identified unambiguously. The radial and translational dynamics, excited by a 2.25 MHz ultrasound wave, is recorded with an ultrahigh speed camera at 15 × 10⁶ frames/s. The time-resolved measurements reveal a quasisteady component of the translational velocity, at an average translational Reynolds number 〈Ret〉 ≈ 0.5, and an oscillatory component at the same frequency as the radial pulsations, as predicted by existing models. Since the coating enforces a no-slip boundary condition, an increased viscous dissipation is expected due to the oscillatory component, similar to the case of an oscillating rigid sphere that was first described by Stokes ["On the effect of the internal friction of fluids on the motion of pendulums," Trans. Cambridge Philos. Soc. 9, 8 (1851)]. A history force term is therefore included in the force balance, in the form originally proposed by Basset and extended to the case of time-dependent radius by Takemura and Magnaudet ["The history force on a rapidly shrinking bubble rising at finite Reynolds number," Phys. Fluids 16, 3247 (2004)]. The instantaneous values of the hydrodynamic forces extracted from the experimental data confirm that the history force accounts for the largest part of the viscous force. The trajectories of the bubbles predicted by numerically solving the equations of motion are in very good agreement with the experiment.