Transient effects in the translation of bubbles insonated with acoustic pulses of finite duration

The translation of a bubble under the action of an acoustic forcing finds applications in fields ranging from drug delivery to sonoluminesce. This phenomenon has been widely studied for cases where the amplitude of the forcing remains constant over time. However, in many practical applications, the duration of the forcing is not long enough for the bubble to attain a constant translational velocity, mainly due to the effect of the history force. Here, we develop a formulation, valid in the limit of very viscous flow and small-amplitude acoustic forcing, that allows us to describe the transient dynamics of bubbles driven by acoustic pulses consisting of a finite number of cycles. We also present an asymptotic solution to this theory for the case of a finite-duration sinusoidal pressure pulse. This solution takes into account both the history integral term and the transient period that the bubble needs to achieve steady radial oscillations, being the former dominant during most of the acceleration process. Moreover, by introducing some additional assumptions, we derive a simplified formula that describes fairly well the time evolution of the bubble velocity. Using this solution we show that the convergence to the steady translational velocity, given by the so-called Bjerknes force, occurs rather slowly, namely as $\tau^{-1/2}$, which explains the slow convergence of the bubble velocity and stresses the importance of taking into account the history force.