Observations of the dynamics of two coated bubbles in ultrasound. (a) Layout of the experiment (side view). A microscope objective is used to focus the laser traps (optical tweezers) and for transmission imaging. The direction of incidence of the ultrasound beam is orthogonal to the line of centers . (b) Frames from an ultrahigh speed time series of bubble dynamics (top view). The recording is taken at , corresponding to an interframe time of 70 ns. Here only every second frame is shown. The black crosses in the first and last frames indicate the initial positions of the bubble centers. The distance between the bubbles decreases due to the secondary Bjerknes force. White scale bar: (enhanced online). [URL: http://dx.doi.org/10.1063/1.3227903.1]10.1063/1.3227903.1
System of coordinates. is the radius of bubble and is its position on the line of centers . denotes the position of a fluid element relative to bubble .
(a) Time evolution of the radii obtained by image tracking. The solid symbols represent experimental data points and the lines represent the resampled and filtered radius-time curves. The bubbles oscillate in phase and have relative radial excursions . (b) Time evolution of the distance between the centers, . The solid symbols represent experimental data points and the line represents the resampled and filtered distance-time curve. [(c) and (d)] Time evolution of the Reynolds numbers computed from the experimental radial and translational dynamics. Only the values for bubble 1 are plotted for clarity. The radial Reynolds number is below 25 with a time average . The translational Reynolds number is below 5 with a time average .
Comparison of models for the viscous force. The values are computed for one bubble from the experimental values of the radius and position and their derivatives. Dashed line: experimental value from the force balance . The solid symbols show the values corresponding to the experimental data points. Solid line: model including quasisteady drag and history force . Dashed-dotted line: a model including only quasisteady drag , neglecting history force, largely underestimates viscous dissipation.
Time evolution of the distance between the centers of the bubbles. Dashed line: experimental value (after resampling and filtering). The solid symbols are the measured data points. Solid line: prediction using the model including history force and quasisteady drag, . The shaded area represents the tolerance (±5%) on the prediction due to the systematic experimental uncertainty on the resting radius of a bubble. Inset: comparison of the model including only the quasisteady drag (dashed-dotted line) with the model including history force, (solid line).
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