A bubble hit by an ultrasound wave consisting of pulsed ultrasound with three peaks in this case. The wave propagates from below and travels with the speed of sound, , in an upwards direction. At , the wave front, indicated by a horizontal line, first makes contact with the bottom part of the bubble. The bubble will start to interact with the sound wave after this instant. The bubble has an initial radius . The wave is not drawn to the same scale as the bubble.
Pulsed ultrasound with increasing intensities of 1000 (pulse 1), 3000 (pulse 2), 5000 (pulse 3), and (pulse 4) as used by Xu et al. (2005) , see also Table I . The number indicated relates to these four pulses (1–4). All four pulses start off with a tensile component that will cause the bubbles to expand before they are forced to collapse by one of the compressive components of the pulse. Each pulse has four tensile parts and three compressive parts. The time scale is relative, due to the finite value of the velocity of sound. The wave arrives earlier at certain parts of the bubble and then progressively moves across the whole bubble. The plot is staggered, in order to show the four pulses in one graph. The plots consist of the sum of the acoustic and the atmospheric pressure: thus all pulses have bar for . In the simulations, every bubble collapsed at times smaller than . The pulses have a main frequency of .
The shape of a microbubble of an initial radius of after it has been hit by pulse 1, which propagates from below. The bubble expands from its initial size at , indicated by the thick solid line at the center of the plot, and expands more or less spherically to a maximum radius, , at . The bubble then starts to collapse aspherically with the development of an upward-directed jet which eventually impacts on the opposite bubble surface at . Mainly shapes in the collapse phase are shown, the corresponding time is also indicated for each curve. The bubble moves upwards and a jet develops in the same direction as the propagation direction of the pulse. The jet attains a velocity of about upon impact (see also Fig. 5 ).
Variation of equivalent bubble radius with time for microbubbles with radii ranging from 1 to interacting with pulse 1. Also indicated is the pressure variation with time for pulse 1 ( , with axis on the right-hand side) at the location . The bubbles obtain maximum radii between 25 and and collapse between 1.23 and . The collapse occurs within the first cycle of the pulsed ultrasound wave.
Jet velocity, , as a function of the initial radius for the four pressure pulses, pulses 1–4. The jet velocity remains approximately constant at a value around for a large range of values of and does not depend significantly on pulse strength. All bubbles collapse during the first cycle of the ultrasound pulse.
Bubble shape just before jet impact for a bubble with initial radius of interacting with pulses 1 and 4. The jet tip width is much wider for pulse 4 and is about twice the value of that for pulse 1 . The vertical displacement of the bubble is also considerably larger for pulse 4 than that of pulse 1 . The jet velocities are comparable, around for both pulses. The initial location of the center of the bubble for both simulations is .
Kelvin impulse, , as a function of the initial radius for pressure pulses 1–4. The Kelvin impulse becomes lower, and even attains negative values, for larger values of . The highest values of are found for the strongest pulse, pulse 4. All bubbles in this plot collapse during the first cycle of the ultrasound pulse.
Bubble shapes for a bubble with initial radius interacting with pulse 1. (a) First contraction stage starting from 0.94 (most outer curve) to (most inner curve). The initial bubble shape is also indicated (dotted line). (b) Second expansion phase from 1.76 (most inner curve) to (most outer curve). (c) Second contraction stage from 2.24 (most outer curve) to (most inner curve). A ring jet is formed at the sides of the bubble, which impacts at and causes the bubble to break up in two parts.
Oscillation of a bubble with initial radius interacting with pulse 4. The positions of the bubble surface on the -axis, and (upper and lower value of the bubble on the axis of symmetry) are indicated as a function of time. The bubble undergoes three oscillations before it collapses with an upward directed jet. pulse 4 is also indicated as a function of time. At jet impact occurs: the upper and lower -curves touch at this instant. The shape of the bubble just before collapse can be found in Fig. 10 .
Collapse of a bubble with initial radius interacting with pulse 4. The shape of the bubble during the phase leading to the final collapse is shown here from 3.46 (most outer curve) to (most inner curve). The bubble assumes very complicated shapes and finally develops an upwards jet, whereas a ring jet is also being created. The final upwards jet has a velocity of prior to impact.
Effect of surface tension: Velocity of the bottom part of the bubble (jet velocity) as a function of time for a bubble where surface tension has been taken into account (solid line) and for a bubble with surface tension neglected (dashed line). The initial radius of the bubble is and it is hit by pulse 1 (similar to the case of Fig. 3 ).
The effect of surface tension on the displacement. The -position of the bottom node on the axis of symmetry , the position where the jet occurs, for a bubble with initial radius and which is hit by pulse 1 as a function of time. Results with inclusion of surface tension, solid line, and without surface tension, dashed line, are shown.
Intensity and peak pressures, both negative and positive, of the first cycle of the pulsed ultrasound waves, pulses 1–4. Their effects on the collapse time , the maximum radius of the microbubbles, and the translation of the bubble center at the moment of jet impact for initial bubble radii and . All other bubbles investigated, ranging from , obtain values between these two extremes.
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