The inertial terms in equations of motion for bubbles in tubular vessels or between plates

Abstract

Equations resembling the Rayleigh-Plesset and Keller-Miksis equations are frequently used to model bubble dynamics in confined spaces, using the standard inertial term RR+3R([middle dot]) (2)/2, where R is the bubble radius. This practice has been widely assumed to be defensible if the bubble is much smaller than the radius of the confining vessel. This paper questions this assumption, and provides a simple rigid wall model for worst-case quantification of the effect on the inertial term of the specific confinement geometry. The relevance to a range of scenarios (including bubbles confined in microfluidic devices; or contained in test chambers for insonification or imaging; or in blood vessels) is discussed.

Models of cylindrical bubble pulsation

“…In applications such as these there is interest in predicting the resonance frequency. 12 A common approach to modeling the dynamics of cylindrical bubbles formed in rigid channels is to acknowledge that the channels are finite in length and assume they open into free space, because then the inertial load of an incompressible liquid is finite and can be approximated analytically for simple geometries; see especially Leighton 14 and the references therein. For the effect of liquid compressibility to be negligible, channel length must be small compared with the acoustic length scale, which for harmonic motion is the wavelength in the liquid.…”

The use of acoustic inversion to estimate the bubble size distribution in pipelines

“…The effect of containment in a pipe on bubble resonance is rarely calculated (Leighton et al 1998a(Leighton et al , 2002. Using the worst case formulations of Leighton (2011), neglect of this effect causes a systematic error which underestimates the bubble inertia associated with its pulsation resonance by 3 per cent for bubble radii of 30 mm, rising to approximately 100 per cent for bubble of radius 1mm. However, the lowest frequency applied to the population is here 15 kHz, corresponding to a resonance with a bubble of radius approximately 200 mm where the inertia is underestimated by 20 per cent.…”

“…Since PWFF conditions never exist in practice in bubbly liquids, strictly speaking no such inversion for BSD from travelling wave attenuations and sound speed has ever done this, although clearly under some circumstances the approximation is sufficiently good (Wilson et al 2005). Note however that the method here does not account for the effects on the bubble dynamics themselves of the wall of a tank (Strasberg 1953;Leighton et al 2002) or pipe (Leighton et al 1995(Leighton et al , 1997(Leighton et al , 1998bLeighton 2011), or the departure from linearity or steady-state conditions (Clarke & Leighton 2000), all of which could be included given sufficient computational resources (Leighton et al 2004). Figure 4a plots Y j , the ratio of the imaginary part to the real part of the complex wavenumber as defined in equation (2.4), for infinite volumes of homogeneous bubbly liquid.…”

Extraterrestrial sound for planetaria: A pedagogical study

“…pulses transmitted between source and sensors on the probe to measure the atmospheric sound speed 77 during descent [17]. Future uses of sound range from the local (consideration of how the 'ears' on a 78 Martian space suit might better be able to warn an astronaut walking downhill of a rockfall behind him/her 79 if the suit microphones are placed on the boot, not the helmet [18]) to the global (using the time taken for 80 man-made or naturally occurring signals to propagate completely around the vast under-ice oceans of 81 moons of Saturn and Jupiter to infer those ocean temperatures [18]). 82 …”

“…Unfortunately at this time the video camera, 570 thermocouples and accelerometer were no longer available. A 'boiling' geyser was generated by filling into the lake (with bubble sizes considerably larger than when they first enter the riser), which here is a 580 reverberant environment (note that in a larger tube bubble dynamics will differ [79][80][81]). With the same 581 experimental arrangement, but with the lake flask empty in order to generate a shooting geyser, the 582 microphone record is shown in Fig.…”