Modelling of quasi-static adiabatic bubble formation, growth and detachment for low Bond numbers

In an effort to lessen the computational expense of bubble growth simulations without compromising its fundamental shape characteristics, an analytical model is developed. It is substantiated using validated numerical results simulating quasi-static adiabatic bubble growth for Bond numbers less than 0.07 in which its characteristic length is the radius of the cavity from which the bubble is issuing. The model's ability to predict shape and size evolution for bubble formations is shown to predict the growth and detachment volume to be in the range from 0.05% for a 0.00137 Bond number to 3% for a 0.06032 Bond number.The model builds upon a recent numerical study which showed that the shape evolution of a quasi-static bubble formation may be idealised as a spherical segment atop a cylindrical neck for low Bond number applications. By incorporating this geometry, the present work's proposed model accounts for bubble shape transformation throughout the bubble growth cycle by including a necking phenomenon in which the bulk of the bubble rises due to an elongating base as it prepares to detach. This is accomplished by introducing: (1) a volume condition which geometrically relates the neck height with the bubble's spherical segment at detachment; (2) a force instability criterion signalling the onset of detachment which relates the size of the bubble to its Bond number and cavity radius; and (3) a neck evolution growth curve. The analytical model ties these relations together with the use of the characteristics of the proposed geometry generating a full description of quasi-static adiabatic bubble growth and detachment for low Bond number formations. The resulting predicted bubble growth characteristics, such as profile, volume, centre of gravity, truncated sphericity and aspect ratio, are presented and discussed with respect to a validated numerical treatment of the problem.