Time-evolutions of bubbling in viscous liquid are characterized by two stages, i.e., the bubble formation stage and the bubble coalescence stage. We have focused on the former stage of bubbling and investigated analytically and numerically bubble formation in viscous liquid decompressed with a constant rate. Bubble formation consists of two important processes, namely nucleation and growth of bubbles due to exsolution of volatile elements from liquid. To describe the growth processes, some authors used the growth model proposed by Toramaru [1995]. This model describes bubble expansion due to decompression in viscous incompressible magma and also time-evolution of volatile concentration. However, it has serious disadvantages; (1) All bubbles are assumed to be equal in size neglecting the bubble size distribution. (2) Bubbles grow in incompressible magma which can be approximated to be extended infinitely. If a bubble sphere enlarges its radius in incompressible fluid, infinite moment is needed to push its surrounding fluid. To overcome these difficulties, we have developed more realistic model of bubble growth, taking into account bubble size distribution and compressibility of surrounding liquid. Numerical solutions show that formation of bubbles changes its behavior abruptly when viscosity of liquid becomes larger than a critical value. If the liquid viscosity is higher than the critical value, bubble growth processes are controlled principally by the liquid viscosity. If the liquid viscosity is lower than the critical values, processes of nucleation and growth of bubbles are swayed by diffusivity of volatile in liquid. Analytical solutions are obtained for the duration time of nucleation and the maximum number density of bubbles. They are consistent with the numerical results in both cases.
In a high viscosity case, it is observed that size distributions of bubbles are well approximated by a power law at the time when the nucleation rate becomes maximum and gradually change to an unimodal size distribution as bubbles grow more. In a case of extreme high viscosity, it maintains an exponentially decreasing function with increasing bubble size. These characteristic time-evolutions of the number density and size distribution of bubbles would be useful in evaluating material quantities such as the diffusivity, the viscosity, and the surface tension of liquid from experimental results. Furthermore our results indicate that bubbles begin to coalesce with each other from conditions that bubbles have almost an equal size. Such information would provide initial conditions in constructing a theoretical model on the bubble coalescence stage.
K. Yamada, Doctor thesis,2005