Bubble rise in molten glasses and silicate melts during heating and cooling cycles

The Hadamard-Rybczynski equation describes the steady-state buoyant rise velocity of an unconfined spherical bubble in a viscous liquid. This solution has been experimentally validated for the case where the liquid viscosity is held constant. Here, we extend this result for non-isothermal conditions, by developing a solution for bubble position in which we account for the time-dependent liquid viscosity, liquid and gas densities, and bubble radius. We validate this solution using experiments in which spherical bubbles are created in a molten silicate liquid by cutting gas cavities into glass sheets, which are stacked, then heated through the glass transition interval. The bubble-bearing liquid, which has a strongly temperature-dependent viscosity, is subjected to various heating and cooling programs such that the bubble rise velocity varies through the experiment. We find that our predictions match the final observed position of the bubble measured in blocks of cooled glass to within the experimental uncertainty, even after the application of a complex temperature-time pathway. We explore applications of this solution for industrial, artistic, and natural volcanological applied problems.