Ostwald ripening of buoyancy-driven microbubbles

Microbubble solutions have a wide range of industrial applications, including heat transfer, agriculture, and water treatment. Therefore, understanding and controlling the variation in bubble sizes is critical. In this study, we develop a theoretical framework for Ostwald ripening of buoyancy-driven microbubbles by introducing a height-dependent size distribution function. For the first time, we show that the population balance equation in steady state can be interpreted within the Lifshitz-Slyozov-Wagner theory when the distribution function is redefined as the density distribution of the buoyancy-induced flux. Notably, in this form of Ostwald ripening, the distribution function approaches a scaled universal distribution, not over time, but as a function of height. We analytically derive the scaled universal distribution and show that the fifth power of the mean radius of the bubbles grows linearly with height.