Effects of split power-law entrainment on bubble fragmentation cascades

Understanding the size distribution of bubbles due to entrainment of gas at a free surface is important to a variety of natural and engineering applications. The bubble-size distribution is the result of multiple processes, and in this work, we focus on the interaction between steady spectral entertainment and turbulence-driven fragmentation at large Weber numbers. We recently demonstrated that a simplified population balance model (S-PBM) with entertainment modeled using a cutoff power-law predicts the shape of observed bubble-size distributions for large bubbles. Analytic analysis shows that, within a weak-entrainment regime, the commonly observed -10/3 power-law size distribution of large-radius bubbles is independent of both the entrainment power-law slope and the fragmentation daughter distribution. Previous work has suggested that entrainment can contain two separate regimes, such as the surface tension-dominant and gravity-dominant regimes of turbulent entertainment, separated by the capillary scale. This work extends our cutoff power-law entertainment model to a split cutoff power-law characterized by two power-law regimes separated at a critical radius. Using S-PBM and Monte Carlo simulations, we show that the shape of the resulting bubble-size distribution is still independent of the entrainment power-law slopes when both are in the weak-entertainment regime.