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The left side is the number density of bubble in unit volume changing over time. Right side are some influencing factors contribute to the variation of number density(there are some other terms like coalescence term or dissolution term, I only list those dominant terms). The first term is the bubble fragmentation. The second term is the bubble degassing. The third term is the bubble entrainment. Different to Navier-Stokes equation, it is a linear equation, which can solve the bubble density in a statistically way efficiently if we can find the empirical expression of left terms.

Comparing different papers, I applied the result from Rui Ni's paper (As shown in the picture). The result ensures the power law(the super hinze bubbles tend to satisfy the slope -10/3,the sub hinze bubbles tend to reach the slope -3/2)

According to some simulation and derivation, I found degassing also satisfy a certain power law.

To ensure the power law, we tried to inject big bubbles with the slope -10/3. I validated it with real random model that the slope will finally diverges to the aimed value.