The code popBalance.py computes the histogram of droplet sizes from the breakup rate. We follow the model of Garrett (2000), in which drops of diameter $d$ break into $m$ equally sized daughter drops. Volume conservation gives the daughter-drop diameter $d' = d / m^{1/3}$.

In steady state, the rate of drop removal equals the rate of creation:

$$ N(d'),\tau(d'),d' = m,N(d),\tau(d),d, $$

where $\tau(d)$ is the lifetime of a drop before breakup, and $N(d)\Delta d$ is the number of drops with diameters in the range $d$ to $d+\Delta d$.

The classical inertial-range estimate for the breakup lifetime is the eddy turnover time:

$$ \tau_d = \frac{d^{2/3}}{\epsilon^{1/3}}, $$

which applies when surface tension is negligible.

The model proposed by Coulaloglou & Tavlarides (1977) and measured by Vela-Martín et al. (2022) accounts for the suppression of breakup near the Hinze scale:

$$ d_H = 0.725,\sigma^{3/5}\rho^{-3/5}\epsilon^{-2/5}. $$

This yields an exponential breakup lifetime:

$$ \tau_{CT} = \frac{d^{2/3}}{14.8,\epsilon^{1/3}} \exp!\left(\frac{-7.8,\sigma}{\rho,\epsilon^{2/3} d^{5/3}}\right). $$

We begin with a single drop of size $8d_H$, which lies in the inertial range, making the subsequent cascade insensitive to the precise starting size.

At each step, the population-balance equation gives $N(d')$ from the known $N(d)$, letting us iteratively solve toward smaller drop sizes using:

• $\tau=\tau_d$ (empty circles)
• $\tau=\tau_{CT}$ (filled circles)

This corresponds physically to injecting drops of size $8d_H$ at a constant rate and letting them break up in turbulence.

Both models converge to the inertial-range prediction:

$$ N(d) \sim (d/d_H)^{-10/3}, $$

because far above $d_H$ both lifetimes scale as $d^{2/3}$, guaranteeing the classical $d^{-10/3}$ steady-state drop-size distribution.

Near $d_H$, however, the Coulaloglou–Tavlarides lifetime predicts thousands of times more drops, as surface tension greatly prolongs the breakup time.

In this analysis we use binary breakups ($m=2$), but the results are insensitive to the choice of $m$ and to any prefactor multiplying the breakup rate (such as the 14.8 in $\tau_{CT}$). This robustness arises because the population balance depends only on the ratio $\tau(d)/\tau(d')$, causing such constants to cancel.

1. C. A. Coulaloglou and L. L. Tavlarides. “Description of Interaction Processes in Agitated Liquid-Liquid Dispersions.” Chemical Engineering Science, 32(11), 1289–1297 (1977). DOI: 10.1016/0009-2509(77)85023-9

2. Chris Garrett, Ming Li, and David Farmer. “The Connection between Bubble Size Spectra and Energy Dissipation Rates in the Upper Ocean.” Journal of Physical Oceanography, 30(9), 2163–2171 (2000). DOI: 10.1175/1520-0485(2000)030<2163:TCBBSS>2.0.CO;2

3. Alberto Vela-Martín and Marc Avila. “Memoryless Drop Breakup in Turbulence.” Science Advances, 8(50), eabp9561 (2022). DOI: 10.1126/sciadv.abp9561