Figure 7 Influence of the interfacial tension on drop breakup frequency. (a) with different drop sizes, ε = 3 m²s^-3; (b) with different turbulent energy dissipation rates, d = 1 mm

For systems with high viscosity, the effect of drop viscosity on drop breakup behaviors needs to be considered.^10,37 The breakup probability and breakup frequency model constructed in Section 3 is only applicable to describe the breakup behavior of low-viscous drops, thus it is necessary to extend the model by considering the effect of drop viscosity. According to the drop oscillation breakup mechanism, the influence of the drop viscosity is mainly reflected in two aspects:

1) The viscous hamper increases with the increase of drop viscosity, which is manifested by the decrease of the oscillation amplitude with increasing the viscosity and thus leads to the increase of drop breakup time. The breakup time considering the effect of drop viscosity can be modeled by Eq. in Section 2.

2) The occurrence of the drop breakup should satisfy the energy constraint, in addition to overcoming the increase in the maximum interfacial energy during the breakup process, it is necessary to consider the effect of the increase in the drop viscous energy.

For a droplet with a size of d , the viscous energy it contains can be expressed as:

Where p_vis is the viscous energy per unit drop volume and can be calculated by Eq.:

Therefore, the increase in viscous energy during the drop breakup process is expressed as:

Based on the above analysis, the breakup constraint is expressed as Eq.:

Then it is obtained that：

Thus, we obtain the extension of the drop breakup probability model by considering the effect of drop viscosity, as shown in Eq..

Where Oh is the Ohnesorge number, which characterizes the relative magnitude of the viscous and interfacial stresses:

Combined with the breakup time model of Eq. in Section 2, the extended breakup frequency model can be expressed as:

In our previous work¹⁰, we investigated the effect of dispersed phase viscosity on drop breakup frequency and obtained the direct experimental data, which is applied to verify the accuracy of Eq.. Figure 8 shows the comparison of the predicted results with the initial breakup experimental data. Compared with the breakup model of Eq. which is constructed for low-viscous drops, Eq. is in better agreement with the experimental data. Moreover, Figure 8 shows that the drop breakup probability gradually decreases as Oh increases.

Based on the analysis in the above sections, we can conclude that for low-viscous drops, drop breakup probability is mainly determined by the Weber number (We_L ), and for intermediate to high viscous drops, the breakup probability is determined by the combined influence of the Weber number (We_L ) and the Ohnesorge number (Oh ).