(646g) A Stress-Balance Based Critical Weber Number for Droplets and Bubbles

In many industrial processes, such as chemical mixing, emulsification, and spray drying, the breakup of droplets and bubbles in turbulent flows is a common yet significant phenomenon. Predicting when breakup occurs is crucial for modeling the evolution of size distributions in multiphase systems, as it governs transport and reaction rates. Several models have been proposed to describe the breakup behavior, most of which rely on a critical Weber number to define the threshold beyond which droplets or bubbles go through breakage under turbulent conditions. The Weber number We_c represents the ratio of inertial to interfacial stresses and is commonly used to predict whether deformation from turbulent flow will result in breakup. However, many of these models overlook the complex interplay of fluid properties, especially; viscosity on breakup dynamics, leading to inaccuracies in their predictions.

A critical Weber number, the concept first introduced by Martínez-Bazán et al. (1999), is described as the threshold below which droplet or bubble breakup does not occur in turbulent flows. Later studies, including those by Rivière et al. and Farsoiya et al., have also characterized the critical Weber number as the point beyond which deforming stresses from turbulence exceed the cohesive stresses—primarily interfacial tension and internal viscous resistance—holding the droplet or bubble together. The critical Weber number We_c remains a central parameter for predicting the onset of breakup and is widely used in population balance models and turbulent multiphase simulations.

Despite the physical complexity of breakup processes, most existing models define the critical Weber number using simplified or empirical values. For example, according to Rivière et al. (2021), the critical Weber number for bubbles is roughly We_c = 3, indicating that breakdown does not take place below this value. In the same way, Farsoiya et al. (2023) suggest We_c = 2.5 for low-viscosity fluids, with the critical threshold increasing exponentially with the viscosity ratio. However, these thresholds are not derived from a physical stress balance and instead result from fitting trends in numerical data. When compared against experimental observations, these values frequently fail to capture early-stage breakup events. Notably, breakups have been observed at Weber numbers below 2.5, indicating that the assumed thresholds may mischaracterize actual breakup behavior, for both low and high viscosity droplets as well as for bubbles.

To address this issue, we propose a physically derived, stress-based formulation of the critical Weber number that links breakup behavior directly to measurable fluid properties. By analyzing the interplay between deformation stress in the continuous phase and the sum of interfacial and internal viscous stresses resisting deformation, we arrive at the following criterion: We_c=1+Ca.

The above equation includes the Capillary Number Ca, which describes the ratio of internal viscous stress to interfacial stress. Rather than depending on a set or empirically adjusted value, this formulation enables the critical Weber number to change systematically with fluid viscosity and surface tension. The model captures both the low-viscosity and high-viscosity limits naturally. In areas where there is little internal viscosity, the Capillary number is small, and the model reduces to We_c ≈1, consistent with early breakup events. In contrast, for high-viscosity systems, Ca becomes significant, raising the threshold and reflecting delayed breakup due to viscous resistance.

We validated this formulation by comparing its predictions with both experimental datasets and DNS results available in the literature. In particular, we analyzed data from both droplet and bubble breakup studies across a wide range of viscosity ratios. The comparisons demonstrate that the We_c=1+Ca model consistently captures the onset of breakup, including events occurring at Weber numbers well below those predicted by existing models. Existing model critical weber numbers, such as, We_c = 2.5 or 3, fail to account for early breakages, both in low and high viscous fluid droplets and even for bubbles.

In contrast to previous models that rely on empirically assigned critical Weber numbers, our formulation is grounded in a physical stress balance. It offers a logical interpretation of stress and defining the critical weber number after which the deformation stress exceeds the total cohesive stress acting on the droplet and does the breakage. Instead of relying on particular flow conditions or simulation results, this offers a more comprehensive knowledge of breakup that is based on the fundamental fluid properties.

The findings from this work are significant. Our physically derived critical Weber number, and the analysis using both literature and our own data, show that the common assumption of no breakup below a predefined threshold (e.g., We_c=2.5) is not valid for all systems. In real systems, turbulence is not always homogeneous, and the mean shear is often present - conditions under which early breakups can occur. All that highlights the need for a model that reflects the true physical processes involved. The stress-balance formulation introduced here bridges this gap and offers a more accurate, physically consistent, and experimentally validated framework for predicting droplet and bubble breakup in turbulent flows.

This advancement has direct effects for the accuracy in estimating the modeling of population balance modeling, in the design optimization of emulsification processes, and in CFD model simulations of multiphase flows. Because the We_c=1+Ca model captures early breakups and considers the effect of viscosity more accurately, it offers a flexible and robust tool for researchers and engineers working on droplet and bubble breakup in turbulent fluid systems.