Coupled capillary-cavitation model for secondary droplet breakup in high-intensity ultrasonic field

A coupled physics-based model is developed to describe secondary breakup of levitating droplets in a high-intensity ultrasonic standing wave field. The model uniquely integrates acoustic radiation pressure-induced droplet deformation, parametric excitation of capillary waves, radial oscillations and collective interaction of cavitation bubbles, and energy accumulation in the system. The methodology involves simultaneous numerical solution of the Mathieu-Hill equation, the Rayleigh-Plesset equation with inter-bubble pressure coupling, and energy conservation principles. A critical gas volume fraction α∼cr∼≈ 0.35-0.38 is identified as the bifurcation threshold between two distinct breakup regimes. Below this value, capillary-wave fragmentation dominates, yielding a fragmentation time of ∼250-300 ms and a minimum droplet size limited by the capillary wavelength (∼24.7 μm for water at 22 kHz). Above the threshold, collective bubble collapse prevails, reducing fragmentation time to ∼10-15 ms and producing droplets an order of magnitude smaller (∼1 μm), as predicted by a Weber-criterion formulation incorporating bubble wall collapse velocity. Analytical expressions for threshold acoustic pressures required to achieve minimum droplet size are derived and validated numerically. For thin disk-shaped droplets (kh<< 1, relevant for droplets < 0.5 mm), the critical excitation pressure is frequency-independent and can reach 84 kPa, whereas for larger droplets it ranges between 15-19 kPa at 22-44 kHz. Predicted final droplet radii for water (28 μm), glycerol (22 μm), and ethanol (18 μm) at 35 kPa and 22 kHz agree quantitatively with experimental data. The proposed model offers a predictive framework for controlling aerosol dispersity by tailoring gas content, initial droplet size, and acoustic field parameters, thereby enabling energy-efficient, contactless atomization technologies with tunable droplet characteristics.