Numerical simulation of surface wave transition thresholds for sessile droplets under vertical vibration: Roles of dimensionless parameters

Surface-wave transitions in vertically vibrated sessile droplets are important to ultrasonic atomization, vibration-assisted deicing, and interfacial transport phenomena. Although several droplet oscillation modes have been identified, the critical transition from directly forced zonal waves to azimuthally non-uniform meridional waves remains difficult to quantify using two-dimensional axisymmetric models. In this study, single-droplet vibration experiments and three-dimensional numerical simulations are combined to investigate the surface-wave characteristics and transition thresholds of hemispherical sessile droplets. The effects of dimensionless displacement amplitude ΔD, Weber number (We), and Reynolds number (Re) are systematically examined. The results show that ΔD mainly controls disturbance growth. Once ΔD exceeds a finite critical value, the droplet transitions from an axisymmetric zonal-wave state to a non-axisymmetric meridional-wave state, while the selected wavelength remains nearly unchanged. Increasing We shortens the selected wavelength in agreement with the capillary-inertial scaling λ/D0∝We−1/3 and lowers the meridional-wave threshold. Increasing Re reduces viscous dissipation, shortens the meridional wave instability time and drives ΔD,c toward a finite, We-dependent high-Re plateau, which represents the effective inviscid-limit threshold under the corresponding operating conditions. A critical-threshold map over Re = 300–1 × 106 and We = 50–5000 further shows that meridional instability is governed jointly by disturbance persistence and capillary smoothing. These results provide a consistent physical framework for interpreting zonal-to-meridional wave transitions in vertically vibrated sessile droplets under the conditions considered here.