Effect of initial Rayleigh mode on drop deformation and breakup under impulsive acceleration

One of the fundamental ways of representing a droplet shape is through its Rayleigh-mode decomposition, in which each mode corresponds to a distinct surface-energy content. The influence of these modes on free oscillation dynamics has been studied extensively; however, their role in droplet deformation, breakup, and fragmentation under impulsive acceleration remains largely unexplored. Here we systematically quantify how prescribed initial axisymmetric Rayleigh modes affect the deformation and breakup of an impulsively accelerated drop. Using experimentally validated, VOF-based multiphase direct numerical simulations, we isolate the coupled effects of finite-amplitude surface oscillation modes and the associated initial surface-energy state by initializing drops with well-defined $(n,0)$ modes (and phases) while conserving volume at finite amplitudes. We show that breakup is governed not simply by the initial drag of the imposed shape, but by the dynamic coupling between the free modal oscillations and the forced aerodynamic (or shear-driven) deformation: constructive superposition can strongly amplify deformation, whereas destructive superposition can stabilize the drop even under otherwise disruptive forcing. Across all systems studied, the outcome is controlled by how efficiently the external work is partitioned into recoverable oscillatory energy versus centre-of-mass translation and viscous dissipation, with viscosity and density ratio acting as key mediators that respectively damp modal interactions and restrict the time window for energy uptake.

💡 Research Summary

This paper investigates how prescribed initial Rayleigh surface‑oscillation modes influence the deformation and breakup of a liquid drop subjected to an impulsive acceleration. While the role of Rayleigh modes in free oscillations is well documented, their impact under external aerodynamic or shear forcing has received little attention. The authors systematically initialize drops with axisymmetric (n, 0) Rayleigh modes—specifically n = 2, 3, 4—at a finite amplitude (A = 0.3 D) and two extreme phase angles (0 and π). To preserve the drop volume, a corrective n = 0 term is added, and the base radius is solved from a cubic volume‑conservation equation.

Three representative fluid systems are examined: (i) a water drop in air at the breakup threshold (ρ≈815, Oh_d≈1.3×10⁻³, We≈12), (ii) a highly viscous water drop (Oh_d≈1.3×10⁻¹, We≈18), and (iii) a water drop in another liquid with comparable density and viscosity (ρ≈10, Oh_d≈1.3×10⁻³, We≈10). The simulations are performed with the open‑source Basilisk VOF solver, which resolves the incompressible two‑phase Navier‑Stokes equations and implements surface tension via the Continuum Surface Force model.

The key findings are as follows. First, when the natural Rayleigh period (2π/ω_n) is comparable to the aerodynamic deformation time scale τ_D, the imposed mode can either constructively amplify the forced deformation (if the initial phase aligns with the external loading) or destructively suppress it (if the phase is opposite). This phase‑dependent interaction can shift the outcome from stable deformation to catastrophic breakup. Second, liquid viscosity strongly damps the initial oscillations and modifies mode coupling; high‑viscosity drops dissipate oscillatory energy rapidly, reducing constructive interference and raising the critical Weber number for breakup. Third, the density ratio controls how much of the external work is converted into translational kinetic energy of the drop’s centre of mass versus surface deformation; larger density ratios favour centre‑of‑mass motion, thereby extending the time window for energy uptake and delaying breakup.

An energy‑budget analysis decomposes the external work into recoverable oscillatory energy, centre‑of‑mass translation, and viscous dissipation. Breakup occurs when a substantial fraction (>~40 %) of the work feeds the oscillatory mode, whereas dominance of translation or dissipation (>~50 %) leads to deformation without fragmentation. This framework extends traditional Weber‑number‑based breakup criteria by incorporating the initial surface‑energy state and modal dynamics.

The authors conclude that initial Rayleigh modes and their phases are decisive parameters in impulsively accelerated drop dynamics, with viscosity and density ratio acting as mediators of modal interaction and energy partitioning. They suggest future work on non‑axisymmetric modes, multi‑mode initial conditions, and experimental validation to broaden the applicability of their findings to spray atomization, raindrop breakup, and fire‑retardant dispersal.