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

Abstract

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.

Figures (22)

• Figure 1: The harmonic modes for the fundamental oscillation $n=2$ at an extreme phase. The $(2,0)$ "zonal" mode is axisymmetric, while the $(2,1)$ "tesseral" and $(2,2)$ "sectoral" modes are non-axisymmetric.
• Figure 2: The figure shows 3-dimensional renders of the two extreme deformation phases, $0$ and $\pi$, for the $(2,0)$, $(3,0)$, and $(4,0)$ axisymmetric Rayleigh oscillation modes, which are the focus of the current study.
• Figure 3: The original Rayleigh mode description fails to conserve the volume of the drop for non-infinitesimal amplitudes, resulting in a drop with a volume greater than the volume of the parent sphere. This error in drop shape and subsequent increase in drop volume is represented by the black outer region for each case, while the light blue inner region represents the drop shape corrected using equations \ref{['eq:rayleigh_shape_n0_general_main']} and \ref{['eq:volume_conservation_final_main']}. All four drops result from the uncorrected Rayleigh modes on a sphere of diameter $D=1$ (volume is $\pi/6 \approx 0.523$). $A = 0.5$ is the amplitude of the Rayleigh mode(s), $\mathcal{V}$ is the volume of the deformed drop, and $E_\mathcal{V}$ is the error in the volume of the deformed drop.
• Figure 4: (a) The computational domain for the problem under consideration. (b) The three oscillation modes and the corresponding two initial phases of amplitude $0.3$ imposed on a sphere of diameter $1$.
• Figure 5: The impulsive acceleration of a prolate drop in a vertical wind tunnel for two different wind tunnel velocities is simulated in both axisymmetric and three-dimensions using Basilisk. Time ($t^*$) is scaled by the deformation timescale $\tau_D = \sqrt{\rho}(D/V_0)$, and length is scaled by the drop diameter ($8$ mm).