Collision and rebound of small droplets in an incompressible continuum gas

We study the head-on collision between two weakly deformable droplets, each of radius a (in the range 10–150 μm), moving towards one another with characteristic impact speeds ± U ′ c . The liquid comprising the drop has density ρ d and viscosity μ d . The collision takes place in an incompressible continuum gas with ambient density ρ g [Lt ] ρ d , ambient pressure p ′ ∞ and viscosity μ g [Lt ] μ d The gas–liquid interface is surfactant free with interfacial tension σ. The Weber number based on the drop density, We d ≡ ρ d U ′ 2 c a /σ [Lt ] 1 and the capillary number based on the gas viscosity, Cag g ≡ μ g U ′ c σ [Lt ] 1. The Reynolds number characterizing flow inside the drops satisfies Re d ≡ a U ′ c ρ d /μ d [Gt ] We 1/2 d and the Stokes number characterizing the drop inertia, St ≡ 2 We d (9 Ca g ) −1 ≡ 2(ρ d U ′ c a μ −1 g )/9 is O (1) or larger. We first analyse a simple model for the rebound process which is valid when St [Gt ] 1 and viscous dissipation in both the gas and in the drop can be neglected. We assume that the film separating the drops only serves to keep the interfaces from touching by supplying a constant excess pressure 2σ/ a . A singular perturbation analysis reveals that when ln( We −1/4 d ) [Gt ] 1, rebound occurs on a time scale t ′ b = &23frac; 1/2 π aWe 1/2 d ln 1/2 ( We −1/4 d ) U ′ −1 c . Numerical results for Weber numbers in the range O (10 −6 ) − O (10 −1 ) compare very well to existing experimental and simulation results, indicating that the approximate treatment of the bounce process is applicable for We d < 0:3. In the second part of the paper we formulate a general theory that not only models the flow inside the drop but also takes into account the evolution of the gap width separating the drops. The drop deformation in the near-contact inner region is determined by solving the lubrication equations and matching to an outer solution. The resulting equations are solved numerically using a direct, semi-implicit, matrix inversion technique for capillary numbers in the range 10 −8 to 10 −4 and Stokes numbers from 2 to 200. Trajectories are mapped out in terms of Ca g and the parameter χ = ( We d / Ca g ) 1/2 so that St ≡ 2/9χ 2 . For small Stokes numbers, the drops behave as nearly rigid spheres and come to rest without any significant rebound. For O (1) Stokes numbers, the surfaces deform noticeably and a dimple forms when the gap thickness is approximately O ( aCa 1/2 ). The dimple extent increases, reaches a maximum and then decreases to zero. Meanwhile, the centroids of the two drops come to rest momentarily and then the drops rebound, ex