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 ρd and viscosity μd. The collision takes place in an incompressible continuum gas with ambient density ρg [Lt ] ρd, ambient pressure p′∞ and viscosity μg [Lt ] μd The gas–liquid interface is surfactant free with interfacial tension σ. The Weber number based on the drop density, Wed ≡ ρdU′2ca/σ [Lt ] 1 and the capillary number based on the gas viscosity, Cagg ≡ μgU′cσ [Lt ] 1. The Reynolds number characterizing flow inside the drops satisfies Red ≡ aU′cρd/μd [Gt ] We1/2d and the Stokes number characterizing the drop inertia, St ≡ 2Wed(9Cag)−1 ≡ 2(ρdU′caμ−1g)/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−1/4d) [Gt ] 1, rebound occurs on a time scale t′b = &23frac;1/2πaWe1/2dln1/2 (We−1/4d)U′−1c. 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 Wed < 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 Cag and the parameter χ = (Wed/Cag)1/2 so that St ≡ 2/9χ2. 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(aCa1/2). 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, executing oscillatory motions before finally coming to rest. As the Stokes number increases with Cag held fixed, more energy is stored as deformation energy and the maximum radial extent of the dimple increases accordingly. For St [Gt ] 1, no oscillations in the centroid positions are observed, but the temporal evolution of the minimum gap thickness exhibits two minima. One minimum occurs during the dimple evolution process and corresponds to the minimum attained by the dimple rim. The second minimum occurs along the axis of symmetry when the dimple relaxes, a tail forms and then retracts. A detailed analysis of the interface shapes, pressure profiles and the force acting on the drops allows us to obtain a complete picture of the collision and rebound process.