Collision and rebound of small droplets in an incompressible continuum gas
Authors: GOPINATH ARVIND , KOCH DONALD L. 
Published: November 2006

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.