Session G13: Drop Interactions

Gravitational Interactions of Small Contaminated Drops ina Temperature Gradient at Finite Stokes Numbers

Relative trajectories are calculated for two sedimenting spherical drops in the presence of a vertical temperature gradient with exact methods for determining the hydrodynamic forces at finite Stokes number and low Reynolds number. The drops are covered with incompressible surfactant, and thermal convection and Brownian motion are negligible. When the Reynolds number is small, fluid inertia is negligible, and the hydrodynamic forces are linear functions of the translational velocities of the drops.  However, at nonzero Stokes numbers, drop inertia must be taken into account, and the hydrodynamic forces do not balance the applied forces. For drops in close approach, lubrication forces and attractive molecular forces are considered. In the absence of attractive molecular forces, inertia leads to asymmetry in the drops’ relative trajectories. Interesting behavior, such as two drops moving in stable, tandem motion with a finite gap, is possible in the dimensionless parameter space. Moreover, retrograde motion is observed in the numerical results, depending on the relative strength of the thermocapillary and gravitational driving forces. An important application is to raindrop growth, where the model is applicable to water droplets with radii between 10 and 30 μm.  

Oscillations of a hydrodynamic crystal lattice

When brought into close proximity, two or more droplets bouncing on a vertically vibrating fluid bath exhibit a rich spectrum of dynamical behaviour. In particular, pairs of such droplets have been shown to destabilize into oscillatory, orbiting, and promenading modes, while a collection of multiple droplets can form crystal-like lattices with highly regular structure. In this talk, we present the results of a combined experimental and theoretical study in we characterize and rationalize the behaviour of a circular chain of bouncing drops, reminiscent of a one-dimensional crystal lattice. As the vibrational forcing is increased, the stationary chain first destabilizes into an oscillatory mode and then into a striking soliton-like disturbance, before ``melting'' at sufficiently high acceleration of the bath. Similarities with models such as the Toda lattice, used to model crystal vibrations in solid-state physics, are discussed.   

Drop Squeezing through Interparticle Constrictions with Insoluble Surfactant

Despite the prevalence of surfactants in confined biological and subsurface settings, their
influence on drop dynamics under tight squeezing conditions remains largely uncharacterized.
Using a fully three-dimensional boundary-integral algorithm, the interfacial behavior of
surfactant-laden drops squeezing through tight constrictions in uniform far-field flow is
modeled under an extensive range of fluid and and surfactant properties. A characteristic
aspect of this confined and contaminated multiphase system is the rapid development of
steep surfactant-concentration gradients during the onset of drop squeezing, due to the
interplay between drop hydrodynamics and Marangoni stresses. The presence of surfactant,
even at low degrees of contamination, is found to significantly decrease the critical capillary
number for droplet trapping, due to the accumulation of surfactant at the downwind pole of the
drop and its subsequent elongation. Surfactant transport is enhanced by low drop-to-medium
viscosity ratios, at which extremely sharp concentration gradients form during various stages
of the squeezing process. Increasing the degree of contamination decreases drop squeezing
times, up to a maximum value above which the addition of surfactant negligibly affects
squeezing dynamics.   

Non-continuum hydrodynamic interactions between settling inertial droplets

As cloud droplets coalesce, the air squeezed out of the gap gives rise to a continuum lubrication force that diverges with decreasing separation, thus preventing collisions from taking place. For separations comparable to the mean free path of the gas, the non-continuum nature of the force leads to collisions in finite time. This work is a continuation of Sundararajakumar & Koch (J.Fluid M.313:283-308,1996) who analyzed the non-continuum lubrication force for the normal motion of two spheres. We extend that work to include tangential motions over the full range of Knudsen numbers (ratio of the gas mean-free path to the average droplet radius) as a function of the gap thickness. The hydrodynamic resistivity functions are obtained using a uniformly valid approximation that includes non-continuum lubrication forces at small separations and continuum hydrodynamic interactions at larger separations. These hydrodynamic forces are used to calculate the collision rate of inertial droplets settling without a background flow. Using trajectory analysis, the collision efficiency is found for different droplet size ratios, Knudsen numbers and Stokes numbers.  Eventually, this analysis will be combined with direct numerical simulations to incorporate the effects of fluid turbulence.