The effect of slight deformation on droplet coalescence in linear flows

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Abstract
A trajectory analysis is used to determine the effect of small deformations and van der Waals attractions on the collision efficiency of two non-Brownian drops freely suspended in a linear flow at small Reynolds number. Simple shear flow and uniaxial compressional and extensional flow are considered. Treating the capillary number (Ca) as a small parameter permits an approach similar to matched asymptotic expansions. For Ca≪1, the analysis shows that the deformation is mainly axisymmetric and that the tangential motion of the drops in apparent contact is unaffected to leading order by the small deformation. A comparison with full three-dimensional boundary-integral calculations confirms the accuracy of the asymptotic approach. In the dimensionless parameter space, results for the collision efficiency are mapped for four parameters: Ca, size ratio, drop-to-medium viscosity ratio, and a dimensionless Hamaker parameter. For spherical drops in uniaxial compression and extension, the collision efficiencies are identical due to the reversibility of Stokes flow. When small deformation is introduced, however, the collision efficiencies are lower for compression than for extension. For slightly deformable drops in simple shear flow, the critical capture cross section upstream is no longer a circle, in contrast to the behavior of spherical drops in the absence of van der Waals forces. For all flow types, a key result is that the collision efficiency decreases rapidly from the corresponding value for spherical drops, as the capillary number increases beyond a critical value, due to small deformations. Consequently, droplet growth by coalescence will be arrested when the drops reach a prescribed size, as shown by population dynamics simulations for a model physical system.