The collision rate of small drops in linear flow fields

Abstract

A dilute dispersion containing small, force-free drops of one fluid dispersed in a second, immiscible in a linear flow field is considered for small Reynolds numbers and large Péclet numbers under isothermal conditions. The emphasis of our analysis is on the effects of pairwise drop interactions on their collision rate, as described by the collision efficiency, using a trajectory analysis. Simple shear flow and uniaxial extensional or compressional flow are considered. For both flows, the collision efficiency decreases with increasing drop viscosity due to the effects of hydrodynamic interactions. It also decreases as the ratio of the smaller drop radius to the larger radius decreases. For uniaxial flow, finite collision rates are predicted in the absence of interdroplet forces for all finite values of the drop size ratio and the ratio of the viscosities of the drop and suspending medium. In contrast, several kinds of relative trajectories exist for a pair of drops in simple shear flow, including open trajectories, collision trajectories, and closed and semi-closed trajectories, in the absence of interdroplet forces. When the ratio of small to large drop diameters is smaller than a critical value, which increases with increasing drop viscosity, all of the relative trajectories that start with the two drops far apart remain open (no collisions), unless in the presence of attractive forces. Attractive van der Walls forces are shown to increase the collision rates.

Nonsingular boundary integral method for deformable drops in viscous flows

“…More general and accurate expressions for the interaction force and angles can be obtained. 32,34 However, for the present case ͑equal drops at ϭ1͒ we found expression ͑25͒ sufficiently accurate. A larger error, at Caϭ0.025, can be accumulated during the time integration for the angle ␤ and also due to the choice of its initial value.…”

The effect of geometrical confinement on coalescence efficiency of droplet pairs in shear flow

“…First, due to hydrodynamic interactions, the droplet trajectories will deviate from the undisturbed ones causing only a fraction of the droplets for which the distance between their undisturbed trajectories in the vorticity or in the velocity gradient direction is less than the sum of the droplet radii to collide. By using trajectory analysis, Wang et al [11] theoretically calculated the collision efficiency for spherical droplets in simple linear flows, which is the percentage of droplets that would still collide taking into account the hydrodynamic interactions between the two droplets. This trajectory analysis assumes that the droplets are nondeformable.…”

Experimental Investigation of the Orthokinetic Coalescence Efficiency of Droplets in Simple Shear Flow

“…The mobility functions are very complex functions that describe the effect of the hydrodynamic interactions between droplets and depend on S * , λ, and the radii ratio q = R 1 /R 2 (10). is the interdroplet potential scaled with the Hamakar constant, A, defined as (10)…”