Particle collection by permeable drops

The interaction of a small, solid particle with a nearby fluid droplet covered with a thin, permeable membrane is considered for linear ambient flows under conditions where viscous forces dominate. A bispherical-coordinate solution was developed to describe the relative motion along and normal to the line-of-centers between the particle and drop. The effect of the permeability of the drop membrane is relatively weak, except in near contact where the lubrication pressure in the narrow gap between the particle and drop surfaces can cause significant permeation of fluid across the membrane, as further described by a lubrication analysis. These results were then used in a trajectory analysis to predict particle collection rates by permeable drops in a dilute suspension undergoing uniaxial extensional (or compressional) flow. Even a small amount of permeation allows for a large increase in the particle collision rate with the drop interface. For example, the collision efficiency (collision rate divided by that in the absence of hydrodynamic interactions) is 0.17 for equisized particles and drops with a dimensionless permeability $k/a=0.0001$ (where $k$ is the membrane permeability per unit thickness, and $a$ is the reduced radius of the drop and particle). In contrast, it is identically zero (due to the singular lubrication resistance) for smooth, impermeable spheres in the absence of attractive molecular forces and fluid slip.

Figure 1

Coordinate system for the axisymmetric motion of a spherical particle relative to a spherical drop centered at the origin.

Figure 2

Example trajectories for $a_p/a_d=0.25$ and $k/a=0.0001$ in uniaxial extension. The limiting or grazing trajectory is shown with the particle at its ends; only those trajectories inside the limiting trajectory end in contact. The trajectories are symmetric about the $x_3$ axis, and all lengths are nondimensionalized using the drop radius.

Figure 3

Schematic of the lubrication region of close contact between a particle and a membrane-covered drop.

Figure 4

Mobility function along the line-of-centers for a spherical solid particle approaching a spherical permeable drop due to a linear flow field. The solid curves are numerical results using bispherical coordinates, the dashed curves are the far-field results for impermeable spheres using Eq. (45), and the dotted lines are the near-field results using Eq. (51) for $K^{*}\ll 1$.

Figure 5

Lubrication force for a solid particle interacting with a permeable drop, normalized by that for two solid particles. The solid curve is the numerical solution, the dotted curve is from Eq. (59), and the dashed curve is from Eq. (61).

Figure 6

Relative mobility function normal to the line-of-centers for a spherical particle approaching a permeable drop in a linear flow field. The solid curves are numerical results using bispherical coordinates, and the dashed curve is the far-field result for impermeable spheres using Eq. (47). For each size ratio, results are shown for $K^{*}=0$ (bottom curve) and 0.1 (top curve), with $K^{*}={10}^{-3}$ and ${10}^{-5}$ also included for $\lambda =0.5$ and 0.25.

Figure 7

Collision efficiency of a solid, spherical particle and a permeable, spherical drop in uniaxial extensional flow versus size ratio for different dimensionless permeabilities.

Figure 8

Collision efficiency of a solid, spherical particle and a permeable, spherical drop in uniaxial extensional flow versus dimensionless permeability for different size ratios.