The concept of ‘drift’, which has been exploited in many high Reynolds number and inviscid flow problems, is here applied to examine transport by a spherical viscous droplet (of radius $a$) moving in a Stokes flow.
In an unbounded flow, the velocity in the direction of translation of a spherical droplet is positive everywhere because streamlines, in the fluid frame of reference, ‘close’ at infinity. Fluid particles are displaced a positive distance, $X$, forward, which is expressed in terms of the initial distance from the stagnation streamline $\rho_0$. Asymptotic expressions are developed for $X$ in the limits of $\rho_0/a\,{\ll}\,1$ and $\gg 1$. The nature of the singularity of the centreline displacement changes from $O(-a\log(\rho_0/a))$ to $O(a^2/\rho_0)$ as the viscosity of the droplet, compared to the ambient fluid, increases. By employing a mass-conservation argument, asymptotic expressions are calculated for the partial drift volume, $D_p$, associated with a circular material surface of radius $\rho_m$ which starts far in front of a droplet that translates a finite distance. Since the velocity perturbation decays slowly with distance from the droplet, $D_p$ tends to become unbounded as $\rho_m$ increases, in contrast to inviscid flows.
The presence of a rigid wall ensures that the velocity perturbation decays sufficiently rapidly that fluid particles, which do not lie on the stagnation streamline, are displaced a finite distance away from the wall. The distortion of a material surface lying a distance $h_L$ above a wall, by the droplet, starting a distance $h_S$ from the wall and moving away, is studied. The volume transported away from the wall, calculated using a multipolar flow approximation, is $D_p = \pi h_L^2 a(3\lambda+2)/(\lambda+1)$, and is weakly dependent on the starting position of the droplet, in accordance with numerical results. When the material surface is close to the wall ($h_L/a \ll 1$), the volume transported away from a wall is significantly smaller than for inviscid flows because the no-slip condition on the rigid wall tends to inhibit ‘scouring’. When the material surface is far from the wall ($h_L/a\gg 1$), the viscously dominated flow transports a larger volume of fluid away from the wall because the flow decays slowly with distance from the droplet.
These results can be generalized to arbitrarily shaped bodies, since the transport processes are dominated by the strength of the Stokeslet. The effect of boundaries and inertia on fluid transport processes is briefly discussed.