We study, both theoretically and experimentally, low-Reynolds number motion of a deformable drop sliding down an inclined wall. The drop is non-wetting and supported by a lubricating layer of liquid between the drop and the wall. This phenomenon has diverse applications in microfluidics and biology. We are particularly interested in steady shapes and velocities, and also in the conditions leading to drop breakup.

The problem is parametrized by the Bond number B = Drga²/s, (where Dr is the density difference, a is the non-deformed drop radius, and s is the surface tension), drop-to-medium viscosity ratio l, and the wall inclination angle from horizontal, q. For theoretical study of the problem from first principles, we developed a novel three-dimensional, multipole-accelerated boundary-integral code. The major difficulty is that gravity pushes the drop very close to the wall, and there is a lubrication region (fairly extended even at moderate Bond numbers), usually requiring high resolution and a large total number of boundary elements. Also, for small viscosity ratios (bubbles), relevant to many experimental data, a large number of boundary-integral iterations is required per time step.  In addition, reaching a steady state requires many thousand time steps, which makes multipole acceleration very essential in this study, especially at small Bond numbers, when the lubrication region is highly localized. Our method incorporates the Green function for the half-space adjacent to the wall, and drop surface partitioning into a large number of non-overlapping “patches,” each containing a large number of mesh nodes. Interaction between the  patches and/or their mirror images (with respect to the wall) are handled by multipole expansions/reexpansions combined (in rare cases) with direct node-to-node summations. For 20000 - 40000 boundary elements on a drop, the computational gain over the standard boundary-integral method is from one to two orders of magnitude, thus making long-time  simulations for such large systems with arbitrary viscosity ratios substantially more feasible.

The main quantities of interest are the steady-state drop velocity and the drop-wall clearance. Unexpectedly, viscous drops maintain smaller separations and deform more than bubbles in steady motion at a fixed Bond number over a large range of inclination angles. Gap profiles are calculated and show pronounced dimpling in the middle part for viscous drops, as compared to bubbles. In general, the drop velocity U, when scaled with Drga²/m, is a non-monotonic function of the Bond number. For example, at q = 30°, U(B) has a maximum for homoviscous drops (l = 1) , whereas non-monotonic behavior is not observed for bubbles under the same conditions. Our study also includes calculations of the critical conditions for the steady-state to occur in a wide range of parameters;  when the Bond number exceeds the critical value, the drop experiences continual elongation and eventual breakup

This  computational study is complemented by physical experiments. A glass tank is used, large enough to neglect the end effects, and a plexiglass plate serves as the plane wall.  The plate has suitable length for the drops to reach steady velocities. The suspending phase is a viscous seed oil, while the drop phase is either ultra-pure water or a silicon oil, depending on the desired viscosity ratio. Bond number variation is accomplished by varying the droplet volume. Droplets are delivered using micro-syringes, and have radii in the range 0.1-0.8 cm. Experiments show a favorable agreement with theory for the steady-state drop velocity and confirm a non-monotonic behavior of U(B). Comparison between theory and experiments has also been  made regarding  the critical conditions for drop breakup.