(429e) Gravity-Induced Trapping of a Deformable Drop in a Three-Dimensional Constriction

The phenomenon of gravity-induced trapping of a deformable drop in a constriction is relevant to many technical applications, including drop infiltration into a highly porous surface and oil recovery. Of particular interest is determining the critical conditions delineating the boundary between drop trapping and drop squeezing though the constriction. Determining these conditions by dynamical boundary-integral simulations is especially difficult for three-dimensional gravity-induced drop motion in comparison to flow-induced squeezing¹. As critical conditions are approached for gravity-induced simulations, the solution becomes very lubrication-sensitive, and the drop-solid clearance is extremely small. Therefore, a special static algorithm for calculating trapped configurations is meaningful to avoid very costly boundary-integral calculations.

One difficulty of a static algorithm is that the wetted area of drop-solid contact is not known a priori. For an axisymmetric geometry (e.g., drop trapped in a toroidal constriction), it is possible to integrate the static Young-Laplace differential equation from the tips along the drop arc length, for both the sessile and pendant portions, and calculate the free parameters in the equations by Newton-Raphson iterations. The free parameters satisfy pressure continuity throughout the drop and the total drop volume constraint². Such an approach, however, can not be generalized for three-dimensional constrictions, for example, when an axisymmetric constriction is tilted relative to the gravity vector. The present method, suitable for three-dimensional trapping conditions, is to utilize an artificial, ?time-dependent' process of drop evolution to achieve the static state. The specially designed normal ?velocity' of the drop surface contains both a local deviation from the Young-Laplace static condition and the drop-solid clearance along the normal of the drop. Asymptotically at large simulation times, a trapped state is reached, where the open parts of the surface satisfy the Young-Laplace condition, and the wetted portions of the drop shape in contact with the solid boundaries are found automatically. For near-critical conditions, there is a severe surface mesh distortion, during the three-dimensional trapping process, and the algorithm employs high-resolution surface triangulations, with the mesh quality being maintained by ?passive mesh stabilization,' mesh relaxation and topological transformations through node reconnections. The artificial ?time-dependent' process is reminiscent of the swelling technique for generating a start-up configuration of a highly-concentrated emulsion in a granular material³.

A key dimensionless parameter is the Bond number, representing the ratio of gravitational and interfacial-tension forces. For Bond numbers above the critical value, the artificial ?time-dependent' process results in drop motion through the hole of the constriction, with an observed minimum in the r.m.s. normal velocity of the drop. The critical Bond number is found through extrapolation, by linear fitting the Bond numbers greater than the critical value versus the minimum r.m.s. normal velocity. Simulation results, including static, trapped drop surfaces and critical Bond numbers for circular and parabolic ring constrictions, show excellent agreement with results obtained from the highly-accurate, axisymmetric Young-Laplace method. Also, the results from the three-dimensional trapping algorithm for a three-sphere constriction are compared to boundary-integral simulation results. Critical Bond number trends are explored for the following constrictions: circular ring (with and without tilt), parabolic ring (with and without tilt), three-sphere constriction, and four-sphere constriction. The critical Bond number, below which trapping occurs, is independent of the drop-to-medium viscosity ratio, but it increases with increasing drop-to-hole size, due to the requirement of greater deformation for a drop to squeeze through a smaller hole.

1. Zinchenko, A. Z., & Davis, R. H., A boundary-integral study of a drop squeezing through interparticle constrictions. J. Fluid Mech. 564, 227?266 (2006).

2. Ratcliffe, T. J., Zinchenko, A. Z., & Davis, R. H., Buoyancy-induced squeezing of deformable drop through an axisymmetric ring constriction. Phys. Fluids Submitted.

3. Zinchenko, A. Z., & Davis, R. H., Algorithm for direct numerical simulation of emulsion flow through a granular material. J. COMPUT PHYS. 227, 7841?7888 (2008).