Simulations of gravity-induced trapping of a deformable drop in a three-dimensional constriction

Abstract

An efficient algorithm is developed to determine the three-dimensional shape of a deformable drop trapped under gravity in a constriction, employing an artificial evolution to a steady state. During the simulation, the drop surface is advanced using a rationally-devised normal “velocity”, based on local deviation from the Young–Laplace equation and the adjacent solid shape, to approach the trapped drop shape. The artificial “time-dependent” evolution of the drop to the static, trapped shape requires that the free portions of the drop interface eventually satisfy the Young–Laplace equation, and the drop-solid contact portions of the drop interface conform to the solid surface. The significant advantage of this solution method is that a simple, numerically-efficient “velocity” is used to construct the evolution to the steady state; the coated areas where the drop is in near contact with solid boundaries of the constriction do not have to be specified a priori, but are found in the course of the solution. Alternative methods (e.g., boundary integral) based on realistic time-marching would be much more costly for determining the trapped state. Trapping conditions and drop shapes are studied for gravity-induced settling of a deformable drop into a three-dimensional constriction. For conditions near critical, where the trapped-drop steady state ceases to exist, severe surface-mesh distortions are treated by a combination of ‘passive mesh stabilization’, mesh relaxation and topological mesh transformations through node reconnections. For Bond numbers above a critical value, the drop is deformable enough to pass through the hole of the constriction, with no trapping. Critical Bond numbers are determined by linearly fitting minima of the root-mean-squared (rms) surface velocities versus corresponding Bond numbers greater than critical, and then extrapolating the Bond number to where the minimum rms velocity is zero (i.e., the drop becomes trapped). For ring and hyperbolic-tube constrictions, with axes parallel to the gravity vector, the results for trapped drops and critical Bond numbers are in close agreement with those obtained by the previous, highly-accurate axisymmetric method [1]. Also, the three-dimensional Young–Laplace and boundary-integral methods show good agreement for the static shape of a drop trapped in a tilted three-sphere constriction. For all constriction types studied, including circular rings, hyperbolic tubes and agglomerates of three and four spheres, the critical Bond number increases nearly linearly with an increase in the drop-to-hole size ratio. In contrast, the constriction type and tilt angle, which is the angle between the gravity vector and the normal to the plane of the constriction hole, have generally a weaker effect on the critical Bond number.