Department of Chemical and Biological Engineering, University of Colorado, Boulder, CO 80309-0424, USA.
J Colloid Interface Sci. 2012 Oct 1;383(1):167-76. doi: 10.1016/j.jcis.2012.06.014. Epub 2012 Jun 23.
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.
开发了一种有效的算法来确定在受限空间中受重力作用的变形液滴的三维形状，该算法采用人工进化到稳定状态。在模拟过程中，基于局部偏离杨氏-拉普拉斯方程和相邻固体形状的合理设计的正常“速度”来推进液滴表面，以接近捕获的液滴形状。液滴向静态捕获形状的人工“时变”进化要求液滴界面的自由部分最终满足杨氏-拉普拉斯方程，并且液滴-固体界面的液滴与固体表面接触部分符合固体表面。这种解决方案的显著优点是使用简单、数值高效的“速度”来构建到稳定状态的进化；液滴与受限空间的固体边界接近接触的涂覆区域不必预先指定，而是在解决方案过程中找到。基于现实时间推移的替代方法（例如边界积分）对于确定捕获状态将更加昂贵。研究了重力诱导变形液滴进入三维受限空间沉降时的捕获条件和液滴形状。对于接近临界条件的情况，其中捕获液滴的稳定状态不再存在，通过节点重新连接的被动网格稳定化、网格松弛和拓扑网格变换的组合来处理严重的表面网格变形。对于大于临界值的 Bond 数，液滴具有足够的可变形性，可以通过受限空间的孔，而不会被捕获。临界 Bond 数是通过将均方根（rms）表面速度的最小值与大于临界值的相应 Bond 数进行线性拟合来确定的，然后将 Bond 数外推到 rms 速度为零的位置（即液滴被捕获）。对于与重力矢量平行的轴的环形和双曲管限制，捕获液滴和临界 Bond 数的结果与以前高度精确的轴对称方法[1]获得的结果非常吻合。此外，对于在倾斜的三球限制中捕获的液滴的静态形状，三维杨氏-拉普拉斯和边界积分方法显示出良好的一致性。对于研究的所有限制类型，包括圆形环、双曲管和三个和四个球体的聚集体，临界 Bond 数几乎与液滴与孔尺寸比的增加呈线性增加。相比之下，限制类型和倾斜角度（即重力矢量与限制孔平面法线之间的角度）通常对临界 Bond 数的影响较小。
