Dvinsky A S, Popel A S
Creare Inc., P.O. Box 71, Hanover, NH 03755, U.S.A.
Department of Biomedical Engineering, School of Medicine, Johns Hopkins University, Baltimore, MD 21205, U.S.A.
Comput Fluids. 1987;15(4):391-404. doi: 10.1016/0045-7930(87)90031-4.
A general numerical scheme for solution of two-dimensional Stokes equations in a multiconnected domain of arbitrary shape [1,2] is applied to the motion of a rigid circular cylinder between plane parallel boundaries. Numerically generated boundary-conforming coordinates are used to transform the flow domain into a domain with rectilinear boundaries. The transformed Stokes equations in vorticity-stream function form are then solved on a uniform grid using an iterative algorithm. In Part I coefficients of the resistance matrix representing the forces and torque on the cylinder due to its translational motion parallel or perpendicular to the boundaries or due to rotation about its axis are calculated. The solutions are obtained for a wide range of particle radii and positions across the channel. It is found that the force on a particle translating parallel to the boundaries without rotation exhibits a minimum at a position between the channel centerline and the wall and a local maximum on the centerline. The resistance matrix is utilized to calculate translational and angular velocities of a free particle settling under gravity in a vertical channel. It is found that the translational velocity has a maximum at some lateral position and a minimum on the centerline; the particle angular velocity is opposite in sign to that of a particle rolling along the nearer channel wall except when the gap between the particle and the wall is very small. These results are compared with existing analytical solutions for a small cylindrical particle situated on the channel centerline, and with solutions of related 3-D problems for a spherical particle in a circular tube and in a plane channel. It is shown that the behavior of cylindrical and spherical particles in a channel in many cases is qualitatively different. This is attributed to different flow patterns in these two cases. The motion of a spherical particle in a circular tube has qualitative and quantitative features similar to those for a cylindrical particle in a plane channel.
一种用于求解任意形状多连通域中二维斯托克斯方程的通用数值格式[1,2]被应用于刚性圆柱体在平面平行边界之间的运动。利用数值生成的边界拟合坐标将流动域转换为具有直线边界的域。然后使用迭代算法在均匀网格上求解以涡度 - 流函数形式表示的变换后的斯托克斯方程。在第一部分中，计算了阻力矩阵的系数，该矩阵表示由于圆柱体平行或垂直于边界的平移运动或绕其轴的旋转而作用在圆柱体上的力和扭矩。针对通道内各种颗粒半径和位置获得了求解结果。发现平行于边界平移而不旋转的颗粒上的力在通道中心线与壁之间的位置处呈现最小值，而在中心线上呈现局部最大值。利用阻力矩阵计算了垂直通道中在重力作用下自由沉降颗粒的平移速度和角速度。发现平移速度在某个横向位置处有最大值，而在中心线上有最小值；颗粒角速度的符号与沿较近通道壁滚动的颗粒的角速度符号相反，除非颗粒与壁之间的间隙非常小。将这些结果与位于通道中心线上的小圆柱形颗粒的现有解析解以及圆形管和平面通道中球形颗粒的相关三维问题的解进行了比较。结果表明，在许多情况下，通道中圆柱形颗粒和球形颗粒的行为在定性上是不同的。这归因于这两种情况下不同的流动模式。圆形管中球形颗粒的运动具有与平面通道中圆柱形颗粒的运动相似的定性和定量特征。
