Periodic sedimentation of three particles in periodic boundary conditions

Solutions of the equations of Stokesian dynamics for point particles are found for periodic boundary conditions with three particles per unit cell of a simple cubic lattice. Two particles per cell move with equal velocity, but three particles per cell usually lead to irregular motion. Special situations are of interest, where initially the distance vector of one pair is parallel to one of the horizontal axes of the cubic lattice and the members of this pair are at equal distance to the third particle. By symmetry the configuration keeps this character during the motion, and numerically the motion is found to be periodic. In our numerical work we have studied mostly the case where the initial triangle is horizontal and equilateral. We have shown that the periodic motion is neutrally stable for sizes of the initial triangle less than the critical size. Such stable solutions of the equations of Stokesian dynamics are of relevance to the theory of sedimentation. In these solutions the particles move coherently in a complicated fashion with the same mean sedimentation velocity and a periodic internal motion of the three-particle cluster. If initially the particles are sufficiently widely separated, but the motion is still stable, the mean sedimentation velocity is less than that of a single particle. In this case the solution describes a situation of hindered settling. If the initial triangle is too large, with the side length larger than the critical size, the two base particles team up with partners in neighboring cells, and we get separate motion of a base pair and a single particle with different mean vertical velocities and with periodic motions superimposed. The columns of horizontal pairs pass the columns of apex particles. The corresponding solutions are unstable. (author)