Kinematics of a Free Particle Moving Between Two Parallel Walls

Abstract

The understanding of some physical phenomena involved in the transport of free particles such as fibers during injection processes is an important issue. To answer some of the questions arising in such problems, we study here numerically the quasi-steady kinematics of a free cylindrical solid particle moving in a Newtonian fluid confined between two parallel plane walls taking the hydrodynamic interactions into account. This is achieved by the use of the resistance matrix technique relating the kinematics of the particle to the forces and the torques exerted on the particle and to the dissipation induced by the motion of this particle. Our approach is confirmed by asymptotical developments and by a comparison with other authors in some cases. The solutions of three practical problems are given. In the first one, the sedimentation of the particle is studied. It is found that the maximum settling velocity of the free particle is obtained at a position off the symmetry plane. The cylinder is observed to rotate counter intuitively against the direction of rolling along the adjacent wall. Moreover the angular velocity has an influence on the settling velocity when the concentration is very high. The second problem concerns the transport of a neutrally buoyant cylindrical particle in a Poiseuille flow. This study reveals that there are relative translational and angular velocities between the free particle and the undisturbed fluid particle contrary to the commonly admitted hypothesis used in several models and numerical codes. Finally the third problem is a combination of the two previous situations: the transport of a non-neutrally buoyant particle in a Poiseuille flow. Depending on the ratio of the buoyancy forces to the viscous ones, different solutions are possible and exposed. Other problems can also be solved with this approach which is less time-consuming than complex methods such as DNS.