Interaction between micro-particles in Oseen flows by the method of fundamental solutions

Abstract

The aim of this study is to investigate interactions between two identical spherical particles in viscous flows using the method of fundamental solutions (MFS). Oseen flow is considered when the Reynolds number is small. Axis-symmetric problems are studied where the flow is parallel to the axis joining the centre of the two spheres. Fluid velocity and pressure can be represented by a linear combination of fundamental solutions, i.e. ring-distributed point forces on a fictitious boundary outside of the fluid domain. The densities of these point forces are determined by enforcing boundary conditions on the capillary wall and the surface of the particles. The MFS approach avoids boundary integrations and, therefore, achieves a much simpler system matrix. Interactions between two spherical particles in a boundless flow field as well as in a small capillary with a parabolic flow are investigated to illustrate the simplicity and accuracy of the method. Forces acting on each of the two particles are calculated at different Reynolds numbers and with varying distances between the two particles. It is found that for small but non-zero Reynolds numbers, the drag force on the leading particle is always bigger than that on the trailing particle. This agrees to experimental observations that blood cells in capillaries form columns of cells in steady state. Furthermore, it is found that the difference in the drag force on the leading and the trailing particles is bigger at larger Reynolds number. At given Reynolds number, the difference in dragging forces decreases as the distance between the two particles increases. The Oseen flow approximation enables the application of the MFS in studying interactions between particles in small but non-zero Reynolds number flows. The proposed MFS method is simple to implement and gives satisfactory results.