Slow viscous flow of two particles in a cylindrical tube - Yao, Xin (2016)
The motion of small particles in a cylindrical tube at low Reynolds number can be found in many fields, ranging from the industrial and environmental applications such as fluidization and filtration, to biomedical applications such as drug delivery in blood vessels. The major challenges in the elucidation of the hydrodynamic behavior of these systems are particle interactions within the tube, which can facilitate analysis of the aggregation and collision of particles. More specifically, the fundamental knowledge of these mechanisms provides significant insight into modelling heat and mass transfer processes of particles. However, most of the current studies on this motion concentrate on a single particle or particles along the axis with same size. The mathematical treatment of multi particles interactions in a tube is still unclear. A logical beginning towards illustration of these behaviors is addressed in this thesis by considering the flow dynamics of two spherical particles with arbitrary positions traveling within a cylindrical tube at low Reynolds number. Such case is not only of fundamental interest of aforementioned applications, but also can significantly contribute to understand hydrodynamic interactions of multi particles flow field in the tube. In order to analyze the detailed hydrodynamic interaction between the two particles, we developed a mathematical model adopting the method of reflections. In general, when the two particles travel within a cylindrical tube, each of them has the translational velocity together with the rotational velocity. Due to the no-slip, three boundary conditions are applied for the entire flow field: the surfaces of the two particles and the wall of the tube. To make the flow field analytically solvable, the method of reflections is utilized to treat the boundary conditions separately. We employ the Lamb’s general solution based on spherical harmonics and cylindrical harmonics to solve the flow field around the particles and the flow within the tube, respectively. By employing this mathematical procedure, we compute the hydrodynamic force coefficients of the particles which are dependent on the distance among the cylinder wall and the two particles. The hydrodynamic forces are also a function of particle velocities and background velocity. Our results are in agreement with the existing theory of a single particle traveling in the tube when the distance between the two particles increases. We found that the particle-particle interaction can be neglected when the separation distance is three times larger than the sum of particles radii. Furthermore, the direction of Poiseuille Flow, particle position relative to the axis and particle size can make the two particles attract and repel. Unlike the single particle case, the two particles can move laterally due to the hydrodynamic interaction. Such analysis can give us insights to understand the mechanisms of collision and aggregation of particles.