## Efficient analysis of general creeping motion of a sphere inside a cylinder

##### Abstract

Creeping motion of a sphere in viscous fluid bounded by a cylindrical surface is crucial in many physical and physiological processes. In this thesis, I describe a very efficient method to comprehensively solve a classical flow problem involving suspended colloidal particle inside a conduit. A general solution technique is proposed to determine the hydrodynamic interactions between such spherical particle and cylindrical confinement.
In order to efficiently solve the Stokesian flow-field around a sphere inside a cylinder, we formulate a general mathematical procedure which can be used to find reflection relations for a vector field at simple surfaces. Unlike other recently developed efficient algorithms, the present technique does not require any translational symmetry. The method is based on the expansion of a vector field in terms of general separable basis functions. It enables us to solve a second order linear vector equation with specified conditions at disconnected bodies defined by planar, cylindrical and spherical boundaries. Thus, one can extend the outlined methodology to describe multiparticle hydrodynamic interactions in a cylinder or an annulus.
The main focus of this article is to provide a complete description of the dynamics of a spherical particle in a cylindrical vessel. For this purpose, we consider motion of the sphere in both quiescent fluid and pressure-driven parabolic flow. First, we determine the force and torque on a translating-rotating particle in quiescent fluid in terms of general friction coefficients. Then we assume an impending parabolic flow, and calculate the force and the torque on a fixed sphere as well as the linear and angular velocities of a freely moving particle. The results are presented for different radial positions of the particle and different ratios between the sphere and the cylinder radius. Because of the generality of the procedure, there is no restriction in relative dimensions, particle-position and direction of motion. For the limiting cases of geometric parameters, our results agree with the ones obtained by past researchers using different asymptotic methods.