A Boundary integral method applied to Stokes flow
John A. Roumeliotis · UNSWorks (University of New South Wales, Sydney, Australia) · 2022
This dissertation is an examination of the application of the boundary integral equation method to describe axi-symmetric particle motion in Stokes flow. In integral form, the Stokes flow equations describe the flow at any point using a surface distribution of singularities over given boundaries. For rigid particles, the strength of the singularity distribution is unknown resulting in a Fredholm integral equation of the first kind. For free surfaces, it is the surface velocity which is unknown and this results in an equation of the second kind. The axi-symmetric integral equations are two-dimensional, linear and exhibit a logarithmic singularity via the presence of complete elliptic integrals of the first and second kind. The work in this dissertation can be divided into two parts. The first is a theoretical investigation of Fredholm integral equations of the first kind and the second part is a study of numerical techniques to simulate axi-symmetric particle (rigid and drop) dynamics in Stokes flow. The theoretical investigation is undertaken to identify and examine the major issues involved in the numerical inversion of first kind equations. Two solution techniques are compared in terms of their stability and accuracy. One is an expansion-collocation method, the other is based on interpolation-collocation. We show that first kind equations are ill-conditioned and that this manifests in instability with high frequency unknowns. It is shown that the expansion approach can fail for this precise reason and a Shanks transformation (Shanks 1955) is employed to avoid higher order terms as well as increase convergence rate. The interpolation method uses Hermite interpolation polynomials to allow the nodal behaviour of the unknown to be furnished. We show that this method is sensitive to collocation and present a method to find an optimal collocation strategy. This is done by employing the Peano kernel theory to develop a weighted trapezoid-like integral inequality. Analysis of the inequality bound reveals an optimal gridding scheme, as well as an indication of favourable collocation points. The Peano kernel theory is expanded to account for more general functions and we describe a method to obtain weighted (or product) composite quadrature rules that share the same abundance of