A Boundary Integral Method applied to Stokes Flow
|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 error bounds as Newton-Cotes type rules but have the advantage of being more accurate. We report that, in this case, interpolation-collocation is more accurate and stable than expansion-collocation.
The main results of this work are the presentation and analysis of a highly accurate and efficient algorithm to study axi-symmetric particle interactions in Stokes flow. For rigid particles, a Hermite polynomial method is used to calculate the particle traction. The numerical integration is performed using a combination of Gauss-Legendre and Gauss-Log quadrature in conjunction with a highly accurate polynomial approximation of the complete elliptic integrals. We show that the method is stable and performs well for simple spheroidal geometries, but due to the ill-conditioned nature of the integral representation, is subject to instability for more complicated particle shapes. Two methods, based on particle curvature, are presented that increase stability and, not least, increase accuracy.
Buoyant liquid drop deformation is studied via inversion of the second kind integral equation. This representation is not subject to the ill-conditioning of the first kind equation and we focus on other aspects of the numerical implementation. They are the representation of deformable surfaces, integration and time-stepping of the dynamic simulation. Results for one, two and three drop interactions are given as well as for a toroidal drop simulation. The algorithm is compared to those given in other studies. Three drop and toroidal drop simulations have not been reported previously (to the author's knowledge) and these are compared to the earlier results as well as experimental results reported elsewhere.
[Note by John Roumeliotis]:
This is my PhD dissertation submitted to the School of Mathematics and Statistics, University College (Australian Defence Force Academy), University of New South Wales in March 2000. Chapter 3 and Appendix C may be of interest to the inequalities community. The PDF file contains 189 pages and is approx 2.5Mb in size.