The boundary integral equation discussed in section 2.3, provides us with a representation of a flow in terms of a dual distribution of a Green's function G and its associated stress tensor T; the two individual distributions are called the single-layer potential and the double-layer potential respectively. Odqvist (1930) noted that each of these potentials expresses an acceptable Stokes flow which, in principle, may be used independently to represent a given flow. This observation provides the basis for a new class of boundary integral methods called generalized or indirect boundary integral methods. In these methods, the flow is expressed simply in terms of a single-layer or a double-layer potential with an unknown density distribution. Imposing boundary conditions yields integral equations of the first or second kind for the densities of the distributions.

Odqvist (1930) studied the properties of the single-layer and double-layer potentials, and investigated the solutions of the integral equations that arise from generalized boundary integral representations. His work is discussed by Ladyzhenskaya (1969, Chapter 3) and Kim & Karrila (1991, Part 4, Chapter 15). In this chapter we summarize the main properties of the hydrodynamic potentials and then proceed to investigate the extent to which generalized boundary integral methods are capable of representing various types of internal and external flow.

It will be helpful to keep in mind throughout this chapter that boundary integral methods leading to integral equations of the second kind are highly preferable over those leading to integral equations of the first kind.

The goal of this book is to bring together classical and recent developments in the field of boundary integral and singularity methods for steady and unsteady Stokes flow. The targeted audience includes graduate students of engineering and applied mathematics, as well as academic and industrial researchers in the field of fluid mechanics. The material was selected so that the book may serve both as a reference monograph and as a textbook in an advanced course of fluid mechanics or computational fluid dynamics. The prerequisites are introductory fluid mechanics, real analysis, and numerical methods. Each section of every chapter is followed by a number of theoretical or computer problems whose objectives are to complement the theory, indicate extensions, and provide further insights. The references were chosen so as to provide the reader with a convenient entry to the immense literature of boundary integral, boundary element, and singularity methods.

The author would like to express his appreciation to Professor Sangtae Kim of the University of Wisconsin for giving him access to his recent work. Insightful discussions with Dr Mark Kennedy provided the motivation for developing and restating a number of results. The typescript benefitted in many different ways from the superb work of Dr Susan Parkinson, technical editor of the Cambridge University Press. Thanks are due to Dr Seppo Karilla for offering constructive comments on the early typescript.

The author would like to take this opportunity to thank Ms Audrey Hill for her indispensable encouragement and support.

Introduction

In the preceding chapters we developed integral representations of a flow in terms of boundary distributions of Green's functions and their associated stress tensor. Physically, these representations involve boundary distributions of point forces, point sources, and point force dipoles. When we applied the representations at the boundaries of the flow and imposed the required boundary conditions, we obtained Fredholm integral equations of the first or second kind for the unknown boundary functions. Two advantages of solving these equations instead of the primary differential equations are reduction of the dimensionality of the mathematical problem with respect to the physical problem by one unit, and efficient treatment of infinite flows.

Now, in section 6.5 we saw that one important aspect of the numerical treatment of the integral equations resulting from boundary integral representations is the accurate computation of the singular single-layer and double-layer integrals. True, the singularities of the corresponding kernels may be subtracted off or eliminated using a suitable identity or changing the variables of integration, but this places burdens on the computer implementation and increases the cost of the computations. One ad-hoc way to circumvent this difficulty is to move the pole of the Green's function and its associated stress tensor from the boundary D onto a surface exterior to the flow, as illustrated in Figure 7.1.1. Of course, this is an arbitrary remedy that lacks physical foundation, but nevertheless it appears worth consideration. Furthermore, taking this action one step further, we may replace the continuous surface distribution of the singularities with line or point-wise distributions.

General procedures

Analytical solutions to the Fredholm integral equations that arise from boundary integral representations are feasible only for a limited number of boundary geometries and types of flow. To compute the solution under general conditions, we must resort to approximate methods. Reviewing the available strategies for solving the Fredholm integral equations of mathematical physics, we find a variety of approximate functional and numerical methods having varying degrees of accuracy and sophistication (Kantorovich & Krylov 1958, Chapter 2; Atkinson 1976, 1980, 1990; Baker, 1977, Delves & Mohamed 1985). In a popular class of methods, originated by Nyström (1930), we produce a numerical solution by carrying out the following steps.

In the first step, we trace the domain of the integral equation D with a network of marker points or nodes, and approximate D using a set of N boundary elements that are defined with respect to the nodes (see section 6.2).

In the second step, we approximate the unknown boundary velocity u, surface force f, or density of the hydrodynamic potential q over each boundary element using a truncated polynomial expansion in terms of properly defined surface variables. The set of all local expansions contains M unknown coefficients. When the unknown boundary function is assumed to be constant over each element, M is equal to the number of elements N; otherwise, M is greater than N. To facilitate the procedure, it is convenient to identify the coefficients of the expansion over an element with the values of the unknown function at the corresponding nodes.

Introduction

Flows involving interfaces between two different fluids occur in a variety of natural, engineering, and biomechanical applications. Two examples are the flow of a suspension of bubbles, drops, or biological cells, and the flow of a liquid film over a solid surface.

The significance of an interface for the behaviour of a flow is two-fold. From a kinematical standpoint, the interface marks the permanent boundary between two adjacent regions of flow with distinct physical constants. Fluid parcels that are located away from the interface are required to reside in the bulk of the flow at all times. From a dynamical standpoint, the interface is a singular surface of concentrated force. To elucidate this interpretation we note that, in general, the surface forces acting on the two sides of an interface have different values. The difference between these values, termed the discontinuity in the surface force, Δf, depends upon the physical properties of the fluids and the structure and thermodynamic properties of the interface. This dependence may be expressed in terms of a constitutive relationship that may involve a number of physical constants, including the densities of the fluids, surface tension, surface viscosity, surface elasticity, and surface modules of bending and dilatation. An interface is active when Δf is finite, and inactive or passive when Δf = 0. An active interface plays a leading role in determining the dynamics of the flow, whereas a passive interface is simply advected by the ambient flow.

The notion of best approximation was introduced into mathematical analysis by the work of P. L. Chebyshev, who in the 1850s considered some of the properties of polynomials with least deviation from a fixed continuous function (see e.g. [2B]). Since then the development of approximation theory has been closely connected with this notion. In contrast to the early investigations which concentrated on the best approximation of individual functions, since the 1930s more effort has been put into the approximation of classes of functions with prescribed differential or difference properties. A variety of extremal problems naturally arises in this field and the solution of these problems led to the concept of exact constants.

The most powerful methods for solving the extremal problems for the best approximation of functional classes are based on the duality relationships in convex analysis. In this chapter we deal mainly with such relationships. The theorems proved in Sections 1.3-1.5 connect different extremal problems. Their solutions are of independent importance, but we shall also use them as a starting point for obtaining exact solutions in particular cases. General results connected with the best approximation of a fixed element from a metric (in particular from a normed) space and the formulation of extremal problems for approximation of a fixed set are given in Sections 1.1-1.2.

In real situations the problem of approximating the function f(i) consists of replacing it following a given rule by a closed (in different senses) function φ(t) from an a priori fixed set N and estimating the error. The final result and the difficulties in obtaining it depend heavily on the choice of the approximating set N and the method of approximation, i.e. rules which determine how the function φ corresponds to f.

In choosing the approximating set, besides ensuring the necessary precision, one also needs to have functions φ which are simple and easy to study and calculate. Algebraic polynomials and (in the periodic case) trigonometric polynomials possess the simplest analytic structure. The main focus in approximation theory since it became an independent branch of analysis has been directed at the problem of approximation by polynomials. But, around the 1960s, spline functions started to play a more prominent role in approximation theory. They have definite advantages, in comparison with polynomials, for computer realizations and, moreover, it turns out that they are the best approximation tool in many important cases.

In this chapter we give the general properties of polynomials and polynomial splines that it will be necessary to know in the rest of the book. Let us note first that this introductory material is different in nature for polynomials and splines. The reason lies not only in their differing structures but also in the fundamental difference in the linear methods used for polynomial or spline approximation. Considering polynomials (algebraic or trigonometric), our main attention is concentrated, besides the classical Chebyshev theorem, on the linear methods based on the Fourier series and their analogs.

In Chapters 4 to 7 we considered approximation problems in which the approximating set (or the sequence of sets) is fixed. In particular, exact results for estimating the approximation error in functional classes by elements of finite dimensional polynomial or spline subspaces were obtained. Is it possible to improve these estimates by changing the approximating subspace to another of the same dimension? And which estimates cannot be improved on the whole set of N-dimensional approximating subspaces?

Here, we are referring to the problem stated in Section 1.2 of finding the N-widths of functional classes M in a normed space X. The exact results from the previous chapters give upper bounds for the N-widths in the corresponding cases and now our attention will be concentrated on lower bounds for Kolmogorov N-widths. But a lower bound for the best approximations of the class M which is simultaneously valid for all N-dimensional approximating subspaces can only be obtained by using some very general and deep result. Borsuk's topological Theorem 2.5.1 stated in Section 2.5 turns out to be a suitable result in many cases.

In order to make the application of this theorem both possible and effective, one has to identify an (N + 1) parametric set MN+1 in M whose N-width is not less than the N-width of M. This depends on the metric of the space X and on the way of defining M. In the various cases the role of MN+1 may, for example, be played by a ball in subspaces of polynomials or splines, as some sets of perfect splines or their generalizations.