TABLE OF CONTENTS The boundary-integral representation developed in Section 6.9 provides us with a powerful method for computing Stokes flows by solving integral equations for functions that are defined over the boundaries. The important benefit of this approach is that the dimensionality of the computational problem is reduced by one unit: Computing a three-dimensional or a two-dimensional flow reduces to solving an integral equation over a two-dimensional or a one-dimensional domain representing the boundaries.
To derive the boundary-integral equation we examine the behavior of the boundary-integral representation as the field point x 0 approaches the boundary D either from the side of the flow or from the external side. Examining the singularity of the single-layer integral shows that it remains continuous as x 0 crosses D. Considering the double-layer integral, we find that if the boundary D is a Lyapunov surface, which means that it has a continuously varying normal vector, and the velocity over D varies in a continuous manner
where the plus sign applies when the point x 0 approaches D from the side of the flow, which is indicated by the direction of the normal vector, and the minus sign otherwise; PV designates the principal value of the double-layer potential defined as the value of the improper double-layer integral computed when the point x 0 is located on D. One way to derive Eq. (6.10.1) is to use the first integral identity shown in Eq. (6.9.11) as...
