F The Boundary Integral Method

The solution of the integral equations must be done numerically in virtually all cases. In this book, we do not discuss the strictly numerical issues of solving these equations. There have been excellent books and review papers written that cover this topic at a level of depth that could not be emulated here (see, for example, the excellent book of Pozrikidis7). Instead, the focus is on explaining the principles of using (8-117) or (8-119) to formulate the boundary-integral equations. 

The key to using (8-117) or (8-119) as the basis for derivation of the boundary-integral equations is that they must be applied at the boundaries of the flow domain where boundary conditions are specified. For this purpose, we need to understand the behavior of the single- and double-layer terms as we approach a boundary, 3D. The single-layer terms are continuous in the entire fluid domain, including boundaries, provided only that the latter are sufficiently smooth. However, the double-layer terms are not continuous at 3D, but suffer a jump.11 In particular, let us define a function W(x) 

) and We(xs.) denote, respectively, the limiting values of W(x) as x approaches Xs e 3D from inside and outside the fluid domain. The function Wv(xv) is W(x) evaluated exactly at the point x = xs e 3D. The reader may well be puzzled why we should care about the value of W(x) for points x that lie out of the fluid domain. This will become evident when we consider the case of applying (8-117) or (8-119) at a liquid-liquid interface. 

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