TABLE OF CONTENTS The solution of linear, elliptic, homogeneous partial differential equations may be represented in terms of boundary integrals involving the unknown function and its derivatives (Stakgold, 1967, 1968). One example is the boundary integral representation of harmonic functions discussed in Section 2.3. Another example is Somigliana's identity for the displacement field in the theory of linear elastostatics (Love, 1944, p. 245; Phan-Thien and Kim, 1994). In the case of Stokes flow, we obtain a boundary integral representation involving the boundary values of the velocity and traction.
A convenient starting point for deriving the boundary integral representation is the Lorentz reciprocal identity (6.8.1) applied for a particular flow of interest with velocity u and modified stress ?, and the flow due to a point force with strength b located at a point x 0, with velocity and modified stress
Substituting these expressions into Eq. (6.8.1), setting ? ? = ? and discarding the arbitrary constant b, we obtain the equation
which is valid everywhere except at the singular point x 0.
Let us now select a control volume V c that is bounded by the closed, singly, or multiply connected surface D, which may be composed of interior fluid surfaces, fluid interfaces, or solid surfaces, as illustrated in Figure 6.9.1, and place the point force outside the control volume. Noting that the function within the square brackets in Eq. (6.9.2) is nonsingular throughout V c
