Residues of Integrals with Three-Dimensional Multipole Singularities, with Application to the Lagally Theorem
IOWA INST OF HYDRAULIC RESEARCH IOWA CITY
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An important mathematical relation in the theory of three-dimensional irrotational flow is the Gauss-Green transformation between volume and surface integrals. A very useful result of this transformation, Greens third formula, requires, in its derivation, the evaluation of the limit of a singular integral over the surface of a sphere as the radius of the sphere approaches zero. Since, in this case, the singularity is due to the potential of a source at the center of the sphere, its limit may be called the residue of a source. Similarly, limits of integrals over the surface of a sphere of vanishingly small radius, with higher-order derivatives of the source potential in the integrand, will be called residues for multipoles. The latter occur in the derivation of the Lagally theorem for the force and moment acting on a body moving in an irrotational flow when multipoles are present in the hydrodynamic singularity system within the body. In contrast to the very simple derivation of the residue occurring in Greens third formula, the evaluation of the multipole residues was a challenging application of the theory of spherical harmonics. The derivations of a set of multipole residues, which were required but not included in the aforementioned references, is in this document.
- Numerical Mathematics
- Fluid Mechanics