Volume integrals of the products of spherical harmonics and their application to viscous dissipation phenomena in fluids

Abstract

The equations governing the conversion of kinetic energy into heat in moving viscous media are formulated as volume integrals of products of spherical harmonics. Although the formulation of the fundamental equations is classical, difficulties in the integration of certain products of generalized spherical harmonics over a sphere have permitted heretofore the treatment of only two cases. The closed, form evaluation of eight fundamental types of definite integrals of the product of spherical harmonics, some of them new, or at least missing in the literature, makes possible for the first time the evaluation of these volume integrals in closed form for arbitrary order and index. Explicit details are given for the rates of energy dissipation produced by viscous motions characterized by spheroidal as well as toroidal symmetry.