Use of complex cross-flow representation in nonplanar oscillating-surface theory.

Abstract

In the formulation of the nonplanar oscillating-surface problem, a complex variable is introduced instead of the two transversal coordinates. Thereby, a compact relation is obtained between the complex transversal velocity, which replaces the two transversal velocity components, and the chordwise integrals, whose integrands are products of the pressure jump and part of the kernel. A particular approach for numerical evaluation of these integrals is also considered. The remaining spanwise integrals are evaluated through interpolation by the aid of Jacobi polynomials. This interpolation, together with the complex representation, leads to a generalization of the Multhopp procedure. Corresponding integration coefficients are determined by Jacobi's functions of the second kind for complex arguments.