Stokes flow past an ellipsoid in a special ambient flow field: A marvelous cubic equation for a quadratic ambient field

Abstract

AbstractA three‐dimensional eigenspace appears in the analysis of an ellipsoid immersed in quadratic ambient viscous flow fields. To celebrate the mathematical achievements of Babatunde Ogunnaike, we focus on mathematical details of the eigenvalues and eigenfunctions of the double‐layer operator that appears in the integral representation of velocity fields as a function of the shape parameters of the ellipsoid. Three special quadratic ambient fields that lack an ambient pressure gradient (toroidal fields) form a decoupled subsystem with eigenvalues associated with three real roots of a cubic equation. The discriminant of this cubic vanishes at discrete points in the parameter space of the ellipsoidal shape revealing repeated roots (and thus repeated eigenvalues) of the cubic. The discrete nature of these points is illustrated with explicit derivation of the discriminant's level set contours in the region near the repeated eigenvalues for the sphere.