Ellipsoidal Microhydrodynamics without Elliptic Integrals and How To Get There Using Linear Operator Theory: A Note on Weighted Inner Products

In this research note we revisit the topic of microhydrodynamics of an ellipsoid in rigid body motion to arrive at the final resolution of a 140-year-old “mystery” that was featured in the dedication paper on the same topic in the Doraiswami Ramkrishna Festschrift. There, the initial focus was on the role of the theory of self-adjoint operators as the framework for proving that the surface tractions on a sphere had to be a constant multiple of the same rigid body motions of the boundary conditions. The ellipsoid was then considered as a simple example to illustrate the loss of this behavior for nonspherical particles. That goal was accomplished because for an ellipsoid, n·x, the dot product of the surface normal n and the point x on the ellipsoid surface, is the required nonconstant multiplier. The simplicity of this result is striking and has been noticed throughout its history with a number of authors remarking on the lengthy algebraic manipulations required to prove this simple result. In keeping with ...