Improper integrals as a puzzle for creeping flow around an ellipsoid

Abstract

The general ellipsoid immersed in the quadratic ambient velocity field Hijkxjxk with the origin at the ellipsoid center will exhibit an additional Faxén law contribution to the hydrodynamic drag because of the ambient pressure gradient. Here we reexamine the solution of S. Kim and P. Arunachalam [“The general solution for an ellipsoid in low-Reynolds-number flow,” J. Fluid Mech. 178, 535–547 (1987)] for Stokes flow by focusing more carefully on the system of equations and boundary condition on the disturbance velocity field. All of the unknown coefficients (quadrupole moments) in the disturbance velocity were determined by Kim and Arunachalam from a linear system of equations that arise from matching the quadratic terms in the disturbance and ambient fields. Note that when matching the boundary conditions, they interpreted the constant terms in the disturbance velocity field as a complicated but formal expression for the drag on the ellipsoid. Kim and Arunachalam did not explore this further and instead opted for the simpler expression for the drag from a direct application of the Faxén law to the quadratic ambient field. Years later, we provide the resolution of the “mystery” of two seemingly different expressions for the hydrodynamic drag on the ellipsoid by proving their equivalence. One interesting consequence is the emergence of rather formidable integrals that are identically zero. Uniqueness theorems and related properties for Stokes flow provide a guarantee that the two expressions for the drag are equivalent, but we also provide a direct proof via the properties of elliptic integrals and thus provide formal closure to the problem of the ellipsoid in a quadratic ambient field.