New insights on the Stokes paradox for flow in unbounded domains

Stokes flow equations, used to model creeping flow, are a commonly used simplification of the Navier–Stokes equations. In infinite domains, they admit a fundamental paradox. In this work we review this paradox and present new insights related to recent research. We approach the paradox from three different points of view: modern functional analysis, numerical simulations, and classical analytic techniques. The first approach yields a novel, rigorous derivation of the paradox. We also show that relaxing the Stokes no-slip condition (by introducing a Navier friction condition) in one case resolves the Stokes paradox but gives rise to d’Alembert’s paradox. The Stokes paradox has previously been resolved by Oseen, who showed that it is caused by a limited validity of Stokes’ approximation. We show that the paradox still holds for the Reynolds–Orr equations describing kinetic energy flow instability, meaning that flow instability steadily increases with domain size. We refer to this as an instability paradox.