New Insights on the Stokes Paradox for Flow in Unbounded Domains

TL;DR

This paper reviews the Stokes paradox in unbounded flow domains, introduces new analytical insights, and explores how modifications to boundary conditions affect flow stability and paradox resolution.

Contribution

It provides a novel, rigorous derivation of the Stokes paradox and examines the effects of boundary condition modifications on flow stability and paradox resolution.

Findings

Relaxing no-slip condition can resolve the paradox but causes d'Alembert's paradox.

The paradox persists in Reynolds--Orr equations, indicating increasing flow instability with domain size.

New analytical approaches deepen understanding of flow instability and paradox resolution.

Abstract

Stokes flow equations, used to model creeping flow, are a commonly used simplification of the Navier--Stokes equations. The simplification is valid for flows where the inertial forces are negligible compared to the viscous forces. In infinite domains, this simplification leads to a fundamental paradox. In this work we review the Stokes 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's 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.