A simple resolution of Stokes' paradox?

TL;DR

This paper presents a novel solution to Stokes' paradox by establishing a global stream function satisfying the Navier-Stokes equations at low Reynolds numbers, resolving the paradox and aligning with empirical drag data.

Contribution

It introduces a mathematically rigorous approach using a perturbative stream function and Helmholtz-Laplace approximation to resolve Stokes' paradox.

Findings

Existence of a global stream function satisfying boundary conditions.

Demonstration of the instability of the original paradox.

Good agreement of calculated drag with experimental data.

Abstract

This paper proposes a solution to Stokes' paradox for asymptotically uniform viscous flow around a cylinder. The existence of a {\it global} stream function satisfying a perturbative form of the two-dimensional Navier-Stokes equations for low Reynolds number is established. This stream function satisfies the appropriate boundary conditions on both the cylinder and at infinity, but nevertheless agrees with Stokes' original results at finite radius as the Reynolds number tends to zero. The Navier-Stokes equations are satisfied to a power-log power of the Reynolds number. The drag on the cylinder is calculated from first principles and the free parameter of the approach can be chosen to give good agreement with data on drag. In this revised working paper we put our approach on a firmer mathematical basis using the Helmholtz-Laplace equation as a linear approximation to the Navier-Stokes system. In so doing we demonstrate the instability of the original paradox. We also demonstrate the absence of a paradox of Stokes-Whitehead class, and give further theoretical constraints on the free parameters of the model.