# Wilhelm Oseen and Stokes' paradox

In the 1840s George Gabriel Stokes investigated the motion of a sphere through a viscous fluid. He published On the theories of the internal friction of fluids in motion in 1845. There were problems with Stokes' equations which became known as Stokes paradox which Stokes himself puzzled over in 1851. In 1911, Carl Wilhelm Oseen published an improved approximation in his paper Über die Stokes'sche Formel und über eine verwandte Aufgabe in der Hydrodynamik. The equations are also given in Oseen's book Neuere Methoden Und Ergebnisse In Der Hydrodynamik (1927).

We give below (1) an extract taken from page 622 of the book William Graebel, Engineering Fluid Mechanics (CRC Press, 2018) about Stokes paradox, and (2) an extract from Oseen's 1911 paper Über die Stokes'sche Formel und über eine verwandte Aufgabe in der Hydrodynamik.

According to Stokes' approximation of the Navier-Stokes equations at very low Reynolds numbers, no flow is possible for flow past an infinitely long cylinder of any shape in an infinite stream. This is the most subtle of these paradoxes and was not satisfactorily explained until 1955. Stokes had published a linearized approximate form of the Navier-Stokes equations and used them to successfully determine the flow past a sphere in a uniform stream. Other three-dimensional shapes could also be computed, but applications to two-dimensional flows led to contradictions. (A leading early numerical analyst did in fact publish a "solution" in the 1920s, which shows that a little knowledge can sometimes be embarrassing.) In 1927 Carl Wilhelm Oseen (1879-1944, Upsala, Sweden) published a modified approximate form of the Navier-Stokes equations that linearized the convective acceleration terms, and in Lamb's later editions of Hydrodynamics a partial explanation is given based on Oseen's results. Nevertheless, the question as to why Stokes' equation was satisfactory for three-dimensional flows while Oseen's equation was necessary for two-dimensional flows remained a puzzle for decades, and was not satisfactorily settled until 1955.

2. Oseen's 1911 paper

Stokes' formula for the resistance experienced by a sphere moving at a constant, infinitely low speed in a viscous, incompressible liquid was proven by its author ...

An attempt to improve Stokes' formulas by taking into account the neglected square terms was made by Mr Whitehead in 1888. The attempt failed due to the fact that it was impossible to determine correction terms for [certain expressions]. Mr Whitehead is inclined to relate this to the fact that when a body moves in a liquid, vortices tend to occur, but emphasizes that his result is not sufficient to prove that such vortices always occur when a sphere moves.

As is well known, Stokes' formula plays a major role in recent corpuscular theory. This has led to an experimental test of the same. This has shown that for bodies which are spherical in shape with sufficient accuracy, the formula is precise within certain limits in terms of size and velocity, a fact that clearly speaks against the assumption that the movement of a sphere always causes vortices.

The question of the type of movement which the progressive movement of a body causes in a liquid has recently been discussed in detail by Mr F W Lanchester. This author assumes that there is a limit speed for each body below which the movement of the fluid caused by the movement of the body is singularity-free, without justifying this assumption by analysing the equations of motion.

Before we go any further, we need to understand why Mr Whitehead's work has led to a negative result. The reason is simply that in deriving his formula, Stokes neglected terms that are of crucial importance at a great distance from the sphere.

Last Updated April 2020