D'Alembert's Paradox - Open Questions

Open Questions

To verify, as Prandtl suggested, that a vanishingly small cause (vanishingly small viscosity for increasing Reynolds number) has a large effect – substantial drag — may be very difficult.

The mathematician Garrett Birkhoff in the opening chapter of his book Hydrodynamics from 1950, addresses a number of paradoxes of fluid mechanics (including d'Alembert's paradox) and expresses a clear doubt in their official resolutions:

"Moreover, I think that to attribute them all to the neglect of viscosity is an unwarranted oversimplification The root lies deeper, in lack of precisely that deductive rigor whose importance is so commonly minimized by physicists and engineers."

In particular, on d'Alembert's paradox, he considers another possible route to the creation of drag: instability of the potential flow solutions to the Euler equations. Birkhoff states:

"In any case, the preceding paragraphs make it clear that the theory of non-viscous flows is incomplete. Indeed, the reasoning leading to the concept of a "steady flow" is inconclusive; there is no rigorous justification for the elimination of time as an independent variable. Thus though Dirichlet flows (potential solutions) and other steady flows are mathematically possible, there is no reason to suppose that any steady flow is stable."

In his 1951 review of Birkhoff's book, the mathematician James J. Stoker sharply criticizes the first chapter of the book:

"The reviewer found it difficult to understand for what class of readers the first chapter was written. For readers that are acquainted with hydrodynamics the majority of the cases cited as paradoxes belong either to the category of mistakes long since rectified, or in the category of discrepancies between theory and experiments the reasons for which are also well understood. On the other hand, the uninitiated would be very likely to get the wrong ideas about some of the important and useful achievements in hydrodynamics from reading this chapter."

In the second and revised edition of Birkhoff's Hydrodynamics in 1960, the above two statements no longer appear.

The importance and usefulness of the achievements, made on the subject of the d'Alembert paradox, are reviewed by Stewartson thirty years later. His long 1981 survey article starts with:

"Since classical inviscid theory leads to the patently absurd conclusion that the resistance experienced by a rigid body moving through a fluid with uniform velocity is zero, great efforts have been made during the last hundred or so years to propose alternate theories and to explain how a vanishingly small frictional force in the fluid can nevertheless have a significant effect on the flow properties. The methods used are a combination of experimental observation, computation often on a very large scale, and analysis of the structure of the asymptotic form of the solution as the friction tends to zero. This three-pronged attack has achieved considerable success, especially during the last ten years, so that now the paradox may be regarded as largely resolved."

For many paradoxes in physics, their resolution often lies in transcending the available theory. In the case of d'Alembert's paradox, the essential mechanism for its resolution was provided by Prandtl through the discovery and modelling of thin viscous boundary layers – which are non-vanishing at high Reynolds numbers.