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D'Alembert's paradox

Jean le Rond d'Alembert (1717-1783)
From experiments it is known that there is always – except in case of superfluidity – a drag force for a body placed in a steady fluid onflow. The figure shows the drag coefficient Cd for a sphere as a function of Reynolds number Re, as obtained from laboratory experiments. The dark line is for a sphere with a smooth surface, while the lighter line is for the case of a rough surface. The numbers along the line indicate several flow regimes and associated changes in the drag coefficient:
•2: attached flow (Stokes flow) and steady separated flow,
•3: separated unsteady flow, having a laminar flow boundary layer upstream of the separation, and producing a vortex street,
•4: separated unsteady flow with a laminar boundary layer at the upstream side, before flow separation, with downstream of the sphere a chaotic turbulent wake,
•5: post-critical separated flow, with a turbulent boundary layer.

In fluid dynamics, d'Alembert's paradox (or the hydrodynamic paradox) is a contradiction reached in 1752 by French mathematician Jean le Rond d'Alembert.[1] d'Alembert proved that – for incompressible and inviscid potential flow – the drag force is zero on a body moving with constant velocity relative to the fluid.[2] Zero drag is in direct contradiction to the observation of substantial drag on bodies moving relative to fluids, such as air and water; especially at high velocities corresponding with high Reynolds numbers. It is a particular example of the reversibility paradox.[3]

d’Alembert, working on a 1749 Prize Problem of the Berlin Academy on flow drag, concluded: "It seems to me that the theory (potential flow), developed in all possible rigor, gives, at least in several cases, a strictly vanishing resistance, a singular paradox which I leave to future Geometers [i.e. mathematicians - the two terms were used interchangeably at that time] to elucidate".[4] A physical paradox indicates flaws in the theory.

Fluid mechanics was thus discredited by engineers from the start, which resulted in an unfortunate split – between the field of hydraulics, observing phenomena which could not be explained, and theoretical fluid mechanics explaining phenomena which could not be observed – in the words of the Chemistry Nobel Laureate Sir Cyril Hinshelwood.[5]

According to scientific consensus, the occurrence of the paradox is due to the neglected effects of viscosity. In conjunction with scientific experiments, there were huge advances in the theory of viscous fluid friction during the 19th century. With respect to the paradox, this culminated in the discovery and description of thin boundary layers by Ludwig Prandtl in 1904. Even at very high Reynolds numbers, the thin boundary layers remain as a result of viscous forces. These viscous forces cause friction drag on streamlined objects, and for bluff bodies the additional result is flow separation and a low-pressure wake behind the object, leading to form drag.[6][7][8][9]

The general view in the fluid mechanics community is that, from a practical point of view, the paradox is solved along the lines suggested by Prandtl.[6][7][8][9][10][11] A formal mathematical proof is lacking, and difficult to provide, as in so many other fluid-flow problems involving the Navier–Stokes equations (which are used to describe viscous flow).

  1. ^ Jean le Rond d'Alembert (1752).
  2. ^ Grimberg, Pauls & Frisch (2008).
  3. ^ Falkovich (2011), p. 32.
  4. ^ Reprinted in: Jean le Rond d'Alembert (1768).
  5. ^ M.J. Lighthill (1956), "Physics of gas flow at very high speeds", Nature, 178 (4529): 343, Bibcode:1956Natur.178..343., doi:10.1038/178343a0 Report on a conference.
  6. ^ a b Landau & Lifshitz (1987), p. 15.
  7. ^ a b Batchelor (2000), pp. 264–265, 303, 337.
  8. ^ a b Schlichting, Hermann; Gersten, Klaus (2000), Boundary-layer theory (8th revised and enlarged ed.), Springer, ISBN 978-3-540-66270-9, pp. XIX–XXIII.
  9. ^ a b Veldman, A.E.P. (2001), "Matched asymptotic expansions and the numerical treatment of viscous–inviscid interaction", Journal of Engineering Mathematics, 39: 189–206, Bibcode:2001JEnMa..39..189V, doi:10.1023/A:1004846400131, S2CID 189820383
  10. ^ Stewartson (1981).
  11. ^ Feynman, R.P.; Leighton, R.B.; Sands, M. (1963), The Feynman Lectures on Physics, Reading, Mass.: Addison-Wesley, ISBN 978-0-201-02116-5, Vol. 2, §41–5: The limit of zero viscosity, pp. 41–9 – 41–10.

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