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In fluid dynamics, d'Alembert's paradox is a contradiction reached by French mathematician Jean le Rond d'Alembert in 1752, who proves that for — incompressible and inviscid — potential flow the drag force is zero on a body moving with constant velocity through the fluid.
Zero drag of inviscid flow is in direct contradiction to measurements finding substantial drag for bodies moving through fluids, such as air and water, especially at high velocities (corresponding with high Reynolds numbers).## Viscous friction: Saint-Venant, Navier and Stokes

## Thin boundary layers: Prandtl

The German physicist Ludwig Prandtl suggested in 1904 that the effects of a thin viscous boundary layer possibly could be the source of substantial drag. Prandtl put forward the idea that a no-slip boundary condition causes — also at high velocities and high Reynolds numbers — a strong variation of the flow speeds over a thin layer near the wall of the body. This leads to the generation of vorticity and viscous dissipation of kinetic energy in the boundary layer. The energy dissipation, which is lacking in the inviscid theories, results for bluff bodies in separation of the flow. The low pressure in the wake region results in a large form drag — larger than the friction drag due to the viscous shear stress at the wall. ## Open questions

## Zero drag in potential flow

### Potential flow

### Zero drag

^{−3} — corresponding to a dipole potential field in case of a three-dimensional body of finite extent — where r is the distance to the centre of the body. The integrand in the volume integral can be rewritten as follows:
## References

### Historical

### Further reading

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D'Alembert concluded, working on a 1749 Prize Problem of the Berlin Academy on flow drag: "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 to elucidate". And a physical paradox indicates flaws in the theory.

Fluid mechanics was thus from start discredited by engineers, 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.

The occurrence of the paradox is due to the neglect of the effects of viscosity, according to scientific consensus. In interplay with scientific experiments, there were large advances on the theory of viscous fluid friction during the 19^{th} century. With respect to the paradox, this culminated in the discovery and description of thin boundary layers by Ludwig Prandtl in 1904. The viscous effects in the thin boundary layers remain also at very high Reynolds numbers — they result in friction drag for streamlined objects, and for bluff bodies the additional result is flow separation and a low-pressure wake behind the object, leading to form drag.

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. A formal mathematical proof is missing, and difficult to provide, as in so many other fluid-flow problems modelled through the Navier–Stokes equations.

First steps towards solving the paradox were made by Saint-Venant, who modelled viscous fluid friction. Saint-Venant states in 1847:

- "But one finds another result if, instead of an ideal fluid — object of the calculations of the geometers of the last century — one uses a real fluid, composed of a finite number of molecules and exerting in its state of motion unequal pressure forces or forces having components tangential to the surface elements through which they act; components to which we refer as the friction of the fluid, a name which has been given to them since Descartes and Newton until Venturi."

However, when the flow problem is put into a non-dimensional form, the viscous Navier–Stokes equations converge for increasing Reynolds numbers towards the inviscid Euler equations, suggesting that the flow should converge towards the inviscid solutions of potential flow theory — having the zero drag of the d'Alembert paradox. Of this, there is no evidence found in experimental measurements of drag and flow visualisations. This again raised questions concerning the applicability of fluid mechanics in the second half of the 19th century.

Evidence that Prandtl´s scenario occurs for bluff bodies in flows of high Reynolds numbers, can for instance be seen in impulsively started flows around a cylinder. Initially the flow resembles potential flow, after which the flow separates near the rear stagnation point. Thereafter, the separation points move upstream, resulting in a low-pressure region of separated flow.

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:

- "...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...".

- "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 critisizes 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."

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."

The three main assumptions in the derivation of d'Alembert's paradox is that the flow is incompressible, inviscid and irrotational. An inviscid fluid is described by the Euler equations, which for an incompressible flow read

- $begin\{align\}$

Hence, we have

- $left(boldsymbol\{u\}\; cdot\; boldsymbol\{nabla\}right)\; boldsymbol\{u\}\; =\; tfrac12\; boldsymbol\{nabla\}\; left(boldsymbol\{u\}\; cdot\; boldsymbol\{u\}right)\; -\; boldsymbol\{u\}\; times\; boldsymbol\{nabla\}\; times\; boldsymbol\{u\}\; =\; tfrac12\; boldsymbol\{nabla\}\; left(boldsymbol\{u\}\; cdot\; boldsymbol\{u\}right)\; qquad\; (1)$

- $boldsymbol\{nabla\}\; left(frac\{partialvarphi\}\{partial\; t\}\; +\; tfrac12\; boldsymbol\{u\}\; cdot\; boldsymbol\{u\}\; +\; frac\; prho\; right)\; =\; boldsymbol\{0\}.$

- $frac\{partialvarphi\}\{partial\; t\}\; +\; tfrac12\; boldsymbol\{u\}\; cdot\; boldsymbol\{u\}\; +\; frac\; prho\; =\; 0,\; qquad\; (2)$

Now, suppose that a body moves with constant velocity v through the fluid, which is at rest infinitely far away. Then the velocity field of the fluid has to follow the body, so it is of the form u(x, t) = u(x − v t, 0), and thus:

- $frac\{partial\; boldsymbol\{u\}\}\{partial\; t\}\; +\; left(boldsymbol\{v\}\; cdot\; boldsymbol\{nabla\}\; right)\; boldsymbol\{u\}\; =\; boldsymbol\{0\}.$

- $frac\{partialvarphi\}\{partial\; t\}\; =\; -boldsymbol\{v\}\; cdot\; boldsymbol\{nabla\}\; varphi\; +\; R(t)\; =\; -boldsymbol\{v\}\; cdot\; boldsymbol\{u\}\; +\; R(t).$

The force F that the fluid exerts on the body is given by the surface integral

- $boldsymbol\{F\}\; =\; -\; int\_A\; p,\; boldsymbol\{n\};\; mathrm\{d\}\; S$

- $p\; =\; -\; rho\; Bigl(frac\{partialvarphi\}\{partial\; t\}\; +\; tfrac12\; boldsymbol\{u\}\; cdot\; boldsymbol\{u\}\; Bigr)\; =\; rho\; Bigl(boldsymbol\{v\}\; cdot\; boldsymbol\{u\}\; -\; tfrac12\; boldsymbol\{u\}\; cdot\; boldsymbol\{u\}\; -\; R(t)\; Bigr),$

- $boldsymbol\{F\}\; =\; -\; int\_A\; p,\; boldsymbol\{n\};\; mathrm\{d\}\; S\; =\; rho\; int\_A\; left(tfrac12\; boldsymbol\{u\}\; cdot\; boldsymbol\{u\}\; -\; boldsymbol\{v\}\; cdot\; boldsymbol\{u\}right)\; boldsymbol\{n\};\; mathrm\{d\}\; S,$

At this point, it becomes more convenient to work in the vector components. The kth component of this equation reads

- $F\_k\; =\; rho\; int\_A\; sum\_i\; (tfrac12\; u\_i^2\; -\; u\_i\; v\_i)\; n\_k\; ,\; mathrm\{d\}\; S.\; qquad\; (3)$

Let V be the volume occupied by the fluid. The divergence theorem says that

- $frac12\; int\_A\; sum\_i\; u\_i^2\; n\_k\; ,\; mathrm\{d\}\; S\; =\; -\; frac12\; int\_V\; frac\{partial\}\{partial\; x\_k\}\; left(sum\_i\; u\_i^2\; right)\; ,mathrm\{d\}\; V.$

- $frac12\; frac\{partial\}\{partial\; x\_k\}\; left(sum\_i\; u\_i^2\; right)\; =\; sum\_i\; u\_i\; frac\{partial\; u\_k\}\{partial\; x\_i\}\; =\; sum\_i\; frac\{partial(u\_iu\_k)\}\{partial\; x\_i\}$

- $-\; frac12\; int\_V\; frac\{partial\}\{partial\; x\_k\}\; left(sum\_i\; u\_i^2\; right)\; ,mathrm\{d\}\; V\; =\; -int\_V\; sum\_i\; frac\{partial(u\_iu\_k)\}\{partial\; x\_i\}\; ,mathrm\{d\}\; V\; =\; int\_A\; u\_k\; sum\_i\; u\_i\; n\_i\; ,mathrm\{d\}\; S.$

- $F\_k\; =\; rho\; int\_A\; sum\_i\; (u\_k\; u\_i\; n\_i\; -\; v\_i\; u\_i\; n\_k)\; ,\; mathrm\{d\}\; S.$

- $F\_k\; =\; rho\; int\_A\; sum\_i\; (u\_k\; v\_i\; n\_i\; -\; v\_i\; u\_i\; n\_k)\; ,\; mathrm\{d\}\; S.$

- $boldsymbol\{v\}\; cdot\; boldsymbol\{F\}\; =\; sum\_i\; v\_i\; F\_i\; =\; 0.$

- . A preprint can be found
## Notes

## External links

- Potential Flow and d'Alembert's Paradox at MathPages

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Last updated on Friday October 10, 2008 at 02:14:54 PDT (GMT -0700)

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