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## D'Alembert's paradox
## Additional recommended knowledge## Solutions to the paradoxThe paradox is considered within the fluid mechanics community to have been resolved by the German physicist Ludwig Prandtl in 1904 who in the short report Recently the following alternative resolution of d'Alembert's paradox was presented [4]: The reason the zero drag potential solution of the Euler equations is not observed in experiments, is that this solution is (exponentially) unstable at separation, and develops into a turbulent Euler solution (with a slip boundary condition and thus no boundary layer prior to separation) with the drag arising from low-pressure tubes of streamwise vorticity generated at separation. This is a completely different resolution from that by Prandtl and it is supported by both theory, computation and experiment. The mathematician Garret Birkhoff in the opening chapter of his book In his 1951 review [7] of Birkhoff's book, the mathematician James J. Stoker sharply critizises the first chapter of the book: " The official standpoint of the fluid mechanics community seems to be that the paradox in principle can been solved along the lines suggested by Prandtl, even if concrete evidence is still to be provided, and the new resolution is (very) controversial. ## Boundary condition: slip or no-slip?Experiments show that the skin friction from a turbulent boundary layer decreases towards zero as Re ## References[1] Jean Le Rond dAlembert, Essai d'une nouvelle the'orie de la re'sistance des fluides, 1752. [2] Ludwig Prandtl, Motion of fluids with very little viscosity, NACA Technical Memorandum 452, 1904. [3] Keith Stewartson, D'Alembert's Paradox, Siam Review, Vol 23(3), pp. 308-343, 1981. [4] Johan Hoffman and Claes Johnson, Computational Turbulent Incompressible Flow, Springer, 2007. [5] Herrman Schlichting, Boundary layer theory, McGraw Hill, 1979. [6] Garret Birkhoff, Hydrodynamics: a study in logic, fact, and similtude, Princeton University Press, 1950. [7] James J. Stoker, Review: Garrett Birkhoff, Hydrodynamics, a study in logic, fact, and similitude, Bull. Amer. Math. Soc. Vol. 57(6), 1951, pp. 497-499. Categories: Fluid dynamics | Fluid mechanics |

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "D'Alembert's_paradox". A list of authors is available in Wikipedia. |