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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 inviscidpotential 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).

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. In interplay with scientific experiments, there were large advances on the theory of viscous fluid friction during the 19 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 — and result in 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, although a formal proof is missing.

Viscous friction: Saint-Venant, Navier and Stokes

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."

Soon after, in 1851, Stokes calculated the drag on a sphere in Stokes flow, known as Stokes law. Stokes flow is the low Reynolds-number limit of the Navier–Stokes equations describing the motion of a viscous liquid.

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.

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.

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.

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. This is illustrated by Stewartson in the long 1981 survey article :

"...great efforts have been made during the last hundred or so years to explain how a vanishingly small frictional force can have a significant effect on the flow properties."

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...".

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

"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."

For many paradoxes in physics, their resolution often lies in transcending the available theory. In the case of d'Alembert's paradox, formal mathematical resolutions are still lacking, but might be found using the mechanisms proposed by Prandtl (thin viscous boundary layers), Birkhoff (inviscid flow instability) or combinations thereof.

Zero drag in potential flow

Potential flow

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

u = 0 (conservation of mass) t u + ( u ) u = 1 ρ p (conservation of momentum) {\displaystyle {\begin{aligned}&{\boldsymbol {\nabla }}\cdot {\boldsymbol {u}}=0&&{\text{(conservation of mass)}}\\&{\frac {\partial }{\partial t}}{\boldsymbol {u}}+\left({\boldsymbol {u}}\cdot {\boldsymbol {\nabla }}\right){\boldsymbol {u}}=-{\frac {1}{\rho }}{\boldsymbol {\nabla }}p&&{\text{(conservation of momentum)}}\end{aligned}}}

where u denotes the flow velocity of the fluid, p the pressure, ρ the density, and is the gradient operator. The assumption that the flow is irrotational means that the velocity satisfies × u = 0.

Hence, we have

( u ) u = 1 2 ( u u ) u × × u = 1 2 ( u u ) ( 1 ) {\displaystyle \left({\boldsymbol {u}}\cdot {\boldsymbol {\nabla }}\right){\boldsymbol {u}}={\tfrac {1}{2}}{\boldsymbol {\nabla }}\left({\boldsymbol {u}}\cdot {\boldsymbol {u}}\right)-{\boldsymbol {u}}\times {\boldsymbol {\nabla }}\times {\boldsymbol {u}}={\tfrac {1}{2}}{\boldsymbol {\nabla }}\left({\boldsymbol {u}}\cdot {\boldsymbol {u}}\right)\qquad (1)}

where the first equality is a vector calculus identity and the second equality uses that the flow is irrotational. Furthermore, for every irrotational flow, there exists a velocity potential φ such that u = φ. Substituting this all in the equation for momentum conservation yields

( φ t + 1 2 u u + p ρ ) = 0 . {\displaystyle {\boldsymbol {\nabla }}\left({\frac {\partial \varphi }{\partial t}}+{\tfrac {1}{2}}{\boldsymbol {u}}\cdot {\boldsymbol {u}}+{\frac {p}{\rho }}\right)={\boldsymbol {0}}.}

Thus, the quantity between brackets must be constant (any t-dependence can be eliminated by redefining φ). Assuming that the fluid is at rest at infinity and that the pressure is defined to be zero there, this constant is zero, and thus

φ t + 1 2 u u + p ρ = 0 , ( 2 ) {\displaystyle {\frac {\partial \varphi }{\partial t}}+{\tfrac {1}{2}}{\boldsymbol {u}}\cdot {\boldsymbol {u}}+{\frac {p}{\rho }}=0,\qquad (2)}

which is the Bernoulli equation for unsteady potential flow.

Zero drag

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(xv t, 0), and thus:

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

Since u = φ, this can be integrated with respect to x:

φ t = ( v ) φ + R ( t ) = v u + R ( t ) . {\displaystyle {\frac {\partial \varphi }{\partial t}}=-\left({\boldsymbol {v}}\cdot {\boldsymbol {\nabla }}\right)\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

F = A p n d S {\displaystyle {\boldsymbol {F}}=-\int _{A}p\,{\boldsymbol {n}}\;\mathrm {d} S}

where A denotes the body surface and n the normal vector on the body surface. But it follows from (2) that

p = ρ ( φ t + 1 2 u u ) = ρ ( v u 1 2 u u R ( t ) ) , {\displaystyle p=-\rho {\Bigl (}{\frac {\partial \varphi }{\partial t}}+{\tfrac {1}{2}}{\boldsymbol {u}}\cdot {\boldsymbol {u}}{\Bigr )}=\rho {\Bigl (}{\boldsymbol {v}}\cdot {\boldsymbol {u}}-{\tfrac {1}{2}}{\boldsymbol {u}}\cdot {\boldsymbol {u}}-R(t){\Bigr )},}

thus

F = A p n d S = ρ A ( 1 2 u u v u ) n d S , {\displaystyle {\boldsymbol {F}}=-\int _{A}p\,{\boldsymbol {n}}\;\mathrm {d} S=\rho \int _{A}\left({\tfrac {1}{2}}{\boldsymbol {u}}\cdot {\boldsymbol {u}}-{\boldsymbol {v}}\cdot {\boldsymbol {u}}\right){\boldsymbol {n}}\;\mathrm {d} S,}

with the contribution of R(t) to the integral being equal to zero.

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

F k = ρ A i ( 1 2 u i 2 u i v i ) n k d S . ( 3 ) {\displaystyle F_{k}=\rho \int _{A}\sum _{i}({\tfrac {1}{2}}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

1 2 A i u i 2 n k d S = 1 2 V x k ( i u i 2 ) d V . {\displaystyle {\frac {1}{2}}\int _{A}\sum _{i}u_{i}^{2}n_{k}\,\mathrm {d} S=-{\frac {1}{2}}\int _{V}{\frac {\partial }{\partial x_{k}}}\left(\sum _{i}u_{i}^{2}\right)\,\mathrm {d} V.}

The right-hand side is an integral over an infinite volume, so this needs some justification, which can be provided by appealing to potential theory to show that the velocity u must fall off as r — 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:

1 2 x k ( i u i 2 ) = i u i u k x i = i ( u i u k ) x i {\displaystyle {\frac {1}{2}}{\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_{i}u_{k})}{\partial x_{i}}}}

where first equality (1) and then the incompressibility of the flow are used. Substituting this back into the volume integral and another application of the divergence theorem again. This yields

1 2 V x k ( i u i 2 ) d V = V i ( u i u k ) x i d V = A u k i u i n i d S . {\displaystyle -{\frac {1}{2}}\int _{V}{\frac {\partial }{\partial x_{k}}}\left(\sum _{i}u_{i}^{2}\right)\,\mathrm {d} V=-\int _{V}\sum _{i}{\frac {\partial (u_{i}u_{k})}{\partial x_{i}}}\,\mathrm {d} V=\int _{A}u_{k}\sum _{i}u_{i}n_{i}\,\mathrm {d} S.}

Substituting this in (3), we find that

F k = ρ A i ( u k u i n i v i u i n k ) d S . {\displaystyle F_{k}=\rho \int _{A}\sum _{i}(u_{k}u_{i}n_{i}-v_{i}u_{i}n_{k})\,\mathrm {d} S.}

The fluid cannot penetrate the body and thus n · u = n · v on the body surface. Thus,

F k = ρ A i ( u k v i n i v i u i n k ) d S . {\displaystyle F_{k}=\rho \int _{A}\sum _{i}(u_{k}v_{i}n_{i}-v_{i}u_{i}n_{k})\,\mathrm {d} S.}

Finally, the drag is the force in the direction in which the body moves, so

v F = i v i F i = 0. {\displaystyle {\boldsymbol {v}}\cdot {\boldsymbol {F}}=\sum _{i}v_{i}F_{i}=0.}

Hence the drag vanishes. This is d'Alembert's paradox.

References and notes

  1. Jean Le Rond d'Alembert, Essai d'une nouvelle théorie de la résistance des fluides, 1752
  2. Grimberg, G.; Pauls, W.; Frisch, U. (2008), "Genesis of d'Alembert's paradox and analytical elaboration of the drag problem", Physica D Nonlinear Phenomena, 237 (14–17): 1878–1886, doi:10.1016/j.physd.2008.01.015. A preprint can be found  . "here". arXiv:0801.3014 . {{cite arXiv}}: Unknown parameter |accessdate= ignored (help); Unknown parameter |version= ignored (help)CS1 maint: extra punctuation (link)
  3. Reprinted in: d'Alembert, Jean le Rond (1768), "Memoir XXXIV", Opuscules Mathématiques, vol. 5, pp. 132–138. {{citation}}: More than one of |section= and |chapter= specified (help)
  4. M.J. Lighthill (1956). "Physics of gas flow at very high speeds". Nature. 178: 343. doi:10.1038/178343a0. Report on a conference.
  5. Feynman, R.P.; Leighton, R.B.; Sands, M. (1963), The Feynman Lectures on Physics, ISBN 0-201-02116-1, Vol. 2, §41–5: The limit of zero viscosity, pp. 41–9 — 41–10.
  6. Saint-Venant, A., "Mémoire sur la théorie de la résistance des fluides. Solution du paradoxe proposé à ce sujet par d'Alembert aux géomètres. Comparaison de la théorie aux expériences", Comptes Rendu des Séances de l'Academie des Science, 24: 243–246, retrieved 15 August 2008
  7. Stokes, G.G. (1851), "On the effect of the internal friction of fluids on the motion of pendulums", Trans. Cambridge Phil. Soc., 9: 8–106. Reprinted in Stokes, G.G., vol. 3 (2nd ed. ed.), Cambridge Univ. Press {{citation}}: |edition= has extra text (help); Missing or empty |title= (help); Unknown parameter |book= ignored (help).
  8. ^ Batchelor, George (2000), An introduction to fluid dynamics, Cambridge Mathematical Library (second paperback ed.), Cambridge University Press, pp. 337–343 & plates, ISBN 978-0-521-66396-0, MR1744638
  9. Prandtl, L. (1904), Motion of fluids with very little viscosity, vol. 452, NACA Technical Memorandum
  10. Keith Stewartson, D'Alembert's Paradox, Siam Review, Vol 23(3), pp. 308-343, 1981
  11. Garrett Birkhoff, Hydrodynamics: a study in logic, fact, and similitude, Princeton University Press, 1950
  12. 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
  13. For instance, the paradox of the constancy of the speed of light in all directions, was solved by the special theory of relativity.
  14. This article follows the derivation in Section 6.4 of Batchelor, George (2000), An introduction to fluid dynamics, Cambridge Mathematical Library (second paperback ed.), Cambridge University Press, ISBN 978-0-521-66396-0, MR1744638.

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