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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.〔Jean le Rond d'Alembert (1752).〕 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.〔Grimberg, Pauls & Frisch (2008).〕 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.〔Falkovich (2011), p. 32.〕 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 (mathematicians - the two terms were used interchangeably at that time ) to elucidate"''.〔Reprinted in: Jean le Rond d'Alembert (1768).〕 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.〔 Report on a conference.〕 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.〔Landau & Lifshitz (1987), p. 15.〕〔Batchelor (2000), pp. 264–265, 303, 337.〕〔 , pp. XIX–XXIII.〕〔 〕 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.〔〔〔〔〔Stewartson (1981).〕〔, Vol. 2, §41–5: The limit of zero viscosity, pp. 41–9 – 41–10.〕 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). ==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.〔. Reprinted in 〕 Stokes flow is the low Reynolds-number limit of the Navier–Stokes equations describing the motion of a viscous liquid.〔The Stokes flow equations have a solution for the flow around a sphere, but not for the flow around a circular cylinder. This is due to the neglect of the convective acceleration in Stokes flow. Convective acceleration is dominating over viscous effects far from the cylinder (Batchelor, 2000, p. 245). A solution can be found when convective acceleration is taken into account, for instance using the Oseen equations (Batchelor, 2000, pp. 245–246).〕 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.〔Batchelor (2000), pp. 337–343 & plates.〕 This again raised questions concerning the applicability of fluid mechanics in the second half of the 19th century. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「D'Alembert's paradox」の詳細全文を読む スポンサード リンク
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