D'Alembert's paradox

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. 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. 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. 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. 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. 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. 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 to elucidate'. 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. 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. 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 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). First steps towards solving the paradox were made by Saint-Venant, who modelled viscous fluid friction. Saint-Venant states in 1847: 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. In the second half of the 19th century, focus shifted again towards using inviscid flow theory for the description of fluid drag—assuming that viscosity becomes less important at high Reynolds numbers. The model proposed by Kirchhoffand Rayleighwas based on the free-streamline theory of Helmholtz and consists of a steady wake behind the body. Assumptions applied to the wake region include: flow velocities equal to the body velocity, and a constant pressure. This wake region is separated from the potential flow outside the body and wake by vortex sheets with discontinuous jumps in the tangential velocity across the interface.In order to have a non-zero drag on the body, the wake region must extend to infinity. This condition is indeed fulfilled for the Kirchhoff flow perpendicular to a plate. The theory correctly states the drag force to be proportional to the square of the velocity.In first instance, the theory could only be applied to flows separating at sharp edges. Later, in 1907, it was extended by Levi-Civita to flows separating from a smooth curved boundary.