D'Alembert's Paradox - Viscous Friction: Saint-Venant, Navier and Stokes

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.