Genesis of d'Alembert's paradox and analytical elaboration of the drag problem

TL;DR

This paper explores the historical development of d'Alembert's paradox, highlighting key contributions from Euler, Borda, d'Alembert, and Saint-Venant, and clarifies the origins of the paradox in fluid dynamics.

Contribution

It provides a detailed historical analysis of the emergence of d'Alembert's paradox and the analytical understanding of drag in ideal fluids.

Findings

Euler nearly proved vanishing drag for arbitrary shapes

Borda linked drag to the principle of live forces

d'Alembert established the paradox for symmetric bodies

Abstract

We show that the issue of the drag exerted by an incompressible fluid on a body in uniform motion has played a major role in the early development of fluid dynamics. In 1745 Euler came close, technically, to proving the vanishing of the drag for a body of arbitrary shape; for this he exploited and significantly extended existing ideas on decomposing the flow into thin fillets; he did not however have a correct picture of the global structure of the flow around a body. Borda in 1766 showed that the principle of live forces implied the vanishing of the drag and should thus be inapplicable to the problem. After having at first refused the possibility of a vanishing drag, d'Alembert in 1768 established the paradox, but only for bodies with a head-tail symmetry. A full understanding of the paradox, as due to the neglect of viscous forces, had to wait until the work of Saint-Venant in 1846.