# Historical Note: d’Alembert and His Paradox

You can well imagine the frustration that Jean le Rond d’Alembert felt in 1744 when, in a paper entitled “Traite de l’equilibre et des mouvements de fluids pour servir de siute au traite de dynamique,” he obtained the result of zero drag for the inviscid, incompressible flow over a closed two-dimensional body. Using different approaches, d’Alembert encountered this result again in 1752 in his paper entitled “Essai sur la resistance” and again in 1768 in his “Opuscules mathematiques.” In this last paper can be found the quote given at the beginning of Chapter 15; in essence, he had given up trying to explain the cause of this paradox. Even though the prediction of fluid – dynamic drag was a very important problem in d’Alembert’s time, and in spite of the number of great minds that addressed it, the fact that viscosity is responsible for drag was not appreciated. Instead, d’Alembert’s analyses used momentum principles in a frictionless flow, and quite naturally he found that the flow field closed smoothly around the downstream portion of the bodies, resulting in zero drag. Who was this man, d’Alembert? Considering the role his paradox played in the development of fluid dynamics, it is worth our time to take a closer look at the man himself.

d’Alembert was born illegitimately in Paris on November 17, 1717. His mother was Madame De Tenun, a famous salon hostess of that time, and his father was Chevalier Destouches-Canon, a cavalry officer. d’Alembert was immediately abandoned by his mother (she was an ex-nun who was afraid of being forcibly returned to the convent). However, his father quickly arranged for a home for d’Alembert—with a family of modest means named Rousseau. d’Alembert lived with this family for the next 47 years. Under the support of his father, d’Alembert was educated at the College de Quatre-Nations, where he studied law and medicine, and later turned to

mathematics. For the remainder of his life, d’Alembert would consider himself a mathematician. By a program of self-study, d’Alembert learned the works of Newton and the Bernoullis. His early mathematics caught the attention of the Paris Academy of Sciences, of which he became a member in 1741. d’Alembert published frequently and sometimes rather hastily, in order to be in print before his competition. However, he made substantial contributions to the science of his time. For example, he was (1) the first to formulate the wave equation of classical physics, (2) the first to express the concept of a partial differential equation, (3) the first to solve a partial differential equation—he used separation of variables—and (4) the first to express the differential equations of fluid dynamics in terms of a field. His contemporary, Leonhard Euler (see Sections 1.1 and 3.18) later expanded greatly on these equations and was responsible for developing them into a truly rational approach for fluid-dynamic analysis.

During the course of his life, d’Alembert became interested in many scientific and mathematical subjects, including vibrations, wave motion, and celestial mechanics. In the 1750s, he had the honored position of science editor for the Encyclopedia— a major French intellectual endeavor of the eighteenth century which attempted to compile all existing knowledge into a large series of books. As he grew older, he also wrote papers on nonscientific subjects, mainly musical structure, law, and religion.

In 1765, d’Alembert became very ill. He was helped to recover by the nursing of Mile. Julie de Lespinasse, the woman who was d’Alembert’s only love throughout his life. Although he never married, d’Alembert lived with Julie de Lespinasse until she died in 1776. d’Alembert had always been a charming gentleman, renowned for his intelligence, gaiety, and considerable conversational ability. However, after Mile, de Lespinasse’s death, he became frustrated and morose—living a life of despair. He died in this condition on October 29, 1783, in Paris.

d’Alembert was one of the great mathematicians and physicists of the eighteenth century. He maintained active communications and dialogue with both Bernoulli and Euler and ranks with them as one of the founders of modem fluid dynamics. This, then, is the man behind the paradox, which has existed as an integral part of fluid dynamics for the past two centuries.