Abstract
After the work of Navier, the Navier-Stokes equation was re-obtained by different arguments by numerous investigators. We have chosen to revisit the approaches of Navier not only because they were pioneering, but also because, unexpectedly, by undergirding his theory on Laplace's new concept of molecular forces - thought to be capable of capturing the effects of viscosity - Navier managed to derive for the first time the ultimate equation for the laminar motion of viscous fluids. A fragile model was thus capable of generating a true prediction in comparison to other, more rigorous models of the Navier-Stokes equation. Navier's derivation appeared in two almost simultaneous publications. In the first one of them, he extended his theory for the motion of elastic solids to the motion of viscous fluids. In the second publication, Navier again derived his equation using Lagrange's method of moments, which could yield the boundary conditions. However, both derivations were not influential, and were neglected by his contemporaries and by specialized publications alike. The fact that his theory could only be applied to slow motion in capillaries may have later discouraged Navier, who abandoned his theory of fluid motion in favour of experiment-based formulations for ordinary applications.
Highlights
The first derivations of the Navier–Stokes equation appeared in two memoirs by Claude-Louis Navier (1785– 1836): Sur les lois des mouvements des fluides, en ayantegarda l’adhesion des molecules [1], here referred to as the 1st memoir, published in the Annales de Chimie et de Physique; and Sur Les Lois du Mouvement des Fluides [2], here referred to as the 2nd memoir, read at the Academy on 18th March, 1822, and which appeared in the Memoires de L’Academie Royale des Sciences de L’Institut de France for the year of 1823
It is possible to say that the equations of motion of fluids had been limited to perfect fluids since 1755, following the publication of the wellknown Euler equation of motion for non-viscous fluids [3]
Saint-Venant [9], in turn, thinking in terms of transverse pressure acting on the faces of the sliding fluid particles, obtained a stress tensor that yields the differential equations of Navier, Cauchy, and Poisson with one constant parameter. All of these 19th century investigators tried to fill the gap between the rational fluid mechanics of the perfect non viscous fluid developed in the 18th century by the Bernoullis (Daniel and Johann), d’Alembert, Euler, and Lagrange, and the actual behaviour of real viscous fluids in hydraulic systems
Summary
The first derivations of the Navier–Stokes equation appeared in two memoirs by Claude-Louis Navier (1785– 1836): Sur les lois des mouvements des fluides, en ayantegarda l’adhesion des molecules [1], here referred to as the 1st memoir, published in the Annales de Chimie et de Physique; and Sur Les Lois du Mouvement des Fluides [2], here referred to as the 2nd memoir, read at the Academy on 18th March, 1822, and which appeared in the Memoires de L’Academie Royale des Sciences de L’Institut de France for the year of 1823. Saint-Venant [9], in turn, thinking in terms of transverse pressure acting on the faces of the sliding fluid particles, obtained a stress tensor that yields the differential equations of Navier, Cauchy, and Poisson with one constant parameter All of these 19th century investigators tried to fill the gap between the rational fluid mechanics of the perfect non viscous fluid developed in the 18th century by the Bernoullis (Daniel and Johann), d’Alembert, Euler, and Lagrange, and the actual behaviour of real viscous fluids in hydraulic systems. These new discoveries show that the impact of Navier’s works on viscous flow was more profound than Navier scholars have so far proposed
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