On the development of the Navier-Stokes equation by Navier

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.