Saint-Venant attended lectures at the Collège de France and the lecture notes he took in Liouville's 1839-40 class still survive. He taught mathematics at the École des Ponts et Chaussées where he succeeded Coriolis.
Saint-Venant worked mainly on mechanics, elasticity, hydrostatics and hydrodynamics. Perhaps his most remarkable work was that which he published in 1843 in which he gave the correct derivation of the Navier-Stokes equations. Anderson writes in :-
Seven years after Navier's death, Saint-Venant re-derived Navier's equations for a viscous flow, considering the internal viscous stresses, and eschewing completely Navier's molecular approach. That 1843 paper was the first to properly identify the coefficient of viscosity and its role as a multiplying factor for the velocity gradients in the flow. He further identified those products as viscous stresses acting within the fluid because of friction. Saint-Venant got it right and recorded it. Why his name never became associated with those equations is a mystery. certainly it is a miscarriage of technical attribution.We should remark that Stokes, like Saint-Venant, correctly derived the Navier-Stokes equations but he published the results two years after Saint-Venant.
Saint-Venant developed a vector calculus similar to that of Grassmann which he published in 1845. He then entered into a dispute with Grassmann about which of the two had thought of the ideas first. Grassmann had published his results in 1844, but Saint-Venant claimed (and there is little reason to doubt him) that he had first developed these ideas in 1832. Again it would appear that Saint-Venant was unlucky. Itard writes in :-
Saint-Venant used this vector calculus in his lectures at the Institut Agronomique, which were published in 1851 as "Principes de mécanique fondés sur la cinématique" . In this book Saint-Venant, a convinced atomist, presented forces as divorced from the metaphysical concept of cause and from the physiological concept of muscular effort, both of which, in his opinion, obscured force as a kinematic concept accessible to the calculus. Although his atomistic conceptions did not prevail, his use of the vector calculus was adopted in the French school system.In the 1850s Saint-Venant derived solutions for the torsion of non-circular cylinders. He extended Navier's work on the bending of beams, publishing a full account in 1864. In 1871 he derived the equations for non-steady flow in open channels.
In 1868 Saint-Venant was elected to succeed Poncelet in the mechanics section of the Académie des Sciences. By this time he was 71 years old, but he continued his research and lived for a further 18 years after this time. At age 86 he translated (with A Flamant) Clebsch's work on elasticity into French and published it as Théorie de l'élasticité des corps solides and Saint-Venant added notes to the text which he wrote himself. Note that Saint-Venant's co-translator A Flamant was a co-author of the obituary notice  for Saint-Venant.
Article by: J J O'Connor and E F Robertson