The difference between them and the closely related Euler equations is that Navier–Stokes equations take viscosity into account while the Euler equations model only inviscid flow. They arise from applying Isaac Newton's second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term-hence describing viscous flow.
They are sometimes accompanied by an equation of state relating pressure, temperature and density.
The Navier–Stokes equations mathematically express momentum balance and conservation of mass for Newtonian fluids. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842–1850 (Stokes). The Navier–Stokes equations ( / n æ v ˈ j eɪ s t oʊ k s/ nav- YAY STOHKS) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Irish physicist and mathematician George Gabriel Stokes.
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