An equation describing the flow of a Newtonian fluid. The Navier–Stokes equation can be written in the form
∂v/∂t + (v grad)v = 1/ρ gradp + η∇2v,
where v is the velocity, ρ is the density, η is the viscosity, and p is the pressure, respectively, of the fluid; ∇2 is the Laplace operator (see Laplace equation). The Navier–Stokes equation describes the flow of fluids, such as air and water, but is not suitable for describing the flow of non-Newtonian fluids. The equation can be derived using fluid mechanics or (in certain cases) from kinetic theory. It requires approximation techniques for a solution in all but the simplest problems. The Navier–Stokes equation was derived by the French engineer and scientist Claude-Louis-Marie-Henri Navier (1785–1836) and the British mathematician and physicist Sir George Gabriel Stokes (1819–1903) and also by the French mathematician Siméon-Denis Poisson (1781–1840) in the first half of the 19th century.