The Navier–Stokes equations are a system of partial differential equations that describe the motion of viscous fluids, named after Claude-Louis Navier (1822) and George Gabriel Stokes (1842–1850), with Siméon Denis Poisson independently reaching the same results. They mathematically express momentum balance for Newtonian fluids by applying Newton's second law, combining a diffusing viscous term with a pressure term, and they generalize the Euler equations by accounting for viscosity. These equations are crucial for modeling a vast array of real-world scenarios, including the design of aircraft and cars, the study of blood flow, power station design, and pollution analysis, and they form the foundation of magnetohydrodynamics when coupled with Maxwell's equations. Despite their practical importance, a purely mathematical challenge remains: the Navier–Stokes existence and smoothness problem—proving that smooth, bounded solutions exist in three dimensions—is unsolved, and the Clay Mathematics Institute has offered a $1 million prize for a solution or counterexample. The solution to the equations is a flow velocity vector field, which assigns a velocity to every point in the fluid at any given time, from which other quantities like pressure and temperature can be derived.