In fluid dynamics, the Navier-Stokes equations are equations, that describe the three-dimensional motion of viscous fluid substances. These equations are named after Claude-Louis Navier (1785-1836) and George Gabriel Stokes (1819-1903). In situations in which there are no strong temperature gradients in the fluid, these equations provide a very good approximation of reality.
The Navier-Stokes equations consists of a time-dependent continuity equation for conservation of mass, three time-dependent conservation of momentum equations and a time-dependent conservation of energy equation. There are four independent variables in the problem, the x, y, and z spatial coordinates of some domain, and the time t.
As can be seen, the Navier-Stokes equations are second-order nonlinear partial differential equations, their solutions have been found to a variety of interesting viscous flow problems. They may be used to model the weather, ocean currents, air flow around an airfoil and water flow in a pipe or in a reactor. The Navier–Stokes equations in their full and simplified forms help with the design of aircraft and cars, the study of blood flow, the design of nuclear reactors and many other things