Navier-Stokes equations
In fluid mechanics, the Navier-Stokes equations are a set of nonlinear partial differential equations that describe the flow of fluids such as liquids and gases. For example: they govern the movement of air in the atmosphere, ocean currents, water flow in a pipe, as well as many other fluid flow phenomena.The equations are derived by considering the mass, momentum and energy balances for an infinitesimal control volume. The variables to be solved for are the velocity components and pressure. The flow is assumed to be differentiable and continuous, allowing these balances to be expressed as partial differential equations. The equations can be converted to equations for the secondary variables vorticity and stream function. Solution depends on the fluid properties viscosity and density and on the boundary conditions of the domain of study. For a derivation of the Navier-Stokes equations, see Further Reading below.
Note that the Navier-Stokes equations can only describe fluid flow approximately and that, at very small scales or under extreme conditions, real fluids made out of mixtures of discrete molecules and other material, such as suspended particles and dissolved gases, will produce different results from the continuous and homogeneous fluids modelled by the Navier-Stokes equations. However, the Navier-Stokes equations are useful for a wide range of practical problems, providing their limitations are borne in mind. The solution of the Navier-Stokes equations is sufficiently accurate alone for cases where the fluid flow is laminar. For turbulent flows special turbulence models must be used that introduce new terms into the equations.
Solution of flow equations by numerical methods is called computational fluid dynamics. There is hope that some problems of this equation can be solved with the help of solution method for flows of any macrostructure.
The Navier-Stokes equations with zero viscosity are known as the Euler equations; there, the emphasis is on compressible flow and shock waves.
In loose usage, "the Navier-Stokes equations" are usually synonymous with "the Navier-Stokes equations in incompressible flow", although the equations above are accurate for compressible flow.
It is a famous open question whether smooth initial conditions always lead to smooth solutions for all times; a $1,000,000 prize was offered in May 2000 by the Clay Mathematics Institute for the answer to this question.
See also:
External Links