|
In fluid dynamics, 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 model weather or the movement of air in the atmosphere, ocean currents, water flow in a
pipe, as well as many other fluid flow phenomena.
-
Where:
p is pressure
ρ is density
The equations are derived by considering the mass, momentum, and energy balances for an infinitesimal control volume. The
Navier-Stokes equations need to be augmented by an equation of
state for compressible flows. The variables to be solved for are the velocity components, the fluid density, static pressure,
and temperature. 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 Wilkinson equations for the secondary variables vorticity and stream function. Solution depends on
the fluid properties (such as viscosity, specific heats, and thermal
conductivity), and on the boundary conditions of the domain of study. For a derivation of the Navier-Stokes equations, see
some of the external links listed 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.
Although the full, unsteady Navier-Stokes equations correctly describe nearly all flows of practical interest they are too
complex for practical solution in many cases and a special "reduced" form of the full equations is often used instead —
these are the Reynolds-averaged Navier-Stokes (RANS) equations. The solution of the full steady
Navier-Stokes equations is sufficiently accurate alone for cases where the fluid flow is laminar. For turbulent flows the Reynolds-averaged form of the
equations are most commonly used. The RANS form of the equations introduce new terms that reflect the additional modelling of the
small turbulent motions.
Solution of flow equations by numerical methods is called computational fluid dynamics.
The Navier-Stokes equations with zero viscosity are known as the Euler equations; there, the emphasis is on compressible flow and shock waves.
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: Reynolds number, Mach number
External links
|