The Navier-Stokes equations are the basic governing equations for a viscous, heat conducting fluid. It is a vector equation obtained by applying Newtons Law of Motion to a fluid element and is also called the momentum equation. It is supplemented by the mass conservation equation, also called continuity equation and the energy equation. Usually, the term Navier-Stokes equations is used to refer to all of these equations.
The instantaneous continuity equation (1), momentum equation (2) and energy equation (3) for a compressible fluid can be written as:
For a Newtonian fluid, assuming Stokes Law for mono-atomic gases, the viscous stress is given by:
Where the trace-less viscous strain-rate is defined by:
The heat-flux, , is given by Fourier's law:
Where the laminar Prandtl number is defined by:
To close these equations it is also necessary to specify an equation of state. Assuming a calorically perfect gas the following relations are valid:
Where , , and are constant.
The total energy is defined by:
Note that the corresponding expression (15) for Favre averaged turbulent flows contains an extra term related to the turbulent energy.
Equations (1)-(9), supplemented with gas data for , , and perhaps , form a closed set of partial differential equations, and need only be complemented with boundary conditions.
Existence and uniqueness
The existence and uniqueness of classical solutions of the 3-D Navier-Stokes equations is still an open mathematical problem. In 2-D, existence and uniqueness of regular solutions for all time have been shown by Jean Leray in 1933. He also gave the theory for the existence of weak solutions in the 3-D case while uniqueness is still an open question.