The Navier-Stokes Equations

The Navier-Stokes equations describe the flow of a Newtonian fluid. In a Cartesian coordinate system, the equations for the i-th component are:

   Eq 1: Compressible momentum (1)
   Eq 2: Compressible mass conservation (2)

The stress tensor in equation (1) is
   Eq 3: Newtonian stress tensor (3)
In these equations a repeated index means to sum over the index (in three dimensional space, sum from 1 to 3).

Equation (1) is derived from Newton's second law; equation (2) expresses that the rate of change of mass in a control volume is determined by the net in-flux (mass is neither created nor destroyed).

For an incompressible flow the equations are simplified to

   Eq 4: Incompressible momentum (4)
   Eq 5: Incompressible mass conservation (5)
The kinematic viscosity has been inserted in equation (4), and is assumed to be a constant material property.

The symbols u and p denote the velocity and the pressure respectively; f in equations (1) and (4) denotes an external force.

For a mathematical derivation and discussion of the Navier-Stokes equations you might look in "A Mathematical Introduction to Fluid Mechanics" by A.J. Chorin and J.E. Marsden, Springer Verlag, 2000.