2015/06/22

Navier-Stokes equation for 3D compressible and incompressible flows

In this blog I would like to present the general form of the Navier-Stokes equation for both incompressible and compressible flows. As in most textbooks you may not find the fully expanded forms in 3D, here you have them all collected. For the first look, it might scare you but after breaking it down, it’s all simple. 

Incompressible flow:
  • Frame invariant form:
  • Dimensional form:
The subscripts imply differentiation with respect to the variables. Primes (‘) indicate that the unknowns are dimensional. The system of equations are solvable, since there are 4 equations and 4 unknowns: 

Fully expanded form to 3D coordinates:

Compressible flow:


  • Frame invariant form:

  • Fully expanded form to 3D coordinates:


Resource:
  • Aerodynamics for Engineering Students, Sixth Edition, E.L. Houghton, (etc.)

The equations were typed into Microsoft Office, unfortunately blogger.com does not have a possibility to type equations so I had to make screenshots of them. You may use them freely as you wish. Beware of the fact that there might be minor mistakes in the formulas, although I checked them several times.

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