Invariant Form of the Navier-Stokes Equations

Most of us learn the fundamentals of fluid mechanics using Cartesian coordinates. Specifically, derivations of the Navier-Stokes equations are done in a Cartesian reference system. This is a valid way for studying such a complicated set of equations as rectangular coordinates do not present us with the nuances of extra terms due to curvature or other effects that are present in curvilinear coordinates. The final form for the momentum equations is concisely written in vector notation for compactness and simplicity. This is given by

Note that this is a vector equation yielding as many equations as there are coordinates.

However, this form is not invariant under a coordinate transformation. Unfortunately, most textbooks in fluid mechanics fail to stress that fact thus leaving the student with the impression that the above equation is applicable in all coordinates. For example, using the above form in Cylindrical coordinates, and upon expansion, will yield the incorrect form of the momentum equations. The problem stems from the nonlinear convective term (second term on the LHS). Fortunately, there is an invariant form that is also more interesting since it explicitly introduces the vorticity into the equations. In brief, the invariant form of the convective term is

Using the above form will render the NS equations invariant under coordinate transformation. It is recommended that this form be used to avoid any ambiguity.

I would like to thank Professor Gary Flandro of UTSI for pointing out this fact time and time again.

Cite as:
Saad, T. "Invariant Form of the Navier-Stokes Equations." Weblog entry from Please Make A Note. http://pleasemakeanote.blogspot.com/2008/05/invariant-form-of-navier-stokes_20.html