Convenient Form of the Navier-Stokes Equations

[NOTE: This post should have been updated a long time ago. I have cleaned it up to avoid any confusion. Thanks.]

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.

In certain cases, it is important to express the convection term using the following identity

This form is very convenient for those of you working with analytical modeling of the NS equations. There was some confusion on whether this identity holds in general curvilinear coordinates. This identity is TRUE in general orthogonal curvilinear coordinates.

I would like to thank Prof. Gary Flandro of UTSI for promoting the use of this form.