It is interesting to know the exact (analytical) solutions of the incompressible Navier-Stokes equations for 3-D rectangular channel (steady state or transient). The solution as an infinite series probably. Need for debugging the codes.

Thank you.

It could be a very challenging task to write here the series solution for this case. You can find it on the White's book "Viscous fluid Flow".

However, if you can't go through it, consider that for unsteady laminar incompressible flows in straight ducts with any kind of costant cross section the Navier-Stokes equations will semplify in the 2D unsteady heat equation for the axial velocity component, the only non zero velocity component.

In this case the pressure gradient has to be considered as a general, known, unsteady forcing term constant in the cross section of the duct, that is:

du/dt = nu ( d2u/dy2 + d2u/dz2 ) + F(t)

where F is at most a function of time and, of course, the only source of unsteadyness. A lot of solution for this problem are also available through the Carslaw and Jaeger book. However, the solution of this problem is very simple if a rectangular cross section is considered.

If you don't want to go through the solution process, also considering that the solution will not effectively be a 3D one, you can consider the following:

Another option is to consider the 2D taylor vortex solution (google it) and apply a simmetry boundary condition in the third direction, actually obtaining a 2D unsteady solution constant in the third direction.

I suggest to follow the second path because the series summation is a very time consuming task...a non necessary one.

FYI

This book seems to contain such a solution ---> http://www.cfdbooks.com/