Exact solutions of the Navier-Stokes equations generated by the eigenfunctions of the Stokes operator

We derive a class of exact solutions to the incompressible unsteady Navier-Stokes equations. These solutions are generated by the eigenfunctions of the Stokes operator and can be applied to studying boundary value problems for the Navier-Stokes equations related to pipe and channel flows. We explore the relation between the spectrum of the Stokes operator, generalized Beltrami flows, and exact solutions of the Navier-Stokes equations. Relevant properties of the Stokes eigenfunctions are also discussed in some detail.