The Eigenfunctions of the Stokes Operator in Special Domains. I

We consider the eigenvalue problem of the Stokes operator in a bounded cylindrical domain of ℝ 3 with homogeneous Dirichlet boundary conditions on the curved part of the boundary and periodical conditions in the main stream direction. We deduce by separation the corresponding systems of ordinary differential equations and solve them explicitly looking for bounded, solenoidal vector fields fulfilling the boundary conditions. The investigation of possible cases yields either the explicit eigenfunctions and eigenvalues, or equations for the determination of the eigenvalues, and a general representation of the eigenfunctions.