### 2000-22

#### The Eigenfunctions of the Stokes Operator in Special Domains III

Series: 2000-22, Preprints

MSC:
34A30 Linear equations and systems, general
34B24 ~Sturm-Liouville theory
35Q30 ~Navier-Stokes equations
76D07 Stokes and related (Oseen, etc.) flows

Abstract:
We consider the eigenvalue problem of the Stokes operator in a bounded
domain of ${\bf R}^{3}$ bounded by two concentrical cylinders with homogeneous
Dirichlet boundary conditions on the curved part of the boundary and periodical conditions in
along the cylinder axis (in $x_{1}$-direction). We deduce by separation the correspondent
systems of ordinary differential equations and solve them explicitly
looking for 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. The completeness of the
the calculated system of eigenfunctions in ${\bf S}$ can be proven
analogous to the corresponding part
in the habilitation of the second author.

Keywords:
Stokes operator, eigenfunctions, concentrical cylinders

This paper was published in:
Submitted to the ZAMM in August 2000