by Lee, Doo-Sung, Rummler, Bernd.

**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