The Eigenfunctions of the Stokes Operator in Special Domains. II

Abstract

We investigate the eigenvalue problem of the Stokes operator in an open bounded rectangular parallelepiped of the ℝ3 supplemented with homogeneous Dirichlet boundary conditions on two face to face lying lateral surfaces and periodical conditions in the directions orthogonal to the other faces. By separation we deduce a corresponding system of ordinary differential equations. We look for solutions as solenoidal vector fields satifying the boundary conditions. The investigation of possible cases yields either the explicit eigenfunctions and eigenvalues or equations for determining the eigenvalues and a general deduced form of the eigenfunctions. In the appendix we give an example of the first eigenvalues of the Stokes operator for fixed l = 2.69.

Micropolar meets Newtonian. The Rayleigh–Bénard problem

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