Numerical study of a rotating fluid in a spheroidal container

The motion of an incompressible, viscous rotating fluid contained in a spheroidal container is studied by a direct numerical simulation in an oblate spheroidal coordinate system. An appropriate formalism is first derived which allows us to expand any scalar field in spherical harmonics and to decompose any vector field into its sphero-poloidal and sphero-toroidal scalar parts. The spinover mode is then considered, by linearizing the equation of motion for the fluid. Boundary conditions specific of the spheroidal geometry are explicited. Finite difference method for the sphero-radial component and spherical harmonics expansion for angular components are considered. A few numerical results are given about the viscous effects on the spinover mode, for different eccentricities of the container, in good agreement with the analytical solution given in the literature.