Spectral decomposition of the Stokes flow operators in the inverted prolate spheroidal coordinates

The eigenfunctions of the axisymmetric Stokes flow operator |E²| in the inverted prolate spheroidal system of coordinates are derived analytically, while an analytical method for the calculation of the eigenfunctions of |E⁴| in the inverted prolate spheroidal system of coordinates is presented. The stream function |ψ| satisfies the Stokes flow equation |E⁴ψ =0| while the equation |E²ψ =0| expresses an irrotational flow. In the prolate spheroidal coordinates, the solution space |²\| is decomposed in separable eigenfunctions and the solution space |⁴\| enjoys a spectral decomposition in semiseparable eigenfunctions. In the inverse prolate spheroidal system, we show that the partial differential operator |E²| admits |R|-separable solutions with |R| being a function of the Euclidean distance, |r|⁠, while the operator |E⁴| admits |R|-semiseparable solutions with |R| being a function of the Euclidean distance on the third, |r³|⁠. We derive the 0-eigenspace of |E²|and we prove that the |⁴\| consists of both: the eigenfunctions and the generalized eigenfunctions of |²\|⁠. Furthermore, by employing the Kelvin transformation, we show that the obtained expressions satisfy the Kelvin theorems as these are applied to the Stokes flow. The obtained eigenfunctions may be used in the mathematical treatment of medical problems, such as the blood plasma flow around erythrocytes.