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

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[1] Hadjinicolaou, M.
[2] Protopapas, E.
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Hadjinicolaou, M. (hadjinicolaou@eap.gr) | 1600年 / Oxford University Press卷 / 80期
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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 psi satisfies the Stokes flow equation E4 =0 while the equation E2 =0 expresses an irrotational flow. In the prolate spheroidal coordinates; the solution space ker E2 is decomposed in separable eigenfunctions and the solution space ker E4 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 ker E4 consists of both: the eigenfunctions and the generalized eigenfunctions of ker E2. 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. © 2015 The Authors;
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