TITLE:
Separability of Stokes Equations in Axisymmetric Geometries
AUTHORS:
Maria Hadjinicolaou, Eleftherios Protopapas
KEYWORDS:
Axisymmetric Creeping Flow, R-Separation, Semiseparation, R-Semiseparation, Irrotational Flow, Analytical Solution
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.8 No.2,
February
26,
2020
ABSTRACT: For Stokes flow in non spherical geometries, when separation of variables fails to derive closed form solutions in a simple product form, analytical solutions can still be obtained in an almost separable form, namely in semiseparable form, R-separable form or R-semiseparable form. Assuming a stream function Ψ, the axisymmetric viscous Stokes flow is governed by the fourth order elliptic partial differential equation E4Ψ = 0 where E4 = E2oE2 and E2 is the irrotational Stokes operator. Depending on the geometry of the problem, the general solution is given in one of the above separable forms, as series expansions of particular combinations of eigenfunctions that belong to the kernel of the operator E2. In the present manuscript, we provide a review of the methodology and the general solutions of the Stokes equations, for almost any axisymmetric system of coordinates, which are given in a ready to use form. Furthermore, we present necessary and sufficient conditions that are serving as criterion for identifying the kind of the separation the Stokes equation admits, in each axisymmetric coordinate system. Additionally, as an illustration of the usefulness of the obtained analytical solutions, we demonstrate indicatively their application to particular Boundary Value Problems that model medical problems.