Author(s)

Maria Hadjinicolaou¹, Eleftherios Protopapas²

¹Applied Mathematics Laboratory, School of Science and Technology, Hellenic Open University, 167 R. Feraiou St., GR-26 222 Patras, Greece.
²School of Applied Mathematical and Physical Sciences, National Technical University of Athens, 9 Heroon Polytechneiou St., GR-15 780 Athens, Greece.

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.

Axisymmetric Creeping Flow, R-Separation, Semiseparation, R-Semiseparation, Irrotational Flow, Analytical Solution

Hadjinicolaou, M. and Protopapas, E. (2020) Separability of Stokes Equations in Axisymmetric Geometries. Journal of Applied Mathematics and Physics, 8, 315-348. doi: 10.4236/jamp.2020.82026.