TITLE:
Necessary and Sufficient Conditions for the Separability and the R-Separability of the Irrotational Stokes Equation and Applications
AUTHORS:
Eleftherios Protopapas, Maria Hadjinicolaou
KEYWORDS:
Axisymmetric Flow, Stokes Operator, Analytical Solution, Separation and R-Separation of Variables, Axisymmetric Orthogonal Coordinate Systems
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.8 No.11,
November
13,
2020
ABSTRACT: In the present manuscript, we formulate and prove rigorously, necessary and sufficient conditions for all kinds of separation of variables that a solution of the irrotational Stokes equation may exhibit, in any orthogonal axisymmetric system, namely: simple separation and R-separation. These conditions may serve as a road map for obtaining the corresponding solution space of the irrotational Stokes equation, in any orthogonal axisymmetric coordinate system. Additionally, we investigate how the inversion of the coordinate system, with respect to a sphere, affects the type of separation. Specifically, we prove that if the irrotational Stokes equation separates variables in an axisymmetric coordinate system, then it R-separates variables in the corresponding inverted coordinate system. This is a quite useful outcome since it allows the derivation of solutions for a problem, from the knowledge of the solution of the same problem in the inverted geometry and vice-versa. Furthermore, as an illustration, we derive the eigenfunctions of the irrotational Stokes equation governing the flow past oblate spheroid particles and inverted oblate spheroidal particles.