Deriving "Eigenflows" in Ellipsoidal Coordinate Systems of Revolution and in their Inverted ones. A Comparative Study

A comparative study of the analytical methods we used and developed for deriving the complete set of solutions for the fourth order Partial Differential Equation (PDE) governing an axisymmetric viscous flow is the subject of the present work. The flow is assumed to pass around objects of different shapes which can be considered either as ellipsoids of revolution, i.e. prolate and oblate spheroids, either as their image under inversion transformation. Choosing each time the appropriate coordinate system provided that the surface of the boundary coincides to one of its coordinate surfaces, we derive the analytic expression of the governing equation E-4 Psi = 0 and calculate the solutions which are given as series expansions in terms of combinations of Gegenbauer functions, either in semiseparable, or R-separable and R-semiseparable forms. Along with the method of semiseparation of variables, we employed the Kelvin inversion as it was applied to Stokes flow for obtaining solutions for the equivalent problems under the Kelvin transformation. The study of the different procedures we followed and the similarity of the obtained analytical results are detailed discussed and compared to those of Charalambopoulos-Dassios and Dassios et al., allowing a better understanding of the structure of the differential operators and their eigenspace and also revealing the effect of the inversion transformation on them. These solutions -eigenflows- have been used in various engineering problems and recently, they were applied to problems arising in biology and medicine such as the blood flow in arteries, the sedimentation of erythrocytes, the relative motion of Low Density Lipoproteins through blood plasma, etc.