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Mathematical Problems in Engineering
Volume 2016, Article ID 9049131, 6 pages
http://dx.doi.org/10.1155/2016/9049131
Research Article

Eigenfunction Expansions for the Stokes Flow Operators in the Inverted Oblate Coordinate System

School of Science and Technology, Hellenic Open University, 11 Sahtouri Street, 26 222 Patras, Greece

Received 3 January 2016; Revised 20 March 2016; Accepted 14 April 2016

Academic Editor: Maurizio Brocchini

Copyright © 2016 Maria Hadjinicolaou and Eleftherios Protopapas. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

When studying axisymmetric particle fluid flows, a scalar function, , is usually employed, which is called a stream function. It serves as a velocity potential and it can be used for the derivation of significant hydrodynamic quantities. The governing equation is a fourth-order partial differential equation; namely, , where is the Stokes irrotational operator and is the Stokes bistream operator. As it is already known, in some axisymmetric coordinate systems, such as the cylindrical, spherical, and spheroidal ones, separates variables, while in the inverted prolate spheroidal coordinate system, this equation accepts -separable solutions, as it was shown recently by the authors. Notably, the kernel space of the operator does not decompose in a similar way, since it accepts separable solutions in cylindrical and spherical system of coordinates, while semiseparates variables in the spheroidal coordinate systems and it -semiseparates variables in the inverted prolate spheroidal coordinates. In addition to these results, we show in the present work that in the inverted oblate spheroidal coordinates, the equation also -separates variables and we derive the eigenfunctions of the Stokes operator in this particular coordinate system. Furthermore, we demonstrate that the equation   -semiseparates variables. Since the generalized eigenfunctions of cannot be obtained in a closed form, we present a methodology through which we can derive the complete set of the generalized eigenfunctions of in the modified inverted oblate spheroidal coordinate system.