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Axisymmetric Stokes' flow in a spherical shell revisited via the Fokas method. Part I: Irrotational flow
Author(s) -
Dassios G.,
Doschoris M.
Publication year - 2011
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1407
Subject(s) - mathematics , mellin transform , mathematical analysis , conservative vector field , integral transform , flow (mathematics) , convergent series , boundary value problem , representation (politics) , boundary (topology) , laplace transform , power series , geometry , physics , politics , political science , compressibility , law , thermodynamics
The axisymmetric irrotational Stokes' flow for a spherical shell is analysed by means of the recently developed Fokas method via the use of global relations. Alternative series and new integral representations concerning a system of concentric spheres, yielding, by a limiting procedure, the Dirichlet or Neumann problems for the interior and the exterior of a sphere, are presented. The boundary value problems considered can be classically solved using either the finite Gegenbauer transform or the Mellin transform. Application of the Gegenbauer transform yields a series representation which is uniformly convergent at the boundary, but not convenient for many applications. The Mellin transform, on the other hand, furnishes an integral representation which is not uniformly convergent at the boundary. Here, by algebraic manipulations of the global relation: (i) a Gegenbauer series representation is derived in a simpler manner, instead of solving ODEs and (ii) an alternative integral representation, different from the Mellin transform representation is derived which is uniformly convergent at the boundary. Copyright © 2011 John Wiley & Sons, Ltd.