Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-16T15:15:51.707Z Has data issue: false hasContentIssue false
  • English
  • Français

The fundamental solutions for irrotational and rotational Stokes flow in spheroidal geometry

Published online by Cambridge University Press:  01 July 2007

G. DASSIOS
G. DASSIOS*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge. e-mail: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

The Stokes operator E2 governs the irrotational axisymmetric Stokes flow and its square governs the corresponding rotational flow. In spheroidal coordinates the elements of the solution space ker E2 enjoy a spectral decomposition into separable eignefunction, while the elements of the ker E4 accept a spectral decomposition in terms of semiseparable eigensolutions involving 3D-by-3D eigenfunctions of the Gegenbauer operator. These spectral characteristics are utilized to construct the fundamental solutions for both the E2 and the E4 operators in spheroidal geometry. The fundamental solution for E2 is expressed in terms of the elements of the irrotational space ker E2, while the fundamental solution for E4 is expressed in terms of the corresponding generalized eigenfunctions alone.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 143 , Issue 1 , July 2007 , pp. 243 - 253
Copyright
Copyright © Cambridge Philosophical Society 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Bird, R. B., Stewart, W. E. and Lightfoot, E. N.. Transport Phenomena (John Wiley and Sons, 1960).Google Scholar
[2]Charalambopoulos, A. and Dassios, G.. Complete decomposition of axisymmetric Stokes flow. Internat. J. Engrg. Sci. 40 (2002), 1099–1111.CrossRefGoogle Scholar
[3]Dassios, G., Hadjinicolaou, M. and Payatakes, A. C.. Generalized eigenfunctions and complete semiseparable solutions for Stokes flow in spheroidal coordinates. Quart. Appl. Math. 52 (1994), 157–191.CrossRefGoogle Scholar
[4]Dassios, G., Payatakes, A. C. and Vafeas, P.. Interrelation between Papkovich–Neuber and Stokes general solutions of the Stokes equations in spheroidal geometry. Quart. Appl. Math. 57 (2004), 181–203.CrossRefGoogle Scholar
[5]Happel, J. and Brenner, H.. Low Reynolds Number Hydrodynamics (Martinus Nijhoff Publishers, 1963).Google Scholar
[6]Kim, S. and Karrila, S. J.. Microhydrodynamics: Principles and Selected Applications (Butterworth–Heinemann, 1991).Google Scholar
[7]Ladyzhenskaya, O. A.. The Mathematical Theory of Viscous Incompressible Flow (Gordon and Breach, 1963).Google Scholar
[8]Leal, L. G.. Laminar Flow and Convective Transport Processes (Butterworth–Heinemann, 1992).Google Scholar
[9]Pozrikidis, C.. Boundary Integral and Singularity Methods for Linearized Viscous Flow (Cambridge University Press, 1992).CrossRefGoogle Scholar
[10]Stokes, G. G.. On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids. Trans. Camb. Phil. Soc. 8 (1845), 287–319.Google Scholar
[11]Stokes, G. G.. On the effect of the internal friction of fluids on the motion of pendulums. Trans. Camb. Phil. Soc. 9 (1851), 8–106.Google Scholar
[12]Xu, X. and Wand, M.. General complete solutions of the equations of spatial and axisymmetric Stokes flow. Quart. J. Mech. Appl. Math. 44 (1991), 537–548.Google Scholar