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Plane poloidal-toroidal decomposition of doubly periodic vector fields. Part 2. The Stokes equations

Published online by Cambridge University Press:  17 February 2009

G. D. McBain
G. D. McBain
Affiliation:
School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Australia; e-mail: geordie.mcbain@aeromech.usyd.edu.au.
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Abstract

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We continue our study of the adaptation from spherical to doubly periodic slot domains of the poloidal-toroidal representation of vector fields. Building on the successful construction of an orthogonal quinquepartite decomposition of doubly periodic vector fields of arbitrary divergence with integral representations for the projections of known vector fields and equivalent scalar representations for unknown vector fields (Part 1), we now present a decomposition of vector field equations into an equivalent set of scalar field equations. The Stokes equations for slow viscous incompressible fluid flow in an arbitrary force field are treated as an example, and for them the application of the decomposition uncouples the conservation of momentum equation from the conservation of mass constraint. The resulting scalar equations are then solved by elementary methods. The extension to generalised Stokes equations resulting from the application of various time discretisation schemes to the Navier-Stokes equations is also solved.

Type
Research Article
Information
The ANZIAM Journal , Volume 47 , Issue 1 , July 2005 , pp. 39 - 50
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Aitken, A. C., Determinants and Matrices, 9th ed. (Oliver and Boyd, Edinburgh, 1956).Google Scholar
[2]Benney, D. J. and Gustavsson, L. H., “A new mechanism for linear and nonlinear hydrodynamic stability”, Stud. Appl. Math. 64 (1981) 185–209.CrossRefGoogle Scholar
[3]Cahouet, J. and Chabard, J.-P., “Some fast 3D finite element solvers for the generalized Stokes problem”, Intl J. Numer. Methods Fluids 8 (1988) 869–895.CrossRefGoogle Scholar
[4]Chandrasekhar, S., Hydrodynamic and Hydromagnetic Stability (Dover, New York, 1981).Google Scholar
[5]Galdi, G. P. and Simader, C. G., “Existence, uniqueness and lq-estimates for the Stokes problem in an exterior domain”, Arch. Rational Mech. Anal. 112 (4) (1990) 291–318.CrossRefGoogle Scholar
[6]Kaplan, W., Elements of Ordinary Differential Equations (Addison-Wesley, Reading, Massachusetts, 1964).Google Scholar
[7]Lamb, H., Hydrodynamics, 6th ed. (Cambridge University Press, Cambridge, 1932).Google Scholar
[8]McBain, G. D., “Plane poloidal-toroidal decomposition of doubly periodic vector fields. Part 1. Fields with divergence”, ANZIAM J. 47 (2005) 21–38.CrossRefGoogle Scholar
[9]McBain, G. D., “Fully developed laminar buoyant flow in vertical cavities and ducts of bounded section”, J. Fluid Mech. 401 (1999) 365–377.Google Scholar
[10]McBain, G. D., “Vapour transport across gas-filled enclosures”, Ph. D. Thesis, James Cook University, Townsville, North Queensland, 1999.Google Scholar
[11]McBain, G. D., “Convection in a horizontally heated sphere”, J. Fluid Mech. 438 (2001) 1–10.CrossRefGoogle Scholar
[12]Munson, B. R. and Joseph, D. D., “Viscous incompressible flow between concentric rotating spheres. Part 1. Basic flow”, J. Fluid Mech. 49 (1971) 289–303.CrossRefGoogle Scholar
[13]Palaniappan, D., Nigam, S. D. and Amaranth, T., “A theorem for a fluid sphere in Stokes flow”, J. Austral. Math. Soc. Ser. B 35 (1994) 335–347.CrossRefGoogle Scholar
[14]Sabbah, C., Forestier, M. Y. and Pasquetti, R., “Filtering of the pressure spurious modes in Pn–Pn spectral approximations of the Stokes system”, Comput. & Fluids 31 (2002) 815–823.CrossRefGoogle Scholar
[15]Sabbah, C. and Pasquetti, R., “A divergence-free multi-domain spectral solver of the Navier–Stokes equations in geometries of high aspect ratio”, J. Comput. Phys. 139 (1998) 359–379.CrossRefGoogle Scholar
[16]Scheichl, R., “Decoupling three-dimensional mixed problems using divergence-free finite elements”, SIAM J. Sci. Comput. 23 (2002) 1752–1776.CrossRefGoogle Scholar
[17]Schmitt, B. J. and von Wahl, W., “Decomposition of solenoidal fields into poloidal fields, toroidal fields and the mean flow. Applications to the Boussinesq equations”, Lecture Notes in Math. 1530 (1992) 291–305.CrossRefGoogle Scholar
[18]Stakgold, I., Green's Functions and Boundary Value Problems, 2nd ed. (Wiley-Interscience, New York, 1998).Google Scholar
[19]Stokes, N., Barton, N. and Gross, L., “Inexact preconditioned Uzawa methods as a general approach to calculations in computational fluid dynamics”, Paper presented to the 11th Computational Techniques and Applications Conference,Sydney, New South Wales, 2005.Google Scholar
[20]Waldmann, L., “Zur Theorie des Gastrennungsverfahrens von Clusius und Dickel”, Die Naturwissenschaften 27 (14) (1939) 230–231.CrossRefGoogle Scholar