ANZIAM J.
47 (2005), 39-50
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Plane poloidal-toroidal decomposition of doubly periodic vector fields. Part 2. The Stokes equations
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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.
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Australian Mathematical Publishing Association Inc.
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