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A NOTE ON NAVIER–STOKES EQUATIONS WITH NONORTHOGONAL COORDINATES

Part of: Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  05 February 2018

J. M. HILL  and
Y. M. STOKES Open the ORCID record for Y. M. STOKES [Opens in a new window]
J. M. HILL
Affiliation:
School of Information Technology and Mathematical Sciences, University of South Australia, Adelaide, SA 5001, Australia email [email protected]
Y. M. STOKES*
Affiliation:
School of Mathematical Sciences, The University of Adelaide, Adelaide, SA 5005, Australia email [email protected]
*
✉[email protected]
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Abstract

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There are many fluid flow problems involving geometries for which a nonorthogonal curvilinear coordinate system may be the most suitable. To the authors’ knowledge, the Navier–Stokes equations for an incompressible fluid formulated in terms of an arbitrary nonorthogonal curvilinear coordinate system have not been given explicitly in the literature in the simplified form obtained herein. The specific novelty in the equations derived here is the use of the general Laplacian in arbitrary nonorthogonal curvilinear coordinates and the simplification arising from a Ricci identity for Christoffel symbols of the second kind for flat space. Evidently, however, the derived equations must be consistent with the various general forms given previously by others. The general equations derived here admit the well-known formulae for cylindrical and spherical polars, and for the purposes of illustration, the procedure is presented for spherical polar coordinates. Further, the procedure is illustrated for a nonorthogonal helical coordinate system. For a slow flow for which the inertial terms may be neglected, we give the harmonic equation for the pressure function, and the corresponding equation if the inertial effects are included. We also note the general stress boundary conditions for a free surface with surface tension. For completeness, the equations for a compressible flow are derived in an appendix.

Keywords

general nonorthogonal coordinatesNavier–Stokes equationsfluid dynamics

MSC classification

Primary: 35Q30: Navier-Stokes equations
Type
Research Article
Information
The ANZIAM Journal , Volume 59 , Issue 3 , January 2018 , pp. 335 - 348
Copyright
© 2018 Australian Mathematical Society 

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