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A NOTE ON NAVIER–STOKES EQUATIONS WITH NONORTHOGONAL COORDINATES

Published online by Cambridge University Press:  05 February 2018

J. M. HILL
Affiliation:
School of Information Technology and Mathematical Sciences, University of South Australia, Adelaide, SA 5001, Australia email jim.hill@unisa.edu.au
Y. M. STOKES*
Affiliation:
School of Mathematical Sciences, The University of Adelaide, Adelaide, SA 5005, Australia email yvonne.stokes@adelaide.edu.au

Abstract

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.

MSC classification

Type
Research Article
Copyright
© 2018 Australian Mathematical Society 

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References

Aris, R., Vectors, tensors and the basic equations of fluid mechanics (Prentice Hall, Englewood Cliffs, NJ, 1962).Google Scholar
Arnold, D. J., Stokes, Y. M. and Green, J. E. F., “Thin-film flow in helically-wound rectangular channels of arbitrary torsion and curvature”, J. Fluid Mech. 764 (2015) 7694;doi:10.1017/jfm.2014.703.CrossRefGoogle Scholar
Batchelor, G. K., An introduction to fluid dynamics (Cambridge University Press, Cambridge, 1967).Google Scholar
Eringen, A. C., Nonlinear theory of continuous media (McGraw-Hill, New York, 1962).Google Scholar
Germano, M., “On the effect of torsion on helical pipe flow”, J. Fluid Mech. 125 (1982) 18; doi:10.1017/S0022112082003206.CrossRefGoogle Scholar
Goldstein, S., Modern developments in fluid mechanics 1 (Clarendon Press, Oxford, 1938).Google Scholar
Kelbin, O., Cheviakov, A. F. and Oberlack, M., “New conservation laws of helically symmetric plane and rotationally symmetric viscous and inviscid flows”, J. Fluid Mech. 721 (2013) 340366; doi:10.1017/jfm.2013.72.CrossRefGoogle Scholar
Lee, S., Stokes, Y. M. and Bertozzi, A. L., “Behavior of a particle-laden flow in a spiral channel”, Phys. Fluids 26 (2014) 043302; doi:10.1063/1.4872035.CrossRefGoogle Scholar
Manoussaki, D. and Chadwick, R. S., “Effects of geometry on fluid loading in a coiled cochlea”, SIAM J. Appl. Math. 61 (2000) 369386; doi:10.1137/S0036139999358404.CrossRefGoogle Scholar
Murata, S., Miyake, Y. and Inaba, T., “Laminar flow in a curved pipe with varying curvature”, J. Fluid Mech. 73 (1976) 735752; doi:10.1017/S0022112076001596.CrossRefGoogle Scholar
Ramsey, A. S., A treatise on hydromechanics, Part II, Hydrodynamics, 4th edn (G. Bell and Sons, London, 1947).Google Scholar
Spain, B., Tensor calculus (Oliver and Boyd, New York, 1960).Google Scholar
Synge, J. L. and Schild, A., Tensor calculus, Volume 5 of Mathematical Expositions (University of Toronto Press, Toronto, 1966).Google Scholar
Tuttle, E. R., “Laminar flow in twisted pipes”, J. Fluid Mech. 219 (1990) 545570;doi:10.1017/S002211209000307X.CrossRefGoogle Scholar
Wang, C. Y., “On the low-Reynolds-number flow in a helical pipe”, J. Fluid Mech. 108 (1981) 185194; doi:10.1017/S0022112081002073.CrossRefGoogle Scholar
Zabielski, L. and Mestel, A. J., “Steady flow in a helically symmetric pipe”, J. Fluid Mech. 370 (1998) 297320; doi:10.1017/S0022112098002006.CrossRefGoogle Scholar
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