Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-07T19:55:24.415Z Has data issue: false hasContentIssue false
  • English
  • Français

A simple model of magnetic fields associated with outflow from a source; new orthogonal polynomials

Published online by Cambridge University Press:  02 December 2015

LAWRENCE K. FORBES
LAWRENCE K. FORBES*
Affiliation:
School of Mathematics and Physics, University of Tasmania, P.O. Box 37, Hobart 7001, Tasmania, Australia email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Outflow from a young star might be regarded as approximately equivalent to flow from a point source. If the fluid consists of charged particles, then the magnetic fields produced are governed by Faraday's law. This simple first approximation yields a linear partial differential equation in spherical polar coordinates, and its solution may be represented as the product of a Legendre polynomial with some function of the radial coordinate. This radial function is shown to involve orthogonal polynomials. Their properties are investigated and recurrence formulae for them are derived. Some of the magnetic fields generated by this simple model are illustrated.

Keywords

Asymptotic methodsMagnetohydrodynamicsOrthogonal polynomials
Type
Papers
Information
European Journal of Applied Mathematics , Volume 27 , Issue 4 , August 2016 , pp. 686 - 698
Copyright
Copyright © Cambridge University Press 2015 

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

[1] Abramowitz, M. & Stegun, I. A. (editors) (1972) Handbook of Mathematical Functions, Dover, New York, p. 1046.Google Scholar
[2] Batchelor, G. K. (1967) An Introduction to Fluid Dynamics, Cambridge, Cambridge University Press, p. 615.Google Scholar
[3] Brenner, M. P., Hilgenfeldt, S. & Lohse, D. (2002) Single-bubble sonoluminescence. Rev. Mod. Phys. 74, 425–484.Google Scholar
[4] Chambers, K. & Forbes, L. K. (2012) The cylindrical magnetic Rayleigh-Taylor instability for viscous fluids. Phys. Plasmas 19, article 102111, 13.CrossRefGoogle Scholar
[5] Dias, F. & Vanden-Broeck, J.-M. (2011) Potential-flow studies of steady two-dimensional jets, waterfalls, weirs and sprays. J. Eng. Math. 70, 165–174.Google Scholar
[6] Forbes, L. K., Hocking, G. C. & Wotherspoon, S. (2004) Salt-water up-coning during extraction of fresh water from a tropical island. J. Eng. Math. 48, 69–91.Google Scholar
[7] Forbes, L. K. (2011) A cylindrical Rayleigh-Taylor instability: Radial outflow from pipes or stars. J. Eng. Math. 70, 205–224.Google Scholar
[8] Forbes, L. K. (2011) Rayleigh-Taylor instabilities in axi-symmetric outflow from a point source. ANZIAM J. 53, 87–121.Google Scholar
[9] Forbes, L. K. (2014) How strain and spin may make a star bi-polar. J. Fluid Mech. 746, 332–367.Google Scholar
[10] Forbes, L. K. & Brideson, M. A. (2014) Exact solutions for interfacial outflows with straining. ANZIAM J. 55, 232–244.Google Scholar
[11] Forbes, L. K. & Hocking, G. C. (1993) Flow induced by a line sink in a quiescent fluid with surface-tension effects. J. Aust. Math. Soc. Ser. B 34, 377–391.Google Scholar
[12] Gómez, L., Rodríguez, L. F. & Loinard, L. (2013) A one-sided knot ejection at the core of the HH 111 outflow. Revista Mexicana Astron. Astro. 49, 79–85.Google Scholar
[13] Hocking, G. C. (1995) Supercritical withdrawal from a two-layer fluid through a line sink. J. Fluid Mech. 297, 37–47.Google Scholar
[14] Hocking, G. C. & Zhang, H. (2014) A note on axisymmetric supercritical coning in a porous medium. ANZIAM J. 55, 327–335.Google Scholar
[15] Kreyszig, E. (2011) Advanced Engineering Mathematics, 10th ed., New York, Wiley.Google Scholar
[16] Lamb, S. H. (1945) Hydrodynamics, 6th ed., New York, Dover.Google Scholar
[17] Lovelace, R. V. E., Romanova, M. M., Ustyugova, G. V. & Koldoba, A. V. (2010) One-sided outflows/jets from rotating stars with complex magnetic fields. Mon. Not. R. Astron. Soc. 408, 2083–2091.CrossRefGoogle Scholar
[18] Mikaelian, K. O. (2005) Rayleigh-Taylor and Richtmyer-Meshkov instabilities and mixing in stratified cylindrical shells. Phys. Fluids 17, article 094105, 13.Google Scholar
[19] Munro, A. I. S. & Forbes, L. K. (2006) Including ionisation in a simple model of single-bubble sonoluminescence. ANZIAM J. 47, 333–358.CrossRefGoogle Scholar
[20] NIST Digital Library of Mathematical Functions. (2015) http://dlmf.nist.gov/ Google Scholar
[21] Rayleigh, L. (1917) On the pressure developed in a liquid during the collapse of a spherical cavity. Phil. Mag. Ser. 6 34, 94–98.Google Scholar
[22] Shivamoggi, B. K. (1986) Theory of Hydromagnetic Stability, New York, Gordon and Breach.Google Scholar
[23] Stahler, S. C. & Palla, F. (2004) The Formation of Stars, Berlin, Wiley - VCH.Google Scholar
[24] Stokes, T. E., Hocking, G. C. & Forbes, L. K. (2003) Unsteady free-surface flow induced by a line sink. J. Eng. Math. 47, 137–160.Google Scholar
[25] Stone, J. M., Gardiner, T. A., Teuben, P., Hawley, J. F. & Simon, J. B. (2008) ATHENA: A new code for astrophysical MHD. Astrophys. J. Suppl. Ser. 178, 137–177.Google Scholar
[26] Tuck, E. O. & Vanden-Broeck, J.-M. (1984) A cusp-like free-surface flow due to a submerged source or sink. J. Aust. Math. Soc. Ser. B 25, 443–450.Google Scholar
[27] Tyvand, P. A. (1992) Unsteady free surface flow due to a line source. Phys. Fluids A 4, 671–676.Google Scholar
[28] Vanden-Broeck, J.-M. (1998) A model for the free-surface flow due to a submerged source in water of infinite depth. J. Aust. Math. Soc. Ser. B 39, 528–538.Google Scholar