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Some special solutions of self similar type in MHD, obtained by a separation method of variables

Published online by Cambridge University Press:  15 August 2002

Michel Cessenat
Affiliation:
CEA/DAM, Centre d'Études de Bruyères-le-Châtel, B.P. 12, 91680 Bruyères-le-Châtel, France.
Philippe Genta
Affiliation:
CEA/DAM, Centre d'Études de Bruyères-le-Châtel, B.P. 12, 91680 Bruyères-le-Châtel, France.
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Abstract

We use a method based on a separation of variables for solving a first order partial differential equations system, using a very simple modelling of MHD. The method consists in introducing three unknown variables Φ1, Φ2, Φ3 in addition to the time variable t and then in searching a solution which is separated with respect to Φ1 and t only. This is allowed by a very simple relation, called a “metric separation equation”, which governs the type of solutions with respect to time. The families of solutions for the system of equations thus obtained, correspond to a radial evolution of the fluid. Solving the MHD equations is then reduced to find the transverse component H of the magnetic field on the unit sphere Σ by solving a non linear partial equation on Σ. Thus, we generalize ideas of Courant-Friedrichs [7] and of Sedov [11], on dimensional analysis and self-similar solutions.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 1999

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