An exact solution for the magnetic diffusion problem with a step-function resistivity model

In the magnetic diffusion problem, a magnetic diffusion equation is coupled by an Ohmic heating energy equation. The Ohmic heating can make the magnetic diffusion coefficient (i.e. the resistivity) vary violently, and make the diffusion a highly nonlinear process. For this reason, the problem is nor...

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Bibliographic Details
Published in:European physical journal plus 2024-04, Vol.139 (4), p.305, Article 305
Main Authors: Xiao, Bo, Wang, Ganghua, Zhao, Li, Feng, Chunsheng, Shu, Shi
Format: Article
Language:English
Subjects:
Applied and Technical Physics
Atomic
Boundary conditions
Complex Systems
Condensed Matter Physics
Diffusion coefficient
Electrical resistivity
Energy
Energy equation
Exact solutions
Heating
Magnetic diffusion
Magnetic fields
Mathematical and Computational Physics
Molecular
Optical and Plasma Physics
Physics
Physics and Astronomy
Regular Article
Self-similarity
Simulation
Theoretical
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Summary:In the magnetic diffusion problem, a magnetic diffusion equation is coupled by an Ohmic heating energy equation. The Ohmic heating can make the magnetic diffusion coefficient (i.e. the resistivity) vary violently, and make the diffusion a highly nonlinear process. For this reason, the problem is normally very hard to be solved analytically. In this article, under the condition of a step-function resistivity and a constant boundary magnetic field, we successfully derived an exact solution for this nonlinear problem. The solution takes four parameters as input: the fixed magnetic boundary B 0 , η S and η L that are resistivities below and above the critical energy density of a material, and the critical energy density e c of the material. The solution curve B ( x ,  t ) possesses the characteristic of a sharp front, and its evolution obeys the usual self-similar rule with the similarity variable x / t .
ISSN:2190-5444
2190-5444
DOI:10.1140/epjp/s13360-024-05086-2