Rotationally symmetric bodies with longitudinal cross sections of parabolic shape are frequently used to model astrophysical objects, such as magnetospheres and other blunt objects, immersed in interplanetary or interstellar gas or plasma flows. We discuss a simple formula for the potential flow of an incompressible fluid around an elliptic paraboloid whose axis of symmetry coincides with the direction of incoming flow. Prescribing this flow, we derive an exact analytical solution to the induction equation of ideal magnetohydrodynamics for the case of an initially homogeneous magnetic field of arbitrary orientation being passively advected in this flow. Our solution procedure employs Euler potentials and Cauchy's integral formalism based on the flow's stream function and isochrones. Furthermore, we use a particular renormalization procedure that allows us to generate more general analytical expressions modeling the deformations experienced by arbitrary scalar or vector-valued fields embedded in the flow as they are advected first toward and then past the parabolic obstacle. Finally, both the velocity field and the magnetic field embedded therein are generalized from incompressible to mildly compressible flow, where the associated density distribution is found from Bernoulli's principle.
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September 2024
Research Article|
September 30 2024
An exact analytical solution for the weakly magnetized flow around an axially symmetric paraboloid, with application to magnetosphere models
Jens Kleimann
;
Jens Kleimann
a)
(Conceptualization, Investigation, Validation, Visualization, Writing – original draft, Writing – review & editing)
1
Theoretische Physik IV, Ruhr-Universität Bochum
, 44780 Bochum, Germany
2
Ruhr Astroparticle and Plasma Physics Center (RAPP Center), Ruhr-Universität Bochum
, 44780 Bochum, Germany
a)Author to whom correspondence should be addressed: [email protected]
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Christian Röken
Christian Röken
(Investigation, Validation, Writing – review & editing)
3
Department of Geometry and Topology, Faculty of Science, University of Granada
, 18071 Granada, Spain
4
Lichtenberg Group for History and Philosophy of Physics, Institut für Philosophie, Universität Bonn
, 53115 Bonn, Germany
Search for other works by this author on:
Jens Kleimann
1,2,a)
Christian Röken
3,4
1
Theoretische Physik IV, Ruhr-Universität Bochum
, 44780 Bochum, Germany
2
Ruhr Astroparticle and Plasma Physics Center (RAPP Center), Ruhr-Universität Bochum
, 44780 Bochum, Germany
3
Department of Geometry and Topology, Faculty of Science, University of Granada
, 18071 Granada, Spain
4
Lichtenberg Group for History and Philosophy of Physics, Institut für Philosophie, Universität Bonn
, 53115 Bonn, Germany
a)Author to whom correspondence should be addressed: [email protected]
Physics of Fluids 36, 093625 (2024)
Article history
Received:
April 26 2024
Accepted:
June 13 2024
Citation
Jens Kleimann, Christian Röken; An exact analytical solution for the weakly magnetized flow around an axially symmetric paraboloid, with application to magnetosphere models. Physics of Fluids 1 September 2024; 36 (9): 093625. https://doi.org/10.1063/5.0215849
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