Ideal magnetohydrodynamic equations on a sphere and elliptic-hyperbolic property
Authors:
Ian Holloway and Sivaguru S. Sritharan
Journal:
Quart. Appl. Math. 79 (2021), 27-53
MSC (2010):
Primary 35M30; Secondary 76J20, 76W05
DOI:
https://doi.org/10.1090/qam/1571
Published electronically:
May 7, 2020
MathSciNet review:
4188623
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Additional Information
Abstract: This work contains the derivation and type analysis of the conical ideal magnetohydrodynamic equations. The 3D ideal MHD equations with Powell source terms, subject to the assumption that the solution is conically invariant, are projected onto a unit sphere using tools from tensor calculus. Conical flows provide valuable insight into supersonic and hypersonic flow past bodies, but are simpler to analyze and solve numerically. Previously, work has been done on conical inviscid flows governed by the Euler equations with great success. It is known that some flight regimes involve flows of ionized gases, and thus there is motivation to extend the study of conical flows to the case where the gas is electrically conducting. To the authors’ knowledge, the conical magnetohydrodynamic equations have never been derived, and so this paper is the first investigation of that system. Among the results, we show that conical flows for this case do exist mathematically and that the governing system of partial differential equations is of mixed type. Throughout the domain it can be either hyperbolic or elliptic depending on the solution.
References
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References
- C. M. Dafermos and E. Feireisl, Evolutionary equations, Handbook of Differential Equations, vol. 2, Elsevier Science, North Holland, 2005.
- Hans De Sterck, Numerical simulation and analysis of magnetically dominated mhd bow shock flows with applications in space physics, Ph.D. thesis, Department of Mathematics, Katholieke Universiteit Leuven, 1999.
- Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845
- Antonio Ferri, Supersonic flow around circular cones at angles of attack, NACA Rep. 1045 (1951), 11. MR 50445
- Jingling Guan and S. S. Sritharan, A hyperbolic-elliptic type conservation law on unit sphere that arises in delta wing aerodynamics, Int. J. Contemp. Math. Sci. 3 (2008), no. 13-16, 721–737. MR 2484962
- Klaus Hoffmann, Henri-Marie Damevin, and Jean-François Dietiker, Numerical simulation of hypersonic magnetohydrodynamic flows, 31st Plasmadynamics and Lasers Conference, 06 2000.
- I. Holloway and S. S. Sritharan, Compressible euler equations on a sphere and elliptic-hyperbolic property, submitted for publication, arXiv:1910.08893v1 (2019).
- I. Holloway and S. S. Sritharan, Numerical solution of compressible euler and magnetohydrodynamic flow past an infinite cone, submitted for publication, arXiv:1910.08896v1 (2019).
- Wolfram Research, Inc., Mathematica, Version 11.3, Champaign, IL, 2018.
- Antony Jameson, Eigenvalues, eigenvectors and symmetrization of the magnetohydrodynamic (mhd) equations, Presented at AFOSR Grantees and Contractors Meeting, August 7, 2006, Atlanta, GA, 6 2006.
- S. Kranc, M. C. Yuen, and A. B. Cambel, Experimental investigation of magnetoaerodynamic flow around blunt bodies, Tech. Report 1393, Prepared by Northwestern University Evanston, IL for National Aeronautics and Space Administration, 1969.
- Peter D. Lax, Hyperbolic partial differential equations, Courant Lecture Notes in Mathematics, vol. 14, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2006. With an appendix by Cathleen S. Morawetz. MR 2273657
- David Lovelock and Hanno Rund, Tensor, differential forms, and variational principles, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1975. MR 0474046
- Kenneth Powell, An approximate riemann solver for magnetohydrodynamics (that works more than one dimension), Tech. Report 94-24, Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton, Virginia, 1994.
- Michael Renardy, Mathematical analysis of viscoelastic flows, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 73, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. MR 1774976
- E. L. Resler Jr. and W. R. Sears, The prospects for magneto-aerodynamics, J. Aero. Sci. 25 (1958), 235–245, 258. MR 93280
- J. S. Shang, Recent research in magneto-aerodynamics., Progress in Aerospace Sciences 37 (2001), no. 1, 1–20.
- J. S. Shang, Solving schemes for computational magneto-aerodynamics, J. Sci. Comput. 25 (2005), no. 1-2, 289–306. MR 2231952, DOI https://doi.org/10.1007/s10915-004-4645-3
- J. H. B. Smith, Remarks on the structure of conical flow, Progress in Aerospace Sciences 12 (1972), 241–272.
- S. S. Sritharan, Nonlinear aerodynamics of conical delta wings, Ph.D. thesis, Applied Mathematics, University of Arizona, 1982.
- S. S. Sritharan, Delta wings with shock-free cross flow, Quart. Appl. Math. 43 (1985), no. 3, 275–286. MR 814226, DOI https://doi.org/10.1090/qam/814226
- S. S. Sritharan and A R. Seebass, Finite area method for nonlinear supersonic conical flows, AIAA Journal 22 (1984), 226–233.
- G. I. Taylor and J. W. MacColl, The Air Pressure on a Cone Moving at High Speeds. I, Proceedings of the Royal Society of London Series A 139 (1933), 278–297.
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Additional Information
Ian Holloway
Affiliation:
Department of Mathematics, Wright State University, Dayton, Ohio 45435
Email:
iancholloway@gmail.com
Sivaguru S. Sritharan
Affiliation:
M. S. Ramaiah University of Applied Sciences, Bengaluru, India
MR Author ID:
226666
ORCID:
0000-0002-1341-0477
Email:
provostsritharan@gmail.com
Received by editor(s):
September 25, 2019
Received by editor(s) in revised form:
March 21, 2020
Published electronically:
May 7, 2020
Additional Notes:
This research was supported in part by an appointment to the Student Research Participation Program at the U.S. Air Force Institute of Technology administered by the Oak Ridge Institute for Science and Education through an interagency agreement between the U.S. Department of Energy and USAFIT
Article copyright:
© Copyright 2020
Brown University