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The solvability of coupling the thermal effect and magnetohydrodynamics field with turbulent convection zone and the flow field. (English) Zbl 1448.76199

Summary: In this paper, the magneto-heating coupling model is studied in details, with turbulent convection zone and the flow field involved. Our main work is to analyze the well-posed property of this model with the regularity techniques. We present the weak formulation of the coupled magneto-heating model and establish the regularity problem. Using Rothe’s method, monotone theories of nonlinear operator, weak convergence theories, we prove that the limits of the solutions from Rothe’s method converge to the solutions of the regularity problem with proper initial data. With the help of the spacial regularity technique, we derive the well-posedness of the original problems when the regular parameter \(\epsilon \longrightarrow 0\). Moreover, with additional regularity assumption for both the magnetic field and temperature variable, we prove the uniqueness of the solutions.

MSC:

76W05 Magnetohydrodynamics and electrohydrodynamics
76F35 Convective turbulence
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
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[1] Bermúdez, A.; Muñoz-Sola, R.; Vázquez, R., Analysis of two stationary magnetohydrodynamics systems of equations including Joule heating, J. Math. Anal. Appl., 368, 444-468 (2010) · Zbl 1189.35243
[2] Brandenburg, A.; Subramanian, K., Astrophysical magnetic fields and nonlinear dynamo theory, Phys. Rep., 417, 1-209 (2005)
[3] Cattaneo, F.; Hughes, D. W., Nonlinear saturation of the turbulent alpha effect where a large scale field is imposed, Phys. Rev. E, 54, 4532-4535 (1996)
[4] Chovan, J.; Slodička, M., Induction hardening of steel with restrained Joule heating and nonlinear law for magnetic induction field: solvability, J. Comput. Appl. Math., 311, 630-644 (2017) · Zbl 1352.35069
[5] Elbashbeshy, Elsayed M. A.; Emam, T. G.; Abdelgaber, K. M., Effects of thermal radiation and magnetic field on unsteady mixed convection flow and heat transfer over an exponentially stretching surface with suction in the presence of internal heat generation/absorption, J. Egyptian Math. Soc., 20, 215-222 (2012) · Zbl 1267.76131
[6] Hossain, M. A.; Gorla, R. S.R., Joule heating effect on magnetohydrodynamic mixed convection boundary layer flow with variable electrical conductivity, Internat. J. Numer. Methods Heat Fluid Flow, 23, 2, 275-288 (2013) · Zbl 1356.76432
[7] Kačur, J., Method of Rothe in Evolution Equations, Lecture Notes in Math., vol. 1192, 23-34 (1986), Springer: Springer Berlin · Zbl 0612.35006
[8] Metaxas, A. C., Foundations of Electroheat: A Unified Approach (1996), Wiley: Wiley New York
[9] Moffatt, H. K., Magnetic Field Generation in Electrically Conducting Fluids (1978), Cambridge University Press: Cambridge University Press Cambridge, UK
[10] Molokov, S.; Moreau, R.; Moffatt, H. K., Magnetohydrodynamics (2007), Springer: Springer Netherlands
[11] Parker, E. N., Cosmical Magnetic Fields (1979), Clarendon Press: Clarendon Press Oxford
[12] Ranjit, N. K.; Shit, G. C., Joule heating effects on electromagnetohydrodynamic flow through a peristaltically induced micro-channel with different zeta potential and wall slip, Phys. A, 482, 458-476 (2017) · Zbl 1495.76134
[13] Sanchez, S.; Fournier, A.; Pinheiro, K. J.; Aubert, J., A mean-field Babcock-Leighton solar dynamo model with long-term variability, An. Acad. Brasil. Ciênc., 86, 1, 11-26 (2014)
[14] Vaĭnberg, M. M., Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations (1973), Halsted Press (A division of John Wiley & Sons)/Israel Program for Scientific Translations: Halsted Press (A division of John Wiley & Sons)/Israel Program for Scientific Translations New York-Toronto, Ont./Jerusalem-London · Zbl 0279.47022
[15] Yin, H. M., Existence and regularity of a weak solution to Maxwell’s equations with a thermal effect, Math. Methods Appl. Sci., 29, 1199-1213 (2006) · Zbl 1109.35110
[16] Zeidler, E., Nonlinear Functional Analysis and Its Applications, II/B: Nonlinear Monotone Operators (1990), Springer-Verlag: Springer-Verlag New York
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