Previous |  Up |  Next


nonlinear elliptic system; magnetohydrodynamics; natural interface conditions; nonlinear heat equation; nonlocal radiation boundary conditions
We consider the problem of influencing the motion of an electrically conducting fluid with an applied steady magnetic field. Since the flow is originating from buoyancy, heat transfer has to be included in the model. The stationary system of magnetohydrodynamics is considered, and an approximation of Boussinesq type is used to describe the buoyancy. The heat sources given by the dissipation of current and the viscous friction are not neglected in the fluid. The vessel containing the fluid is embedded in a larger domain, relevant for the global temperature- and magnetic field- distributions. Material inhomogeneities in this larger region lead to transmission relations for the electromagnetic fields and the heat flux on inner boundaries. In the presence of transparent materials, the radiative heat transfer is important and leads to a nonlocal and nonlinear jump relation for the heat flux. We prove the existence of weak solutions, under the assumption that the imposed velocity at the boundary of the fluid remains sufficiently small.
[1] Bossavit, A.: Electromagnétisme en vue de la modélisation. Springer, Berlin, Heidelberg, New York (2004). MR 1616583
[2] Chandrasekhar, S.: Hydrodynamic and hydromagnetic stability. Dover Publications Inc., New York (1981).
[3] Duvaut, G., Lions, J.-L.: Inéquations en thermoélasticité et magnétohydrodynamique. Archs ration. Mech. Analysis 46 241-279 (1972). MR 0346289 | Zbl 0264.73027
[4] Druet, P.-E.: Higher integrability of the lorentz force for weak solutions to Maxwell's equations in complex geometries. Preprint 1270 of the Weierstrass Institute for Applied mathematics and Stochastics, Berlin (2007). Available in pdf-format at\hfil
[5] Druet, P.-E.: Weak solutions to a stationary heat equation with nonlocal radiation boundary condition and right-hand side in $L^p$ ($p\geq1$). Math. Meth. Appl. Sci. 32 135-166 (2008). DOI 10.1002/mma.1029 | MR 2478911
[6] Galdi, G.-P.: An introduction to the mathematical theory of the Navier-Stokes equations. Vol I. Linearized steady problems. Springer, New York (1994). MR 1284205
[7] Gray, Donald D., Giorgini, A.: The validity of the Boussinesq approximation for liquids and gases. Int. J. Heat Mass Transfer 19 545-551 (1976). DOI 10.1016/0017-9310(76)90168-X | Zbl 0328.76066
[8] Giaquinta, M., Modica, L., Souček, J.: Cartesian Currents in the Calculus of Variations. Vol. I. Cartesian Currents. Springer, Berlin, Heidelberg (1998). MR 1645086
[9] Hansen, O.: The radiosity equation on polyhedral domains. Logos Verlag, Berlin (2002). Zbl 1005.65143
[10] Kufner, A., John, O., Fučik, S.: Function spaces. Academia Prague, Prague (1977). MR 0482102
[11] Klein, O., Philip, P., Sprekels, J.: Modelling and simulation of sublimation growth in sic bulk single crystals. Interfaces and Free Boundaries 6 295-314 (2004). DOI 10.4171/IFB/101 | MR 2095334
[12] Lions, J.-L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris (1969). MR 0259693 | Zbl 0189.40603
[13] Ladyzhenskaja, O. A., Solonnikov, V. A.: Solutions of some non-stationary problems of magnetohydrodynamics for a viscous incompressible fluid. Trudy Mat. Inst. Steklov 59 115-173 (1960), Russian. MR 0170130
[14] Laitinen, M., Tiihonen, T.: Conductive-radiative heat transfer in grey materials. Quart. Appl. Math. 59 737-768 (2001). MR 1866555
[15] Meir, A. J., Schmidt, P. G.: Variational methods for stationary {MHD} flow under natural interface conditions. Nonlinear Analysis. Theory, Methods and Applications 26 659-689 (1996). DOI 10.1016/0362-546X(94)00308-5 | MR 1362743 | Zbl 0853.76095
[16] Meir, A. J., Schmidt, P. G.: On electromagnetically and thermally driven liquid-metal flows. Nonlinear analysis 47 3281-3294 (2001). DOI 10.1016/S0362-546X(01)00445-X | MR 1979224 | Zbl 1042.76597
[17] Naumann, J.: Existence of weak solutions to the equations of stationary motion of heat-conducting incompressible viscous fluids. In Progress Nonlin. Diff. Equs. Appl., volume 64, pages 373-390, Basel, 2005. Birkhäuser. MR 2185227 | Zbl 1122.35115
[18] Rakotoson, J.-M.: Quasilinear elliptic problems with measures as data. Diff. Integral Eqs. 4 449-457 (1991). MR 1097910 | Zbl 0834.35056
[19] Tiihonen, T.: Stefan-Boltzmann radiation on non-convex surfaces. Math. Meth. in Appl. Sci. 20 47-57 (1997). DOI 10.1002/(SICI)1099-1476(19970110)20:1<47::AID-MMA847>3.0.CO;2-B | MR 1429330
[20] Voigt, A.: Numerical Simulation of Industrial Crystal Growth. PhD thesis, {Technische-Universität} München, Germany (2001). Zbl 1009.82001
[21] Zeidler, E.: Nonlinear functional analysis and its applications. II/B. Springer Verlag, New York (1990). MR 1033498 | Zbl 0684.47029
Partner of
EuDML logo