American Institute of Mathematical Sciences

March  2012, 11(2): 675-696. doi: 10.3934/cpaa.2012.11.675

On a heated incompressible magnetic fluid model

 1 Laboratoire de Mathématiques, CNRS UMR 6620, Université Blaise Pascal (Clermont-Ferrand 2), 63177 Aubière cedex, France 2 Centre de Mathématiques Appliquées, CNRS, Ecole Polytechnique, 91128 Palaiseau cedex, France

Received  September 2010 Revised  May 2011 Published  October 2011

In this paper we study the equations describing the dynamics of heat transfer in an incompressible magnetic fluid under the action of an applied magnetic field. The system consists of the Navier-Stokes equations, the magnetostatic equations and the temperature equation. We prove global-in-time existence of weak solutions to the system posed in a bounded domain of $R^3$ and equipped with initial and boundary conditions. The main difficulty comes from the singularity of the term representing the Kelvin force due to magnetization.
Citation: Youcef Amirat, Kamel Hamdache. On a heated incompressible magnetic fluid model. Communications on Pure & Applied Analysis, 2012, 11 (2) : 675-696. doi: 10.3934/cpaa.2012.11.675
References:
 [1] Y. Amirat, K. Hamdache and F. Murat, Global weak solutions to the equations of motion for magnetic fluids, J. Math. Fluid Mech., 10 (2008), 326-351. Google Scholar [2] Y. Amirat and K. Hamdache, Global weak solutions to a ferrofluid flow model, Math. Meth. Appl. Sci., 31 (2007), 123-151. Google Scholar [3] Y. Amirat and K. Hamdache, Strong solutions to the equations of a ferrofluid flow model, J. Math. Anal. Appl., 353 (2009), 271-294. Google Scholar [4] Y. Amirat and K. Hamdache, Weak solutions to the equations of motion for compressible magnetic fluids, J. Math. Pures Appl., 91 (2009), 433-467. Google Scholar [5] Y. Amirat and K. Hamdache, Unique solvability of equations of motion for ferrofluids, Nonlinear Analysis, Series A: Theory, Methods & Applications, 73 (2010), 471-494. Google Scholar [6] H. I. Andersson and O. A. Valnes, Flow of a heated ferrofluid over a stretching sheet in the presence of a magnetic dipole, Acta Mech., 128 (1998), 39-47. Google Scholar [7] P. J. Blennerhasset, F. Lin and P. J. Stiles, Heat transfert through strongly magnetized ferrofluids, Proc. R. Soc. Lond. A, 433 (1991), 165-177. Google Scholar [8] E. Blums, A. Cebers and M. M Maiorov, "Magnetic Fluids,'' Walter de Gryuter & Co., Berlin-New York, 1997. Google Scholar [9] B. A. Finlayson, Convective instability of ferromagnetic fluids, J. Fluid Mech., 40 (1970), 753-767. Google Scholar [10] G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. I. Linearized Steady Problems,'' Springer tracts in Natural Philosophy, 38, Springer Verlag, New-York, 1994. Google Scholar [11] G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. II. Nonlinear Steady Problems,'' Springer tracts in Natural Philosophy, 39, Springer Verlag, 1994. Google Scholar [12] P. Grisvard, "Elliptic Problems in Nonsmooth Domains,'' Monographs and Studies in Mathematics, 24, Pitman, 1985. Google Scholar [13] P. N. Kaloni and J. X. Lou, Convective instability of magnetic fluids, Physical Review E, 70 (2004), 1-12. Google Scholar [14] P. N. Kaloni and J. X. Lou, Convective instability of magnetic fluids under alternating magnetic field, Physical Review E, 71 (2005), 1-12. Google Scholar [15] D. Kinderlehrer and G. Stampacchia, "An Introduction to Variational Inequalities and Their Applications,'' Academic Press, New York, 1980. Google Scholar [16] O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,'' Rev. second edition, Gordon and Breach, 1969. Google Scholar [17] O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,'' Translations Math. Monogr.23, AMS, Providence, R.I., 1968. Google Scholar [18] J. L. Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires,'' Dunod-Gauthier-Villars, 1969. Google Scholar [19] P. L. Lions, "Mathematical Topics in Fluid Mechanics. Volume 1. Incompressible Models,'' Oxford Science Publications, 1996. Google Scholar [20] J. L. Neuringer and R. E. Rosensweig, Ferrohydrodynamics, Phys. Fluids, 7 (1964), 1927-1937. Google Scholar [21] A. Novotný and I. Straškraba, "Introduction to the Mathematical Theory of Compressible Flow,'' Oxford University Press (UK), 2004. Google Scholar [22] Q. Q. A. Pankhurst, J. Connolly, S. K. Jones and J. Dobson, Applications of magnetic nonoparticles in biomedicine, J. Phys. D: Appl. Phys., 36 (2003), R167-R181. Google Scholar [23] R. E. Rosensweig, "Ferrohydrodynamics,'' Dover Publications, Inc., 1997. Google Scholar [24] C. L. Russel, P. J. Blennerhassett and P. J. Stiles, Strongly nonlinear vortices in magnetized ferrofluids, J. Austral. Math. Soc., Ser. B, 40 (1999), 146-171. Google Scholar [25] M. I. Shliomis, Effective viscosity of magnetic suspensions, Sov. Phys. JEPT, 34 (1972), 1291-1394. Google Scholar [26] M. I. Shliomis, Convective instability of a ferrofluid, Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, 6 (1973), 130-135. Google Scholar [27] M. I. Shliomis, Ferrofluids: Magnetically controllable fluids and their applications, Lecture Notes in Physics (Springer-Verlag, Heidelberg), V. {594}, ed. S. Odenbache, 85-111, 2002. Google Scholar [28] M. I. Shliomis and B. L Smorodin, Convective instability of magnetized fluids, Journal of Magnetism and Magnetic Materials, 252 (2002), 197-202. Google Scholar [29] J. Simon, Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure, SIAM J. Math. Anal., 21 (1990), 1093-1117. Google Scholar [30] Z. Tan and Y. Wang, Global analysis for strong solutions to the equations of a ferrofluid flow model, J. Math. Anal. Appl., 364 (2010), 424-436. Google Scholar [31] L. Tartar, Topics in nonlinear analysis, Publications Mathématiques d'Orsay, {78.13}, 1978. Google Scholar [32] R. Temam, "Navier-Stokes Equations,'' 3rd (revised) edition, Elsevier Science Publishers B.V., Amsterdam, 1984. Google Scholar [33] H. C. Torrey, Bloch equations with diffusion terms, Phys. Rev., 104 (1956), 563-565. Google Scholar [34] S. Venkatasubramanian and P. Kaloni, Stability and uniqueness of magnetic fluid motions, Proc. R. Soc. Lond. A, 458 (2002), 1189-1204. Google Scholar [35] M. Zahn, Magnetic fluid and nonoparticle applications to nanotechnology, Journal of Nanoparticle Research, 73 (2001), 73-78. Google Scholar

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References:
 [1] Y. Amirat, K. Hamdache and F. Murat, Global weak solutions to the equations of motion for magnetic fluids, J. Math. Fluid Mech., 10 (2008), 326-351. Google Scholar [2] Y. Amirat and K. Hamdache, Global weak solutions to a ferrofluid flow model, Math. Meth. Appl. Sci., 31 (2007), 123-151. Google Scholar [3] Y. Amirat and K. Hamdache, Strong solutions to the equations of a ferrofluid flow model, J. Math. Anal. Appl., 353 (2009), 271-294. Google Scholar [4] Y. Amirat and K. Hamdache, Weak solutions to the equations of motion for compressible magnetic fluids, J. Math. Pures Appl., 91 (2009), 433-467. Google Scholar [5] Y. Amirat and K. Hamdache, Unique solvability of equations of motion for ferrofluids, Nonlinear Analysis, Series A: Theory, Methods & Applications, 73 (2010), 471-494. Google Scholar [6] H. I. Andersson and O. A. Valnes, Flow of a heated ferrofluid over a stretching sheet in the presence of a magnetic dipole, Acta Mech., 128 (1998), 39-47. Google Scholar [7] P. J. Blennerhasset, F. Lin and P. J. Stiles, Heat transfert through strongly magnetized ferrofluids, Proc. R. Soc. Lond. A, 433 (1991), 165-177. Google Scholar [8] E. Blums, A. Cebers and M. M Maiorov, "Magnetic Fluids,'' Walter de Gryuter & Co., Berlin-New York, 1997. Google Scholar [9] B. A. Finlayson, Convective instability of ferromagnetic fluids, J. Fluid Mech., 40 (1970), 753-767. Google Scholar [10] G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. I. Linearized Steady Problems,'' Springer tracts in Natural Philosophy, 38, Springer Verlag, New-York, 1994. Google Scholar [11] G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. II. Nonlinear Steady Problems,'' Springer tracts in Natural Philosophy, 39, Springer Verlag, 1994. Google Scholar [12] P. Grisvard, "Elliptic Problems in Nonsmooth Domains,'' Monographs and Studies in Mathematics, 24, Pitman, 1985. Google Scholar [13] P. N. Kaloni and J. X. Lou, Convective instability of magnetic fluids, Physical Review E, 70 (2004), 1-12. Google Scholar [14] P. N. Kaloni and J. X. Lou, Convective instability of magnetic fluids under alternating magnetic field, Physical Review E, 71 (2005), 1-12. Google Scholar [15] D. Kinderlehrer and G. Stampacchia, "An Introduction to Variational Inequalities and Their Applications,'' Academic Press, New York, 1980. Google Scholar [16] O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,'' Rev. second edition, Gordon and Breach, 1969. Google Scholar [17] O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,'' Translations Math. Monogr.23, AMS, Providence, R.I., 1968. Google Scholar [18] J. L. Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires,'' Dunod-Gauthier-Villars, 1969. Google Scholar [19] P. L. Lions, "Mathematical Topics in Fluid Mechanics. Volume 1. Incompressible Models,'' Oxford Science Publications, 1996. Google Scholar [20] J. L. Neuringer and R. E. Rosensweig, Ferrohydrodynamics, Phys. Fluids, 7 (1964), 1927-1937. Google Scholar [21] A. Novotný and I. Straškraba, "Introduction to the Mathematical Theory of Compressible Flow,'' Oxford University Press (UK), 2004. Google Scholar [22] Q. Q. A. Pankhurst, J. Connolly, S. K. Jones and J. Dobson, Applications of magnetic nonoparticles in biomedicine, J. Phys. D: Appl. Phys., 36 (2003), R167-R181. Google Scholar [23] R. E. Rosensweig, "Ferrohydrodynamics,'' Dover Publications, Inc., 1997. Google Scholar [24] C. L. Russel, P. J. Blennerhassett and P. J. Stiles, Strongly nonlinear vortices in magnetized ferrofluids, J. Austral. Math. Soc., Ser. B, 40 (1999), 146-171. Google Scholar [25] M. I. Shliomis, Effective viscosity of magnetic suspensions, Sov. Phys. JEPT, 34 (1972), 1291-1394. Google Scholar [26] M. I. Shliomis, Convective instability of a ferrofluid, Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, 6 (1973), 130-135. Google Scholar [27] M. I. Shliomis, Ferrofluids: Magnetically controllable fluids and their applications, Lecture Notes in Physics (Springer-Verlag, Heidelberg), V. {594}, ed. S. Odenbache, 85-111, 2002. Google Scholar [28] M. I. Shliomis and B. L Smorodin, Convective instability of magnetized fluids, Journal of Magnetism and Magnetic Materials, 252 (2002), 197-202. Google Scholar [29] J. Simon, Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure, SIAM J. Math. Anal., 21 (1990), 1093-1117. Google Scholar [30] Z. Tan and Y. Wang, Global analysis for strong solutions to the equations of a ferrofluid flow model, J. Math. Anal. Appl., 364 (2010), 424-436. Google Scholar [31] L. Tartar, Topics in nonlinear analysis, Publications Mathématiques d'Orsay, {78.13}, 1978. Google Scholar [32] R. Temam, "Navier-Stokes Equations,'' 3rd (revised) edition, Elsevier Science Publishers B.V., Amsterdam, 1984. Google Scholar [33] H. C. Torrey, Bloch equations with diffusion terms, Phys. Rev., 104 (1956), 563-565. Google Scholar [34] S. Venkatasubramanian and P. Kaloni, Stability and uniqueness of magnetic fluid motions, Proc. R. Soc. Lond. A, 458 (2002), 1189-1204. Google Scholar [35] M. Zahn, Magnetic fluid and nonoparticle applications to nanotechnology, Journal of Nanoparticle Research, 73 (2001), 73-78. Google Scholar
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