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 and 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.

[2]

Y. Amirat and K. Hamdache, Global weak solutions to a ferrofluid flow model, Math. Meth. Appl. Sci., 31 (2007), 123-151.

[3]

Y. Amirat and K. Hamdache, Strong solutions to the equations of a ferrofluid flow model, J. Math. Anal. Appl., 353 (2009), 271-294.

[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.

[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.

[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.

[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.

[8]

E. Blums, A. Cebers and M. M Maiorov, "Magnetic Fluids,'' Walter de Gryuter & Co., Berlin-New York, 1997.

[9]

B. A. Finlayson, Convective instability of ferromagnetic fluids, J. Fluid Mech., 40 (1970), 753-767.

[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.

[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.

[12]

P. Grisvard, "Elliptic Problems in Nonsmooth Domains,'' Monographs and Studies in Mathematics, 24, Pitman, 1985.

[13]

P. N. Kaloni and J. X. Lou, Convective instability of magnetic fluids, Physical Review E, 70 (2004), 1-12.

[14]

P. N. Kaloni and J. X. Lou, Convective instability of magnetic fluids under alternating magnetic field, Physical Review E, 71 (2005), 1-12.

[15]

D. Kinderlehrer and G. Stampacchia, "An Introduction to Variational Inequalities and Their Applications,'' Academic Press, New York, 1980.

[16]

O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,'' Rev. second edition, Gordon and Breach, 1969.

[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.

[18]

J. L. Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires,'' Dunod-Gauthier-Villars, 1969.

[19]

P. L. Lions, "Mathematical Topics in Fluid Mechanics. Volume 1. Incompressible Models,'' Oxford Science Publications, 1996.

[20]

J. L. Neuringer and R. E. Rosensweig, Ferrohydrodynamics, Phys. Fluids, 7 (1964), 1927-1937.

[21]

A. Novotný and I. Straškraba, "Introduction to the Mathematical Theory of Compressible Flow,'' Oxford University Press (UK), 2004.

[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.

[23]

R. E. Rosensweig, "Ferrohydrodynamics,'' Dover Publications, Inc., 1997.

[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.

[25]

M. I. Shliomis, Effective viscosity of magnetic suspensions, Sov. Phys. JEPT, 34 (1972), 1291-1394.

[26]

M. I. Shliomis, Convective instability of a ferrofluid, Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, 6 (1973), 130-135.

[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.

[28]

M. I. Shliomis and B. L Smorodin, Convective instability of magnetized fluids, Journal of Magnetism and Magnetic Materials, 252 (2002), 197-202.

[29]

J. Simon, Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure, SIAM J. Math. Anal., 21 (1990), 1093-1117.

[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.

[31]

L. Tartar, Topics in nonlinear analysis, Publications Mathématiques d'Orsay, {78.13}, 1978.

[32]

R. Temam, "Navier-Stokes Equations,'' 3rd (revised) edition, Elsevier Science Publishers B.V., Amsterdam, 1984.

[33]

H. C. Torrey, Bloch equations with diffusion terms, Phys. Rev., 104 (1956), 563-565.

[34]

S. Venkatasubramanian and P. Kaloni, Stability and uniqueness of magnetic fluid motions, Proc. R. Soc. Lond. A, 458 (2002), 1189-1204.

[35]

M. Zahn, Magnetic fluid and nonoparticle applications to nanotechnology, Journal of Nanoparticle Research, 73 (2001), 73-78.

show all references

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.

[2]

Y. Amirat and K. Hamdache, Global weak solutions to a ferrofluid flow model, Math. Meth. Appl. Sci., 31 (2007), 123-151.

[3]

Y. Amirat and K. Hamdache, Strong solutions to the equations of a ferrofluid flow model, J. Math. Anal. Appl., 353 (2009), 271-294.

[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.

[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.

[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.

[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.

[8]

E. Blums, A. Cebers and M. M Maiorov, "Magnetic Fluids,'' Walter de Gryuter & Co., Berlin-New York, 1997.

[9]

B. A. Finlayson, Convective instability of ferromagnetic fluids, J. Fluid Mech., 40 (1970), 753-767.

[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.

[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.

[12]

P. Grisvard, "Elliptic Problems in Nonsmooth Domains,'' Monographs and Studies in Mathematics, 24, Pitman, 1985.

[13]

P. N. Kaloni and J. X. Lou, Convective instability of magnetic fluids, Physical Review E, 70 (2004), 1-12.

[14]

P. N. Kaloni and J. X. Lou, Convective instability of magnetic fluids under alternating magnetic field, Physical Review E, 71 (2005), 1-12.

[15]

D. Kinderlehrer and G. Stampacchia, "An Introduction to Variational Inequalities and Their Applications,'' Academic Press, New York, 1980.

[16]

O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,'' Rev. second edition, Gordon and Breach, 1969.

[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.

[18]

J. L. Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires,'' Dunod-Gauthier-Villars, 1969.

[19]

P. L. Lions, "Mathematical Topics in Fluid Mechanics. Volume 1. Incompressible Models,'' Oxford Science Publications, 1996.

[20]

J. L. Neuringer and R. E. Rosensweig, Ferrohydrodynamics, Phys. Fluids, 7 (1964), 1927-1937.

[21]

A. Novotný and I. Straškraba, "Introduction to the Mathematical Theory of Compressible Flow,'' Oxford University Press (UK), 2004.

[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.

[23]

R. E. Rosensweig, "Ferrohydrodynamics,'' Dover Publications, Inc., 1997.

[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.

[25]

M. I. Shliomis, Effective viscosity of magnetic suspensions, Sov. Phys. JEPT, 34 (1972), 1291-1394.

[26]

M. I. Shliomis, Convective instability of a ferrofluid, Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, 6 (1973), 130-135.

[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.

[28]

M. I. Shliomis and B. L Smorodin, Convective instability of magnetized fluids, Journal of Magnetism and Magnetic Materials, 252 (2002), 197-202.

[29]

J. Simon, Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure, SIAM J. Math. Anal., 21 (1990), 1093-1117.

[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.

[31]

L. Tartar, Topics in nonlinear analysis, Publications Mathématiques d'Orsay, {78.13}, 1978.

[32]

R. Temam, "Navier-Stokes Equations,'' 3rd (revised) edition, Elsevier Science Publishers B.V., Amsterdam, 1984.

[33]

H. C. Torrey, Bloch equations with diffusion terms, Phys. Rev., 104 (1956), 563-565.

[34]

S. Venkatasubramanian and P. Kaloni, Stability and uniqueness of magnetic fluid motions, Proc. R. Soc. Lond. A, 458 (2002), 1189-1204.

[35]

M. Zahn, Magnetic fluid and nonoparticle applications to nanotechnology, Journal of Nanoparticle Research, 73 (2001), 73-78.

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