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August  2013, 33(8): 3289-3320. doi: 10.3934/dcds.2013.33.3289

Strong solutions to the equations of flow and heat transfer in magnetic fluids with internal rotations

Youcef Amirat 1, and  Kamel Hamdache 2,

1. 

Laboratoire de Mathématiques, CNRS UMR 6620, Université Blaise Pascal (Clermont-Ferrand 2), 63177 Aubière cedex
2. 

Centre de Mathématiques Appliquées, CNRS, Ecole Polytechnique, 91128 Palaiseau cedex

Received  May 2012 Revised  September 2012 Published  January 2013

In this paper we study the equations of flow and heat transfer in a magnetic fluid with internal rotations, when the fluid is subjected to the action of an external magnetic field. The system of equations is a combination of the Navier-Stokes equations, the magnetization relaxation equation of Bloch type, the magnetostatic equations and the temperature equation. We prove the local-in-time existence of the unique strong solution to the system equipped with initial and boundary conditions and establish a blow-up criterium for strong solutions. We then prove the global-in-time existence of strong solutions, under smallness assumptions on the initial data and the external magnetic field.
Keywords: Navier-Stokes equations, heat transfer, Magnetic fluid, global weak solutions., incompressible flow.
Mathematics Subject Classification: Primary: 35Q35, 76D05; Secondary: 80A2.
Citation: Youcef Amirat, Kamel Hamdache. Strong solutions to the equations of flow and heat transfer in magnetic fluids with internal rotations. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3289-3320. doi: 10.3934/dcds.2013.33.3289
References:
[1]

Y. Amirat and K. Hamdache, Strong solutions to the equations of a ferrofluid flow model, J. Math. Anal. Appl., 353 (2009), 271-294. doi: 10.1016/j.jmaa.2008.11.084.  Google Scholar

[2]

Y. Amirat and K. Hamdache, On a heated incompressible magnetic fluid model, Comm. Pure Appl. Anal., 11 (2012), 675-696. doi: 10.3934/cpaa.2012.11.675.  Google Scholar

[3]

Y. Amirat and K. Hamdache, Heat transfer in incompressible magnetic fluid, J. Math. Fluid Mech., 14 (2012), 217-247. doi: 10.1007/s00021-011-0050-5.  Google Scholar

[4]

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

[5]

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

[6]

G. Carbou and P. Fabrie, Regular solutions for Landau-Lifschitz equation in a bounded domain, Differential and Integral Equations, 14 (2001), 213-229.  Google Scholar

[7]

Y. Cho, H. J. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluids, J. Math. Pures Appl., 83 (2004), 243-275. doi: 10.1016/j.matpur.2003.11.004.  Google Scholar

[8]

R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547. doi: 10.1007/BF01393835.  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. doi: 10.1007/978-1-4612-5364-8.  Google Scholar

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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, New-York, 1994. doi: 10.1007/978-1-4612-5364-8.  Google Scholar

[12]

R. Ganguly, S. Sen and I. K. Puri, Heat transfer augmentation using a magnetic fluid under the influence of a line dipole, J. Magn. Magn. Mater., 271 (2004), 63-73. Google Scholar

[13]

M. Giaquinta and G. Modica, Nonlinear systems of the type of the stationary Navier-Stokes system, J. Reine Angew. Math., 330 (1982), 173-214.  Google Scholar

[14]

Y. Giga and H. Sohr, Abstract $L^p$ estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains, Journal of Functional Analysis, 102 (1991), 72-94. doi: 10.1016/0022-1236(91)90136-S.  Google Scholar

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M. Giga, Y. Giga and H. Sohr, $L^p$ estimates for abstract linear parabolic equations, Proc. Japan Acad. Ser. A Math. Sci., 67 (1991), 197-202.  Google Scholar

[16]

P. Grisvard, "Elliptic Problems in Nonsmooth Domains,'' Monographs and Studies in Mathematics, 24, Pitman (Advanced Publishing Program), Boston, MA, 1985.  Google Scholar

[17]

P. N. Kaloni and J. X. Lou, Convective instability of magnetic fluids, Physical Review, E70 (2004), 1-12. Google Scholar

[18]

P. N. Kaloni and A. Mahajan, Stability of magnetic fluid motions in a saturated porous medium, Z. Angew. Math. Phys., 62 (2011), 529-538. doi: 10.1007/s00033-010-0096-x.  Google Scholar

[19]

H. Kim, A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes equations, SIAM J. Math. Anal., 37 (2006), 1417-1434. doi: 10.1137/S0036141004442197.  Google Scholar

[20]

O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,'' Second English edition, revised and enlarged, Mathematics and its Applications, Vol. 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969.  Google Scholar

[21]

O. A. Ladyžhenskaya, 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

[22]

J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,'' Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[23]

P.-L. Lions, "Mathematical Topics in Fluid Mechanics. Volume 1. Incompressible Models,'' Oxford Lecture Series in Mathematics and its Applications, 3, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1996.  Google Scholar

[24]

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

[25]

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

[26]

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

[27]

R. E. Rosensweig, Ferrofluids: Magnetically controllable fluids and their applications, Lecture Notes in Physics (Springer-Verlag, Heidelberg), S. Odenbach Ed., 594 (2002), 61-84. Google Scholar

[28]

M. I Shliomis, Effective viscosity of magnetic suspension, Sov. Phys. JETP, 44 (1972), 1291-1294. Google Scholar

[29]

M. I. Shliomis and B. L. Smorodin, Convective instability of magnetized ferrofluids, J. Magn. and Magn. Mater., 252 (2002), 197-202. Google Scholar

[30]

M. I. Shliomis, Ferrofluids: Magnetically controllable fluids and their applications, Lecture Notes in Physics (Springer-Verlag, Heidelberg), S. Odenbach Ed., 594 (2002), 85-111. Google Scholar

[31]

M. I. Shliomis, Convective instability of magnetized ferrofluids: Influence of magnetophoresis and soret effect, in "Thermal Nonequilibrium Phenomena in Fluid Mixtures," Lecture Notes in Physics, Springer, Berlin, 584 (2002), 355-371. Google Scholar

[32]

L. Sunil, P. Chand, P. Bharti and A. Mahajan, Thermal convection in micropolar ferrofluid in the presence of rotation, J. of Magn. and Magn. Mater., 320 (2008), 316-324. Google Scholar

[33]

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. doi: 10.1016/j.jmaa.2009.10.032.  Google Scholar

[34]

L. Tartar, "Topics in Nonlinear Analysis," Publications Mathématiques d'Orsay 78, 13, Université de Paris-Sud, Département de Mathématique, Orsay, 1978.  Google Scholar

[35]

R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis,'' Reprint of the 1984 edition, AMS-Chelsea Publishing, Providence, RI, 2001.  Google Scholar

[36]

E. E Tzirtzilakis and N. G. Kafoussias, Biomagnetic fluid flow over a stretching sheet with nonlinear temperature dependent magnetization, Z. angew. Math. Phys., 54 (2003), 551-565. doi: 10.1007/s00033-003-1100-5.  Google Scholar

[37]

M. Zahn, Magnetic fluid and nonoparticle applications to nanotechnology, Journal of Nanoparticle Research, 3 (2001), 73-78. Google Scholar

show all references

References:
[1]

Y. Amirat and K. Hamdache, Strong solutions to the equations of a ferrofluid flow model, J. Math. Anal. Appl., 353 (2009), 271-294. doi: 10.1016/j.jmaa.2008.11.084.  Google Scholar

[2]

Y. Amirat and K. Hamdache, On a heated incompressible magnetic fluid model, Comm. Pure Appl. Anal., 11 (2012), 675-696. doi: 10.3934/cpaa.2012.11.675.  Google Scholar

[3]

Y. Amirat and K. Hamdache, Heat transfer in incompressible magnetic fluid, J. Math. Fluid Mech., 14 (2012), 217-247. doi: 10.1007/s00021-011-0050-5.  Google Scholar

[4]

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

[5]

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

[6]

G. Carbou and P. Fabrie, Regular solutions for Landau-Lifschitz equation in a bounded domain, Differential and Integral Equations, 14 (2001), 213-229.  Google Scholar

[7]

Y. Cho, H. J. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluids, J. Math. Pures Appl., 83 (2004), 243-275. doi: 10.1016/j.matpur.2003.11.004.  Google Scholar

[8]

R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547. doi: 10.1007/BF01393835.  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. doi: 10.1007/978-1-4612-5364-8.  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, New-York, 1994. doi: 10.1007/978-1-4612-5364-8.  Google Scholar

[12]

R. Ganguly, S. Sen and I. K. Puri, Heat transfer augmentation using a magnetic fluid under the influence of a line dipole, J. Magn. Magn. Mater., 271 (2004), 63-73. Google Scholar

[13]

M. Giaquinta and G. Modica, Nonlinear systems of the type of the stationary Navier-Stokes system, J. Reine Angew. Math., 330 (1982), 173-214.  Google Scholar

[14]

Y. Giga and H. Sohr, Abstract $L^p$ estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains, Journal of Functional Analysis, 102 (1991), 72-94. doi: 10.1016/0022-1236(91)90136-S.  Google Scholar

[15]

M. Giga, Y. Giga and H. Sohr, $L^p$ estimates for abstract linear parabolic equations, Proc. Japan Acad. Ser. A Math. Sci., 67 (1991), 197-202.  Google Scholar

[16]

P. Grisvard, "Elliptic Problems in Nonsmooth Domains,'' Monographs and Studies in Mathematics, 24, Pitman (Advanced Publishing Program), Boston, MA, 1985.  Google Scholar

[17]

P. N. Kaloni and J. X. Lou, Convective instability of magnetic fluids, Physical Review, E70 (2004), 1-12. Google Scholar

[18]

P. N. Kaloni and A. Mahajan, Stability of magnetic fluid motions in a saturated porous medium, Z. Angew. Math. Phys., 62 (2011), 529-538. doi: 10.1007/s00033-010-0096-x.  Google Scholar

[19]

H. Kim, A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes equations, SIAM J. Math. Anal., 37 (2006), 1417-1434. doi: 10.1137/S0036141004442197.  Google Scholar

[20]

O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,'' Second English edition, revised and enlarged, Mathematics and its Applications, Vol. 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969.  Google Scholar

[21]

O. A. Ladyžhenskaya, 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

[22]

J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,'' Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[23]

P.-L. Lions, "Mathematical Topics in Fluid Mechanics. Volume 1. Incompressible Models,'' Oxford Lecture Series in Mathematics and its Applications, 3, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1996.  Google Scholar

[24]

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

[25]

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

[26]

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

[27]

R. E. Rosensweig, Ferrofluids: Magnetically controllable fluids and their applications, Lecture Notes in Physics (Springer-Verlag, Heidelberg), S. Odenbach Ed., 594 (2002), 61-84. Google Scholar

[28]

M. I Shliomis, Effective viscosity of magnetic suspension, Sov. Phys. JETP, 44 (1972), 1291-1294. Google Scholar

[29]

M. I. Shliomis and B. L. Smorodin, Convective instability of magnetized ferrofluids, J. Magn. and Magn. Mater., 252 (2002), 197-202. Google Scholar

[30]

M. I. Shliomis, Ferrofluids: Magnetically controllable fluids and their applications, Lecture Notes in Physics (Springer-Verlag, Heidelberg), S. Odenbach Ed., 594 (2002), 85-111. Google Scholar

[31]

M. I. Shliomis, Convective instability of magnetized ferrofluids: Influence of magnetophoresis and soret effect, in "Thermal Nonequilibrium Phenomena in Fluid Mixtures," Lecture Notes in Physics, Springer, Berlin, 584 (2002), 355-371. Google Scholar

[32]

L. Sunil, P. Chand, P. Bharti and A. Mahajan, Thermal convection in micropolar ferrofluid in the presence of rotation, J. of Magn. and Magn. Mater., 320 (2008), 316-324. Google Scholar

[33]

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. doi: 10.1016/j.jmaa.2009.10.032.  Google Scholar

[34]

L. Tartar, "Topics in Nonlinear Analysis," Publications Mathématiques d'Orsay 78, 13, Université de Paris-Sud, Département de Mathématique, Orsay, 1978.  Google Scholar

[35]

R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis,'' Reprint of the 1984 edition, AMS-Chelsea Publishing, Providence, RI, 2001.  Google Scholar

[36]

E. E Tzirtzilakis and N. G. Kafoussias, Biomagnetic fluid flow over a stretching sheet with nonlinear temperature dependent magnetization, Z. angew. Math. Phys., 54 (2003), 551-565. doi: 10.1007/s00033-003-1100-5.  Google Scholar

[37]

M. Zahn, Magnetic fluid and nonoparticle applications to nanotechnology, Journal of Nanoparticle Research, 3 (2001), 73-78. Google Scholar

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