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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 and 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.

[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.

[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.

[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.

[5]

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

[6]

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

[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.

[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.

[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. doi: 10.1007/978-1-4612-5364-8.

[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.

[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.

[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.

[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.

[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.

[16]

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

[17]

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

[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.

[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.

[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.

[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.

[22]

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

[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.

[24]

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

[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.

[26]

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

[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.

[28]

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

[29]

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

[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.

[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.

[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.

[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.

[34]

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

[35]

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

[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.

[37]

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

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.

[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.

[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.

[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.

[5]

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

[6]

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

[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.

[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.

[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. doi: 10.1007/978-1-4612-5364-8.

[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.

[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.

[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.

[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.

[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.

[16]

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

[17]

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

[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.

[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.

[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.

[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.

[22]

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

[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.

[24]

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

[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.

[26]

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

[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.

[28]

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

[29]

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

[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.

[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.

[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.

[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.

[34]

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

[35]

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

[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.

[37]

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

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