Weak solutions to stationary equations of heat transfer in a magnetic fluid

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March 2019, 18(2): 709-734. doi: 10.3934/cpaa.2019035

Weak solutions to stationary equations of heat transfer in a magnetic fluid

Youcef Amirat ^1,, and Kamel Hamdache ^2,

1.	Laboratoire de Mathématiques Blaise Pascal, Université Clermont Auvergne, CNRS UMR 6620, Campus universitaire des Cézeaux, 3, place Vasarely, 63178, Aubière, France
2.	Pôle universitaire Léonard de Vinci. DVRC. 92916 Paris la Défense Cedex

* Corresponding author

Received January 2018 Revised July 2018 Published October 2018

We consider the differential system describing the stationary heat transfer in a magnetic fluid in the presence of a heat source and an external magnetic field. The system consists of the stationary incompressible Navier-Stokes equations, the magnetostatic equations and the stationary heat equation. We prove, for the differential system posed in a bounded domain of $\mathbb{R}^3$ and equipped with Fourier boundary conditions, the existence of weak solutions by using a regularization of the Kelvin force and the thermal power.

Keywords: Magnetic fluid, Navier-Stokes equations, heat equation, steady weak solutions.

Mathematics Subject Classification: Primary: 35Q35, 76D05.

Citation: Youcef Amirat, Kamel Hamdache. Weak solutions to stationary equations of heat transfer in a magnetic fluid. Communications on Pure & Applied Analysis, 2019, 18 (2) : 709-734. doi: 10.3934/cpaa.2019035

References:

[1]	R. Alexandre and C. Villani, On the Boltzmann equation for long-range interactions, Comm. Pure Appl. Math., 55 (2002), 30-70. doi: 10.1002/cpa.10012.
[2]	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.
[3]	Y. Amirat and K. Hamdache, Global weak solutions to the equations of thermal convection in micropolar fluids subjected to Hall current, Nonlinear Analysis, Series A: Theory, Methods & Applications, 102 (2014), 186-207. doi: 10.1016/j.na.2014.02.001.
[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]	B. Ducomet and E. Feireisl, On the dynamics of gaseous stars, Arch. Rational Mech. Anal., 174 (2004), 221-266. doi: 10.1007/s00205-004-0326-5.
[6]	E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, 2004.
[7]	E. Feireisl and D. Prazak, Asymptotic Behavior of Dynamical Systems in Fluid Mechanics, AIMS Series on Applied Mathematics, 4, Springfield, MO, 2010.
[8]	E. Feireisl and J. Málek, On the Navier-Stokes equations with temperature-dependent transport coefficients, Differential Equations and Nonlinear Mechanics, (2006), 1-14.
[9]	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.
[10]	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. doi: 10.1007/978-1-4612-5364-8.
[11]	P. Grisvard, Elliptic Problems in Non Smooth Domains, Pitman, 1985.
[12]	D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980.
[13]	J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod-Gauthier-Villars, 1969.
[14]	P. B. Mucha and M. Pokorny, Weak solutions to equations of steady compressible heat conducting fluids, Mathematical Models and Methods in Applied Sciences, 20 (2010), 785-813. doi: 10.1142/S0218202510004441.
[15]	P. B. Mucha and M. Pokorny, On the steady compressible Navier-Stokes-Fourier system, Commun. Math. Phys, 288 (2009), 349-377. doi: 10.1007/s00220-009-0772-x.
[16]	A. Novotný and I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford University Press, 2004.
[17]	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), 167-181.
[18]	A. Prignet, Conditions aux limites non homogènes pour des problènmes elliptiques avec second membre mesure, Ann. Fac. Sciences Toulouse, 6 (1997), 297-318.
[19]	R. E. Rosensweig, Ferrohydrodynamics, Dover Publications, Inc. 1997.
[20]	R. E. Rosensweig, Basic equations for magnetic fluids with internal rotations, in Ferrofluids: Magnetically Controllable Fluids and Their Applications, Lecture Notes in Physics (SpringerVerlag, Heidelberg), 594, S. Odenbache Ed., (2002), 61-84.
[21]	M. I. Shliomis, in Ferrofluids: Magnetically controllable fluids and their applications, Lecture Notes in Physics (Springer-Verlag, Heidelberg), S. Odenbach Ed., 594 (2002), 85-111.
[22]	R. Temam, Navier-Stokes Equations, 3rd (revised) edition, Elsevier Science Publishers B.V., Amsterdam, 1984.
[23]	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.

show all references

References:

[1]	R. Alexandre and C. Villani, On the Boltzmann equation for long-range interactions, Comm. Pure Appl. Math., 55 (2002), 30-70. doi: 10.1002/cpa.10012.
[2]	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.
[3]	Y. Amirat and K. Hamdache, Global weak solutions to the equations of thermal convection in micropolar fluids subjected to Hall current, Nonlinear Analysis, Series A: Theory, Methods & Applications, 102 (2014), 186-207. doi: 10.1016/j.na.2014.02.001.
[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]	B. Ducomet and E. Feireisl, On the dynamics of gaseous stars, Arch. Rational Mech. Anal., 174 (2004), 221-266. doi: 10.1007/s00205-004-0326-5.
[6]	E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, 2004.
[7]	E. Feireisl and D. Prazak, Asymptotic Behavior of Dynamical Systems in Fluid Mechanics, AIMS Series on Applied Mathematics, 4, Springfield, MO, 2010.
[8]	E. Feireisl and J. Málek, On the Navier-Stokes equations with temperature-dependent transport coefficients, Differential Equations and Nonlinear Mechanics, (2006), 1-14.
[9]	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.
[10]	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. doi: 10.1007/978-1-4612-5364-8.
[11]	P. Grisvard, Elliptic Problems in Non Smooth Domains, Pitman, 1985.
[12]	D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980.
[13]	J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod-Gauthier-Villars, 1969.
[14]	P. B. Mucha and M. Pokorny, Weak solutions to equations of steady compressible heat conducting fluids, Mathematical Models and Methods in Applied Sciences, 20 (2010), 785-813. doi: 10.1142/S0218202510004441.
[15]	P. B. Mucha and M. Pokorny, On the steady compressible Navier-Stokes-Fourier system, Commun. Math. Phys, 288 (2009), 349-377. doi: 10.1007/s00220-009-0772-x.
[16]	A. Novotný and I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford University Press, 2004.
[17]	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), 167-181.
[18]	A. Prignet, Conditions aux limites non homogènes pour des problènmes elliptiques avec second membre mesure, Ann. Fac. Sciences Toulouse, 6 (1997), 297-318.
[19]	R. E. Rosensweig, Ferrohydrodynamics, Dover Publications, Inc. 1997.
[20]	R. E. Rosensweig, Basic equations for magnetic fluids with internal rotations, in Ferrofluids: Magnetically Controllable Fluids and Their Applications, Lecture Notes in Physics (SpringerVerlag, Heidelberg), 594, S. Odenbache Ed., (2002), 61-84.
[21]	M. I. Shliomis, in Ferrofluids: Magnetically controllable fluids and their applications, Lecture Notes in Physics (Springer-Verlag, Heidelberg), S. Odenbach Ed., 594 (2002), 85-111.
[22]	R. Temam, Navier-Stokes Equations, 3rd (revised) edition, Elsevier Science Publishers B.V., Amsterdam, 1984.
[23]	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.

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