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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.  Google Scholar

[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.  Google Scholar

[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.  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]

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.  Google Scholar

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E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, 2004.  Google Scholar

[7]

E. Feireisl and D. Prazak, Asymptotic Behavior of Dynamical Systems in Fluid Mechanics, AIMS Series on Applied Mathematics, 4, Springfield, MO, 2010.  Google Scholar

[8]

E. Feireisl and J. Málek, On the Navier-Stokes equations with temperature-dependent transport coefficients, Differential Equations and Nonlinear Mechanics, (2006), 1-14.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

P. Grisvard, Elliptic Problems in Non Smooth Domains, Pitman, 1985.  Google Scholar

[12]

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

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[16]

A. Novotný and I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford University Press, 2004.  Google Scholar

[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.   Google Scholar

[18]

A. Prignet, Conditions aux limites non homogènes pour des problènmes elliptiques avec second membre mesure, Ann. Fac. Sciences Toulouse, 6 (1997), 297-318.   Google Scholar

[19]

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

[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. Google Scholar

[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. Google Scholar

[22]

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

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  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]

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.  Google Scholar

[6]

E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, 2004.  Google Scholar

[7]

E. Feireisl and D. Prazak, Asymptotic Behavior of Dynamical Systems in Fluid Mechanics, AIMS Series on Applied Mathematics, 4, Springfield, MO, 2010.  Google Scholar

[8]

E. Feireisl and J. Málek, On the Navier-Stokes equations with temperature-dependent transport coefficients, Differential Equations and Nonlinear Mechanics, (2006), 1-14.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

P. Grisvard, Elliptic Problems in Non Smooth Domains, Pitman, 1985.  Google Scholar

[12]

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

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[16]

A. Novotný and I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford University Press, 2004.  Google Scholar

[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.   Google Scholar

[18]

A. Prignet, Conditions aux limites non homogènes pour des problènmes elliptiques avec second membre mesure, Ann. Fac. Sciences Toulouse, 6 (1997), 297-318.   Google Scholar

[19]

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

[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. Google Scholar

[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. Google Scholar

[22]

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

[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.  Google Scholar

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