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