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.
Citation: |
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.
![]() |
|
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.![]() ![]() ![]() |
|
E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, 2004.
![]() ![]() |
|
E. Feireisl and D. Prazak, Asymptotic Behavior of Dynamical Systems in Fluid Mechanics, AIMS Series on Applied Mathematics, 4, Springfield, MO, 2010.
![]() ![]() |
|
E. Feireisl
and J. Málek
, On the Navier-Stokes equations with temperature-dependent transport coefficients, Differential Equations and Nonlinear Mechanics, (2006)
, 1-14.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
P. Grisvard, Elliptic Problems in Non Smooth Domains, Pitman, 1985.
![]() ![]() |
|
D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980.
![]() ![]() |
|
J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod-Gauthier-Villars, 1969.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
A. Novotný and I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford University Press, 2004.
![]() ![]() |
|
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.
![]() |
|
A. Prignet
, Conditions aux limites non homogènes pour des problènmes elliptiques avec second membre mesure, Ann. Fac. Sciences Toulouse, 6 (1997)
, 297-318.
![]() ![]() |
|
R. E. Rosensweig, Ferrohydrodynamics, Dover Publications, Inc. 1997.
![]() |
|
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.
![]() |
|
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.
![]() |
|
R. Temam, Navier-Stokes Equations, 3rd (revised) edition, Elsevier Science Publishers B.V., Amsterdam, 1984.
![]() ![]() |
|
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.![]() ![]() ![]() |