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

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

* 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.

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

[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. PankhurstJ. ConnollyS. 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. PankhurstJ. ConnollyS. 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

[1]

Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241

[2]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348

[3]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136

[4]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020268

[5]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081

[6]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276

[7]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272

[8]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364

[9]

Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020454

[10]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

[11]

Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168

[12]

Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137

[13]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

[14]

Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119

[15]

Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318

[16]

Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456

[17]

Zhouchao Wei, Wei Zhang, Irene Moroz, Nikolay V. Kuznetsov. Codimension one and two bifurcations in Cattaneo-Christov heat flux model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020344

[18]

Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020467

[19]

Mathew Gluck. Classification of solutions to a system of $ n^{\rm th} $ order equations on $ \mathbb R^n $. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246

[20]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (108)
  • HTML views (182)
  • Cited by (0)

Other articles
by authors

[Back to Top]