American Institute of Mathematical Sciences

Advanced
March  2012, 11(2): 675-696. doi: 10.3934/cpaa.2012.11.675

On a heated incompressible magnetic fluid model

Youcef Amirat 1, and  Kamel Hamdache 2,

1. 

Laboratoire de Mathématiques, CNRS UMR 6620, Université Blaise Pascal (Clermont-Ferrand 2), 63177 Aubière cedex, France
2. 

Centre de Mathématiques Appliquées, CNRS, Ecole Polytechnique, 91128 Palaiseau cedex, France

Received  September 2010 Revised  May 2011 Published  October 2011

In this paper we study the equations describing the dynamics of heat transfer in an incompressible magnetic fluid under the action of an applied magnetic field. The system consists of the Navier-Stokes equations, the magnetostatic equations and the temperature equation. We prove global-in-time existence of weak solutions to the system posed in a bounded domain of $R^3$ and equipped with initial and boundary conditions. The main difficulty comes from the singularity of the term representing the Kelvin force due to magnetization.
Keywords: incompressible flow, heat transfer, Magnetic fluid, global weak solutions., Navier-Stokes equations.
Mathematics Subject Classification: Primary: 35Q35, 76D05; Secondary: 80A2.
Citation: Youcef Amirat, Kamel Hamdache. On a heated incompressible magnetic fluid model. Communications on Pure & Applied Analysis, 2012, 11 (2) : 675-696. doi: 10.3934/cpaa.2012.11.675
References:
[1]

Y. Amirat, K. Hamdache and F. Murat, Global weak solutions to the equations of motion for magnetic fluids, J. Math. Fluid Mech., 10 (2008), 326-351. Google Scholar

[2]

Y. Amirat and K. Hamdache, Global weak solutions to a ferrofluid flow model, Math. Meth. Appl. Sci., 31 (2007), 123-151. Google Scholar

[3]

Y. Amirat and K. Hamdache, Strong solutions to the equations of a ferrofluid flow model, J. Math. Anal. Appl., 353 (2009), 271-294. Google Scholar

[4]

Y. Amirat and K. Hamdache, Weak solutions to the equations of motion for compressible magnetic fluids, J. Math. Pures Appl., 91 (2009), 433-467. Google Scholar

[5]

Y. Amirat and K. Hamdache, Unique solvability of equations of motion for ferrofluids, Nonlinear Analysis, Series A: Theory, Methods & Applications, 73 (2010), 471-494. Google Scholar

[6]

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

[7]

P. J. Blennerhasset, F. Lin and P. J. Stiles, Heat transfert through strongly magnetized ferrofluids, Proc. R. Soc. Lond. A, 433 (1991), 165-177. Google Scholar

[8]

E. Blums, A. Cebers and M. M Maiorov, "Magnetic Fluids,'' Walter de Gryuter & Co., Berlin-New York, 1997. Google Scholar

[9]

B. A. Finlayson, Convective instability of ferromagnetic fluids, J. Fluid Mech., 40 (1970), 753-767. Google Scholar

[10]

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. Google Scholar

[11]

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. Google Scholar

[12]

P. Grisvard, "Elliptic Problems in Nonsmooth Domains,'' Monographs and Studies in Mathematics, 24, Pitman, 1985. Google Scholar

[13]

P. N. Kaloni and J. X. Lou, Convective instability of magnetic fluids, Physical Review E, 70 (2004), 1-12. Google Scholar

[14]

P. N. Kaloni and J. X. Lou, Convective instability of magnetic fluids under alternating magnetic field, Physical Review E, 71 (2005), 1-12. Google Scholar

[15]

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

[16]

O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,'' Rev. second edition, Gordon and Breach, 1969. Google Scholar

[17]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,'' Translations Math. Monogr.23, AMS, Providence, R.I., 1968. Google Scholar

[18]

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

[19]

P. L. Lions, "Mathematical Topics in Fluid Mechanics. Volume 1. Incompressible Models,'' Oxford Science Publications, 1996. Google Scholar

[20]

J. L. Neuringer and R. E. Rosensweig, Ferrohydrodynamics, Phys. Fluids, 7 (1964), 1927-1937. Google Scholar

[21]

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

[22]

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), R167-R181. Google Scholar

[23]

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

[24]

C. L. Russel, P. J. Blennerhassett and P. J. Stiles, Strongly nonlinear vortices in magnetized ferrofluids, J. Austral. Math. Soc., Ser. B, 40 (1999), 146-171. Google Scholar

[25]

M. I. Shliomis, Effective viscosity of magnetic suspensions, Sov. Phys. JEPT, 34 (1972), 1291-1394. Google Scholar

[26]

M. I. Shliomis, Convective instability of a ferrofluid, Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, 6 (1973), 130-135. Google Scholar

[27]

M. I. Shliomis, Ferrofluids: Magnetically controllable fluids and their applications, Lecture Notes in Physics (Springer-Verlag, Heidelberg), V. {594}, ed. S. Odenbache, 85-111, 2002. Google Scholar

[28]

M. I. Shliomis and B. L Smorodin, Convective instability of magnetized fluids, Journal of Magnetism and Magnetic Materials, 252 (2002), 197-202. Google Scholar

[29]

J. Simon, Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure, SIAM J. Math. Anal., 21 (1990), 1093-1117. Google Scholar

[30]

Z. Tan and Y. Wang, Global analysis for strong solutions to the equations of a ferrofluid flow model, J. Math. Anal. Appl., 364 (2010), 424-436. Google Scholar

[31]

L. Tartar, Topics in nonlinear analysis, Publications Mathématiques d'Orsay, {78.13}, 1978. Google Scholar

[32]

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

[33]

H. C. Torrey, Bloch equations with diffusion terms, Phys. Rev., 104 (1956), 563-565. Google Scholar

[34]

S. Venkatasubramanian and P. Kaloni, Stability and uniqueness of magnetic fluid motions, Proc. R. Soc. Lond. A, 458 (2002), 1189-1204. Google Scholar

[35]

M. Zahn, Magnetic fluid and nonoparticle applications to nanotechnology, Journal of Nanoparticle Research, 73 (2001), 73-78. Google Scholar

show all references

References:
[1]

Y. Amirat, K. Hamdache and F. Murat, Global weak solutions to the equations of motion for magnetic fluids, J. Math. Fluid Mech., 10 (2008), 326-351. Google Scholar

[2]

Y. Amirat and K. Hamdache, Global weak solutions to a ferrofluid flow model, Math. Meth. Appl. Sci., 31 (2007), 123-151. Google Scholar

[3]

Y. Amirat and K. Hamdache, Strong solutions to the equations of a ferrofluid flow model, J. Math. Anal. Appl., 353 (2009), 271-294. Google Scholar

[4]

Y. Amirat and K. Hamdache, Weak solutions to the equations of motion for compressible magnetic fluids, J. Math. Pures Appl., 91 (2009), 433-467. Google Scholar

[5]

Y. Amirat and K. Hamdache, Unique solvability of equations of motion for ferrofluids, Nonlinear Analysis, Series A: Theory, Methods & Applications, 73 (2010), 471-494. Google Scholar

[6]

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

[7]

P. J. Blennerhasset, F. Lin and P. J. Stiles, Heat transfert through strongly magnetized ferrofluids, Proc. R. Soc. Lond. A, 433 (1991), 165-177. Google Scholar

[8]

E. Blums, A. Cebers and M. M Maiorov, "Magnetic Fluids,'' Walter de Gryuter & Co., Berlin-New York, 1997. Google Scholar

[9]

B. A. Finlayson, Convective instability of ferromagnetic fluids, J. Fluid Mech., 40 (1970), 753-767. Google Scholar

[10]

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. Google Scholar

[11]

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. Google Scholar

[12]

P. Grisvard, "Elliptic Problems in Nonsmooth Domains,'' Monographs and Studies in Mathematics, 24, Pitman, 1985. Google Scholar

[13]

P. N. Kaloni and J. X. Lou, Convective instability of magnetic fluids, Physical Review E, 70 (2004), 1-12. Google Scholar

[14]

P. N. Kaloni and J. X. Lou, Convective instability of magnetic fluids under alternating magnetic field, Physical Review E, 71 (2005), 1-12. Google Scholar

[15]

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

[16]

O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,'' Rev. second edition, Gordon and Breach, 1969. Google Scholar

[17]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,'' Translations Math. Monogr.23, AMS, Providence, R.I., 1968. Google Scholar

[18]

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

[19]

P. L. Lions, "Mathematical Topics in Fluid Mechanics. Volume 1. Incompressible Models,'' Oxford Science Publications, 1996. Google Scholar

[20]

J. L. Neuringer and R. E. Rosensweig, Ferrohydrodynamics, Phys. Fluids, 7 (1964), 1927-1937. Google Scholar

[21]

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

[22]

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), R167-R181. Google Scholar

[23]

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

[24]

C. L. Russel, P. J. Blennerhassett and P. J. Stiles, Strongly nonlinear vortices in magnetized ferrofluids, J. Austral. Math. Soc., Ser. B, 40 (1999), 146-171. Google Scholar

[25]

M. I. Shliomis, Effective viscosity of magnetic suspensions, Sov. Phys. JEPT, 34 (1972), 1291-1394. Google Scholar

[26]

M. I. Shliomis, Convective instability of a ferrofluid, Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, 6 (1973), 130-135. Google Scholar

[27]

M. I. Shliomis, Ferrofluids: Magnetically controllable fluids and their applications, Lecture Notes in Physics (Springer-Verlag, Heidelberg), V. {594}, ed. S. Odenbache, 85-111, 2002. Google Scholar

[28]

M. I. Shliomis and B. L Smorodin, Convective instability of magnetized fluids, Journal of Magnetism and Magnetic Materials, 252 (2002), 197-202. Google Scholar

[29]

J. Simon, Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure, SIAM J. Math. Anal., 21 (1990), 1093-1117. Google Scholar

[30]

Z. Tan and Y. Wang, Global analysis for strong solutions to the equations of a ferrofluid flow model, J. Math. Anal. Appl., 364 (2010), 424-436. Google Scholar

[31]

L. Tartar, Topics in nonlinear analysis, Publications Mathématiques d'Orsay, {78.13}, 1978. Google Scholar

[32]

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

[33]

H. C. Torrey, Bloch equations with diffusion terms, Phys. Rev., 104 (1956), 563-565. Google Scholar

[34]

S. Venkatasubramanian and P. Kaloni, Stability and uniqueness of magnetic fluid motions, Proc. R. Soc. Lond. A, 458 (2002), 1189-1204. Google Scholar

[35]

M. Zahn, Magnetic fluid and nonoparticle applications to nanotechnology, Journal of Nanoparticle Research, 73 (2001), 73-78. Google Scholar

[1]

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
[2]

Youcef Amirat, Kamel Hamdache. Strong solutions to the equations of flow and heat transfer in magnetic fluids with internal rotations. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3289-3320. doi: 10.3934/dcds.2013.33.3289
[3]

Fei Jiang, Song Jiang, Junpin Yin. Global weak solutions to the two-dimensional Navier-Stokes equations of compressible heat-conducting flows with symmetric data and forces. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 567-587. doi: 10.3934/dcds.2014.34.567
[4]

Jingrui Wang, Keyan Wang. Almost sure existence of global weak solutions to the 3D incompressible Navier-Stokes equation. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 5003-5019. doi: 10.3934/dcds.2017215
[5]

Keyan Wang. On global regularity of incompressible Navier-Stokes equations in $\mathbf R^3$. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1067-1072. doi: 10.3934/cpaa.2009.8.1067
[6]

Daniel Coutand, J. Peirce, Steve Shkoller. Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains. Communications on Pure & Applied Analysis, 2002, 1 (1) : 35-50. doi: 10.3934/cpaa.2002.1.35
[7]

Daniel Pardo, José Valero, Ángel Giménez. Global attractors for weak solutions of the three-dimensional Navier-Stokes equations with damping. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3569-3590. doi: 10.3934/dcdsb.2018279
[8]

Jian-Guo Liu, Zhaoyun Zhang. Existence of global weak solutions of $ p $-Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2022, 27 (1) : 469-486. doi: 10.3934/dcdsb.2021051
[9]

Yongfu Wang. Global strong solution to the two dimensional nonhomogeneous incompressible heat conducting Navier-Stokes flows with vacuum. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4317-4333. doi: 10.3934/dcdsb.2020099
[10]

Peter E. Kloeden, José Valero. The Kneser property of the weak solutions of the three dimensional Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2010, 28 (1) : 161-179. doi: 10.3934/dcds.2010.28.161
[11]

Joanna Rencławowicz, Wojciech M. Zajączkowski. Global regular solutions to the Navier-Stokes equations with large flux. Conference Publications, 2011, 2011 (Special) : 1234-1243. doi: 10.3934/proc.2011.2011.1234
[12]

Peixin Zhang, Jianwen Zhang, Junning Zhao. On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 1085-1103. doi: 10.3934/dcds.2016.36.1085
[13]

Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations II: Global existence of small solutions. Evolution Equations & Control Theory, 2012, 1 (1) : 217-234. doi: 10.3934/eect.2012.1.217
[14]

Joelma Azevedo, Juan Carlos Pozo, Arlúcio Viana. Global solutions to the non-local Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021146
[15]

J. Huang, Marius Paicu. Decay estimates of global solution to 2D incompressible Navier-Stokes equations with variable viscosity. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4647-4669. doi: 10.3934/dcds.2014.34.4647
[16]

Stefano Scrobogna. Global existence and convergence of nondimensionalized incompressible Navier-Stokes equations in low Froude number regime. Discrete & Continuous Dynamical Systems, 2020, 40 (9) : 5471-5511. doi: 10.3934/dcds.2020235
[17]

Lihuai Du, Ting Zhang. Local and global strong solution to the stochastic 3-D incompressible anisotropic Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4745-4765. doi: 10.3934/dcds.2018209
[18]

Ciprian Foias, Ricardo Rosa, Roger Temam. Topological properties of the weak global attractor of the three-dimensional Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2010, 27 (4) : 1611-1631. doi: 10.3934/dcds.2010.27.1611
[19]

Yong Yang, Bingsheng Zhang. On the Kolmogorov entropy of the weak global attractor of 3D Navier-Stokes equations:Ⅰ. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2339-2350. doi: 10.3934/dcdsb.2017101
[20]

Xue-Li Song, Yan-Ren Hou. Attractors for the three-dimensional incompressible Navier-Stokes equations with damping. Discrete & Continuous Dynamical Systems, 2011, 31 (1) : 239-252. doi: 10.3934/dcds.2011.31.239

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (91)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy