• Previous Article
    Finite dimensional smooth attractor for the Berger plate with dissipation acting on a portion of the boundary
  • CPAA Home
  • This Issue
  • Next Article
    Elliptic operators with unbounded diffusion coefficients perturbed by inverse square potentials in $L^p$--spaces
November  2016, 15(6): 2329-2355. doi: 10.3934/cpaa.2016039

Steady state solutions of ferrofluid flow models

1. 

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

2. 

Léonard de Vinci, Pôle Universitaire. Research Center, 92916 Paris la Défense Cedex, France

Received  February 2016 Revised  May 2016 Published  September 2016

We study two models of differential equations for the stationary flow of an incompressible viscous magnetic fluid subjected to an external magnetic field. The first model, called Rosensweig's model, consists of the incompressible Navier-Stokes equations, the angular momentum equation, the magnetization equation of Bloch-Torrey type, and the magnetostatic equations. The second one, called Shliomis model, is obtained by assuming that the angular momentum is given in terms of the magnetic field, the magnetization field and the vorticity. It consists of the incompressible Navier-Stokes equation, the magnetization equation and the magnetostatic equations. We prove, for each of the differential systems posed in a bounded domain of $\mathbb{R}^3$ and equipped with boundary conditions, existence of weak solutions by using regularization techniques, linearization and the Schauder fixed point theorem.
Citation: Youcef Amirat, Kamel Hamdache. Steady state solutions of ferrofluid flow models. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2329-2355. doi: 10.3934/cpaa.2016039
References:
[1]

S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II,, \emph{Comm. Pure Appl. Math., 17 (1964), 35. doi: 10.1002/cpa.3160170104.

[2]

Y. Amirat, K. Hamdache and F. Murat, Global weak solutions to the equations of motion for magnetic fluids,, \emph{J. Math. Fluid Mech., 10 (2008), 326. doi: 10.1016/j.matpur.2009.01.015.

[3]

Y. Amirat and K. Hamdache, Global weak solutions to a ferrofluid flow model,, \emph{Math. Meth. Appl. Sci., 31 (2007), 123. doi: 10.1002/mma.896.

[4]

C. Amrouche and N. Seloula, On the Stokes equations with the Navier-type boundary conditions,, \emph{Differ. Equ. & Appl., 3 (2011), 581. doi: dx.doi.org/10.7153/dea-03-36.

[5]

F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models,, Applied Mathematical Sciences, (2013). doi: 10.1007/978-1-4614-5975-0.

[6]

L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes,, \emph{Rendiconti del Seminario Matematico della Universit\`a di Padova, 31 (1961), 308. doi: http://eudml.org/doc/107065.

[7]

P. G. Ciarlet, Mathematical Elasticity,, North-Holland, (1988). doi: 044481776X,9780444817761.

[8]

R. Dautray and J.-L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques,, Vol. 5, (1984).

[9]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. I. Linearized Steady Problems,, Springer tracts in Natural Philosophy, (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, (1994). doi: 10.1007/978-1-4612-5364-8.

[11]

G. D. Gaspari, Bloch equation for conduction-electron spin resonance,, \emph{Phys. Review, 131 (1966), 215. doi: http://dx.doi.org/10.1103/PhysRev.151.215.

[12]

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

[13]

Q. Q. A. Pankhurst, J. Connolly, S. K. Jones and J. Dobson, Applications of magnetic nonoparticles in biomedicine,, \emph{J. Phys. D: Appl. Phys., 36 (2003).

[14]

C. Rinaldi and M. Zahn, Effects of spin viscosity on ferrofluids flow profiles in alternating and rotating magnetic fields,, \emph{Phys. of Fluids, 14 (2002), 2847. doi: http://dx.doi.org/10.1063/1.1485762.

[15]

R. E. Rosensweig, Ferrohydrodynamics,, Dover Publications, (1997).

[16]

R. E. Rosensweig, Basic equations for magnetic fluids with internal rotations,, in \emph{Ferrofluids: Magnetically Controllable Fluids and Their Applications}, 594 (2002), 61.

[17]

P. Shi and S. Wright, $W^{2,p}$ Regularity of the displacement problem for the Lamé system on $W^{2,s}$ domains,, \emph{J. Math. Anal. Appl., 239 (1999), 291. doi: 10.1006/jmaa.1999.6562.

[18]

M. I Shliomis, Effective viscosity of magnetic suspension,, \emph{Sov. Phys. JETP, 44 (1972), 1291.

[19]

M. I Shliomis, Retrospective and issues,, in \emph{Ferrofluids: Magnetically Controllable Fluids and Their Applications}, 594 (2002), 85.

[20]

R. Temam, Navier-Stokes Equations,, 3rd (revised) edition, (1984). doi: 0821827375,9780821827376.

[21]

H. C. Torrey, Bloch equations with diffusion terms,, \emph{Phys. Rev., 104 (1956), 563.

[22]

M. Zahn, Magnetic fluid and nonoparticle applications to nanotechnology,, \emph{Journal of Nanoparticle Research, 3 (2001), 73. doi: 10.1023/A:1011497813424.

show all references

References:
[1]

S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II,, \emph{Comm. Pure Appl. Math., 17 (1964), 35. doi: 10.1002/cpa.3160170104.

[2]

Y. Amirat, K. Hamdache and F. Murat, Global weak solutions to the equations of motion for magnetic fluids,, \emph{J. Math. Fluid Mech., 10 (2008), 326. doi: 10.1016/j.matpur.2009.01.015.

[3]

Y. Amirat and K. Hamdache, Global weak solutions to a ferrofluid flow model,, \emph{Math. Meth. Appl. Sci., 31 (2007), 123. doi: 10.1002/mma.896.

[4]

C. Amrouche and N. Seloula, On the Stokes equations with the Navier-type boundary conditions,, \emph{Differ. Equ. & Appl., 3 (2011), 581. doi: dx.doi.org/10.7153/dea-03-36.

[5]

F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models,, Applied Mathematical Sciences, (2013). doi: 10.1007/978-1-4614-5975-0.

[6]

L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes,, \emph{Rendiconti del Seminario Matematico della Universit\`a di Padova, 31 (1961), 308. doi: http://eudml.org/doc/107065.

[7]

P. G. Ciarlet, Mathematical Elasticity,, North-Holland, (1988). doi: 044481776X,9780444817761.

[8]

R. Dautray and J.-L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques,, Vol. 5, (1984).

[9]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. I. Linearized Steady Problems,, Springer tracts in Natural Philosophy, (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, (1994). doi: 10.1007/978-1-4612-5364-8.

[11]

G. D. Gaspari, Bloch equation for conduction-electron spin resonance,, \emph{Phys. Review, 131 (1966), 215. doi: http://dx.doi.org/10.1103/PhysRev.151.215.

[12]

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

[13]

Q. Q. A. Pankhurst, J. Connolly, S. K. Jones and J. Dobson, Applications of magnetic nonoparticles in biomedicine,, \emph{J. Phys. D: Appl. Phys., 36 (2003).

[14]

C. Rinaldi and M. Zahn, Effects of spin viscosity on ferrofluids flow profiles in alternating and rotating magnetic fields,, \emph{Phys. of Fluids, 14 (2002), 2847. doi: http://dx.doi.org/10.1063/1.1485762.

[15]

R. E. Rosensweig, Ferrohydrodynamics,, Dover Publications, (1997).

[16]

R. E. Rosensweig, Basic equations for magnetic fluids with internal rotations,, in \emph{Ferrofluids: Magnetically Controllable Fluids and Their Applications}, 594 (2002), 61.

[17]

P. Shi and S. Wright, $W^{2,p}$ Regularity of the displacement problem for the Lamé system on $W^{2,s}$ domains,, \emph{J. Math. Anal. Appl., 239 (1999), 291. doi: 10.1006/jmaa.1999.6562.

[18]

M. I Shliomis, Effective viscosity of magnetic suspension,, \emph{Sov. Phys. JETP, 44 (1972), 1291.

[19]

M. I Shliomis, Retrospective and issues,, in \emph{Ferrofluids: Magnetically Controllable Fluids and Their Applications}, 594 (2002), 85.

[20]

R. Temam, Navier-Stokes Equations,, 3rd (revised) edition, (1984). doi: 0821827375,9780821827376.

[21]

H. C. Torrey, Bloch equations with diffusion terms,, \emph{Phys. Rev., 104 (1956), 563.

[22]

M. Zahn, Magnetic fluid and nonoparticle applications to nanotechnology,, \emph{Journal of Nanoparticle Research, 3 (2001), 73. doi: 10.1023/A:1011497813424.

[1]

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

[2]

Grégoire Allaire, Carlos Conca, Luis Friz, Jaime H. Ortega. On Bloch waves for the Stokes equations. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 1-28. doi: 10.3934/dcdsb.2007.7.1

[3]

Vittorino Pata. On the regularity of solutions to the Navier-Stokes equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 747-761. doi: 10.3934/cpaa.2012.11.747

[4]

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

[5]

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

[6]

Fang Li, Bo You, Yao Xu. Dynamics of weak solutions for the three dimensional Navier-Stokes equations with nonlinear damping. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4267-4284. doi: 10.3934/dcdsb.2018137

[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, 2017, 22 (11) : 1-22. doi: 10.3934/dcdsb.2018279

[8]

Chérif Amrouche, María Ángeles Rodríguez-Bellido. On the very weak solution for the Oseen and Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 159-183. doi: 10.3934/dcdss.2010.3.159

[9]

Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602

[10]

Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149

[11]

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 - A, 2014, 34 (2) : 567-587. doi: 10.3934/dcds.2014.34.567

[12]

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

[13]

Peter Anthony, Sergey Zelik. Infinite-energy solutions for the Navier-Stokes equations in a strip revisited. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1361-1393. doi: 10.3934/cpaa.2014.13.1361

[14]

Tomás Caraballo, Peter E. Kloeden, José Real. Invariant measures and Statistical solutions of the globally modified Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 761-781. doi: 10.3934/dcdsb.2008.10.761

[15]

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

[16]

Jochen Merker. Strong solutions of doubly nonlinear Navier-Stokes equations. Conference Publications, 2011, 2011 (Special) : 1052-1060. doi: 10.3934/proc.2011.2011.1052

[17]

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

[18]

Rafaela Guberović. Smoothness of Koch-Tataru solutions to the Navier-Stokes equations revisited. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 231-236. doi: 10.3934/dcds.2010.27.231

[19]

Yukang Chen, Changhua Wei. Partial regularity of solutions to the fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5309-5322. doi: 10.3934/dcds.2016033

[20]

Zhilei Liang. Convergence rate of solutions to the contact discontinuity for the compressible Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1907-1926. doi: 10.3934/cpaa.2013.12.1907

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]