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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 and 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, Comm. Pure Appl. Math., 17 (1964), 35-92. doi: 10.1002/cpa.3160170104. [2] 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. doi: 10.1016/j.matpur.2009.01.015. [3] Y. Amirat and K. Hamdache, Global weak solutions to a ferrofluid flow model, Math. Meth. Appl. Sci., 31 (2007), 123-151. doi: 10.1002/mma.896. [4] C. Amrouche and N. Seloula, On the Stokes equations with the Navier-type boundary conditions, Differ. Equ. & Appl., 3 (2011), 581-607. 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, vol. 183, Springer, 2013. doi: 10.1007/978-1-4614-5975-0. [6] L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes, Rendiconti del Seminario Matematico della Universit\a di Padova, 31 (1961), 308-340. doi: http://eudml.org/doc/107065. [7] P. G. Ciarlet, Mathematical Elasticity, North-Holland, Amsterdam/New York, 1988. doi: 044481776X,9780444817761. [8] R. Dautray and J.-L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques, Vol. 5, Masson, 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, 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] G. D. Gaspari, Bloch equation for conduction-electron spin resonance, Phys. Review, 131 (1966), 215-219. 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, J. Phys. D: Appl. Phys., 36 (2003), R167-R181. [14] C. Rinaldi and M. Zahn, Effects of spin viscosity on ferrofluids flow profiles in alternating and rotating magnetic fields, Phys. of Fluids, 14 (2002), 2847-2870. doi: http://dx.doi.org/10.1063/1.1485762. [15] R. E. Rosensweig, Ferrohydrodynamics, Dover Publications, Inc., 1997. [16] R. E. Rosensweig, Basic equations for magnetic fluids with internal rotations, in Ferrofluids: Magnetically Controllable Fluids and Their Applications, Lecture Notes in Physics (Springer-Verlag, Heidelberg), 594, S. Odenbache Ed., (2002), 61-84. [17] P. Shi and S. Wright, $W^{2,p}$ Regularity of the displacement problem for the Lamé system on $W^{2,s}$ domains, J. Math. Anal. Appl., 239 (1999), 291-305. doi: 10.1006/jmaa.1999.6562. [18] M. I Shliomis, Effective viscosity of magnetic suspension, Sov. Phys. JETP, 44 (1972), 1291-1294. [19] M. I Shliomis, Retrospective and issues, in Ferrofluids: Magnetically Controllable Fluids and Their Applications, Lecture Notes in Physics (Springer-Verlag, Heidelberg), 594, S. Odenbache Ed., (2002), 85-111. [20] R. Temam, Navier-Stokes Equations, 3rd (revised) edition, Elsevier Science Publishers B.V., Amsterdam, 1984. doi: 0821827375,9780821827376. [21] H. C. Torrey, Bloch equations with diffusion terms, Phys. Rev., 104 (1956), 563-565. [22] M. Zahn, Magnetic fluid and nonoparticle applications to nanotechnology, Journal of Nanoparticle Research, 3 (2001), 73-78. doi: 10.1023/A:1011497813424.

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##### 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, Comm. Pure Appl. Math., 17 (1964), 35-92. doi: 10.1002/cpa.3160170104. [2] 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. doi: 10.1016/j.matpur.2009.01.015. [3] Y. Amirat and K. Hamdache, Global weak solutions to a ferrofluid flow model, Math. Meth. Appl. Sci., 31 (2007), 123-151. doi: 10.1002/mma.896. [4] C. Amrouche and N. Seloula, On the Stokes equations with the Navier-type boundary conditions, Differ. Equ. & Appl., 3 (2011), 581-607. 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, vol. 183, Springer, 2013. doi: 10.1007/978-1-4614-5975-0. [6] L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes, Rendiconti del Seminario Matematico della Universit\a di Padova, 31 (1961), 308-340. doi: http://eudml.org/doc/107065. [7] P. G. Ciarlet, Mathematical Elasticity, North-Holland, Amsterdam/New York, 1988. doi: 044481776X,9780444817761. [8] R. Dautray and J.-L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques, Vol. 5, Masson, 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, 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] G. D. Gaspari, Bloch equation for conduction-electron spin resonance, Phys. Review, 131 (1966), 215-219. 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, J. Phys. D: Appl. Phys., 36 (2003), R167-R181. [14] C. Rinaldi and M. Zahn, Effects of spin viscosity on ferrofluids flow profiles in alternating and rotating magnetic fields, Phys. of Fluids, 14 (2002), 2847-2870. doi: http://dx.doi.org/10.1063/1.1485762. [15] R. E. Rosensweig, Ferrohydrodynamics, Dover Publications, Inc., 1997. [16] R. E. Rosensweig, Basic equations for magnetic fluids with internal rotations, in Ferrofluids: Magnetically Controllable Fluids and Their Applications, Lecture Notes in Physics (Springer-Verlag, Heidelberg), 594, S. Odenbache Ed., (2002), 61-84. [17] P. Shi and S. Wright, $W^{2,p}$ Regularity of the displacement problem for the Lamé system on $W^{2,s}$ domains, J. Math. Anal. Appl., 239 (1999), 291-305. doi: 10.1006/jmaa.1999.6562. [18] M. I Shliomis, Effective viscosity of magnetic suspension, Sov. Phys. JETP, 44 (1972), 1291-1294. [19] M. I Shliomis, Retrospective and issues, in Ferrofluids: Magnetically Controllable Fluids and Their Applications, Lecture Notes in Physics (Springer-Verlag, Heidelberg), 594, S. Odenbache Ed., (2002), 85-111. [20] R. Temam, Navier-Stokes Equations, 3rd (revised) edition, Elsevier Science Publishers B.V., Amsterdam, 1984. doi: 0821827375,9780821827376. [21] H. C. Torrey, Bloch equations with diffusion terms, Phys. Rev., 104 (1956), 563-565. [22] M. Zahn, Magnetic fluid and nonoparticle applications to nanotechnology, Journal of Nanoparticle Research, 3 (2001), 73-78. doi: 10.1023/A:1011497813424.
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