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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 & Applied Analysis, 2016, 15 (6) : 2329-2355. doi: 10.3934/cpaa.2016039
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show all references

References:
[1]

Comm. Pure Appl. Math., 17 (1964), 35-92. doi: 10.1002/cpa.3160170104.  Google Scholar

[2]

J. Math. Fluid Mech., 10 (2008), 326-351. doi: 10.1016/j.matpur.2009.01.015.  Google Scholar

[3]

Math. Meth. Appl. Sci., 31 (2007), 123-151. doi: 10.1002/mma.896.  Google Scholar

[4]

Differ. Equ. & Appl., 3 (2011), 581-607. doi: dx.doi.org/10.7153/dea-03-36.  Google Scholar

[5]

Applied Mathematical Sciences, vol. 183, Springer, 2013. doi: 10.1007/978-1-4614-5975-0.  Google Scholar

[6]

Rendiconti del Seminario Matematico della Universit\`a di Padova, 31 (1961), 308-340. doi: http://eudml.org/doc/107065.  Google Scholar

[7]

North-Holland, Amsterdam/New York, 1988. doi: 044481776X,9780444817761.  Google Scholar

[8]

Vol. 5, Masson, 1984.  Google Scholar

[9]

Springer tracts in Natural Philosophy, 38, Springer Verlag, New-York, 1994. doi: 10.1007/978-1-4612-5364-8.  Google Scholar

[10]

Springer tracts in Natural Philosophy, 39, Springer Verlag, 1994. doi: 10.1007/978-1-4612-5364-8.  Google Scholar

[11]

Phys. Review, 131 (1966), 215-219. doi: http://dx.doi.org/10.1103/PhysRev.151.215.  Google Scholar

[12]

Dunod-Gauthier-Villars, 1969.  Google Scholar

[13]

J. Phys. D: Appl. Phys., 36 (2003), R167-R181. Google Scholar

[14]

Phys. of Fluids, 14 (2002), 2847-2870. doi: http://dx.doi.org/10.1063/1.1485762.  Google Scholar

[15]

Dover Publications, Inc., 1997. Google Scholar

[16]

in Ferrofluids: Magnetically Controllable Fluids and Their Applications, Lecture Notes in Physics (Springer-Verlag, Heidelberg), 594, S. Odenbache Ed., (2002), 61-84. Google Scholar

[17]

J. Math. Anal. Appl., 239 (1999), 291-305. doi: 10.1006/jmaa.1999.6562.  Google Scholar

[18]

Sov. Phys. JETP, 44 (1972), 1291-1294. Google Scholar

[19]

in Ferrofluids: Magnetically Controllable Fluids and Their Applications, Lecture Notes in Physics (Springer-Verlag, Heidelberg), 594, S. Odenbache Ed., (2002), 85-111. Google Scholar

[20]

3rd (revised) edition, Elsevier Science Publishers B.V., Amsterdam, 1984. doi: 0821827375,9780821827376.  Google Scholar

[21]

Phys. Rev., 104 (1956), 563-565. Google Scholar

[22]

Journal of Nanoparticle Research, 3 (2001), 73-78. doi: 10.1023/A:1011497813424.  Google Scholar

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