\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Strong solutions to the equations of flow and heat transfer in magnetic fluids with internal rotations

Abstract Related Papers Cited by
  • In this paper we study the equations of flow and heat transfer in a magnetic fluid with internal rotations, when the fluid is subjected to the action of an external magnetic field. The system of equations is a combination of the Navier-Stokes equations, the magnetization relaxation equation of Bloch type, the magnetostatic equations and the temperature equation. We prove the local-in-time existence of the unique strong solution to the system equipped with initial and boundary conditions and establish a blow-up criterium for strong solutions. We then prove the global-in-time existence of strong solutions, under smallness assumptions on the initial data and the external magnetic field.
    Mathematics Subject Classification: Primary: 35Q35, 76D05; Secondary: 80A20.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    Y. Amirat and K. Hamdache, Strong solutions to the equations of a ferrofluid flow model, J. Math. Anal. Appl., 353 (2009), 271-294.doi: 10.1016/j.jmaa.2008.11.084.

    [2]

    Y. Amirat and K. Hamdache, On a heated incompressible magnetic fluid model, Comm. Pure Appl. Anal., 11 (2012), 675-696.doi: 10.3934/cpaa.2012.11.675.

    [3]

    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.

    [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.

    [5]

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

    [6]

    G. Carbou and P. Fabrie, Regular solutions for Landau-Lifschitz equation in a bounded domain, Differential and Integral Equations, 14 (2001), 213-229.

    [7]

    Y. Cho, H. J. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluids, J. Math. Pures Appl., 83 (2004), 243-275.doi: 10.1016/j.matpur.2003.11.004.

    [8]

    R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547.doi: 10.1007/BF01393835.

    [9]

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

    [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.doi: 10.1007/978-1-4612-5364-8.

    [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, New-York, 1994.doi: 10.1007/978-1-4612-5364-8.

    [12]

    R. Ganguly, S. Sen and I. K. Puri, Heat transfer augmentation using a magnetic fluid under the influence of a line dipole, J. Magn. Magn. Mater., 271 (2004), 63-73.

    [13]

    M. Giaquinta and G. Modica, Nonlinear systems of the type of the stationary Navier-Stokes system, J. Reine Angew. Math., 330 (1982), 173-214.

    [14]

    Y. Giga and H. Sohr, Abstract $L^p$ estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains, Journal of Functional Analysis, 102 (1991), 72-94.doi: 10.1016/0022-1236(91)90136-S.

    [15]

    M. Giga, Y. Giga and H. Sohr, $L^p$ estimates for abstract linear parabolic equations, Proc. Japan Acad. Ser. A Math. Sci., 67 (1991), 197-202.

    [16]

    P. Grisvard, "Elliptic Problems in Nonsmooth Domains,'' Monographs and Studies in Mathematics, 24, Pitman (Advanced Publishing Program), Boston, MA, 1985.

    [17]

    P. N. Kaloni and J. X. Lou, Convective instability of magnetic fluids, Physical Review, E70 (2004), 1-12.

    [18]

    P. N. Kaloni and A. Mahajan, Stability of magnetic fluid motions in a saturated porous medium, Z. Angew. Math. Phys., 62 (2011), 529-538.doi: 10.1007/s00033-010-0096-x.

    [19]

    H. Kim, A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes equations, SIAM J. Math. Anal., 37 (2006), 1417-1434.doi: 10.1137/S0036141004442197.

    [20]

    O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,'' Second English edition, revised and enlarged, Mathematics and its Applications, Vol. 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969.

    [21]

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

    [22]

    J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,'' Dunod, Gauthier-Villars, Paris, 1969.

    [23]

    P.-L. Lions, "Mathematical Topics in Fluid Mechanics. Volume 1. Incompressible Models,'' Oxford Lecture Series in Mathematics and its Applications, 3, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1996.

    [24]

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

    [25]

    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.

    [26]

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

    [27]

    R. E. Rosensweig, Ferrofluids: Magnetically controllable fluids and their applications, Lecture Notes in Physics (Springer-Verlag, Heidelberg), S. Odenbach Ed., 594 (2002), 61-84.

    [28]

    M. I Shliomis, Effective viscosity of magnetic suspension, Sov. Phys. JETP, 44 (1972), 1291-1294.

    [29]

    M. I. Shliomis and B. L. Smorodin, Convective instability of magnetized ferrofluids, J. Magn. and Magn. Mater., 252 (2002), 197-202.

    [30]

    M. I. Shliomis, Ferrofluids: Magnetically controllable fluids and their applications, Lecture Notes in Physics (Springer-Verlag, Heidelberg), S. Odenbach Ed., 594 (2002), 85-111.

    [31]

    M. I. Shliomis, Convective instability of magnetized ferrofluids: Influence of magnetophoresis and soret effect, in "Thermal Nonequilibrium Phenomena in Fluid Mixtures," Lecture Notes in Physics, Springer, Berlin, 584 (2002), 355-371.

    [32]

    L. Sunil, P. Chand, P. Bharti and A. Mahajan, Thermal convection in micropolar ferrofluid in the presence of rotation, J. of Magn. and Magn. Mater., 320 (2008), 316-324.

    [33]

    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.doi: 10.1016/j.jmaa.2009.10.032.

    [34]

    L. Tartar, "Topics in Nonlinear Analysis," Publications Mathématiques d'Orsay 78, 13, Université de Paris-Sud, Département de Mathématique, Orsay, 1978.

    [35]

    R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis,'' Reprint of the 1984 edition, AMS-Chelsea Publishing, Providence, RI, 2001.

    [36]

    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.

    [37]

    M. Zahn, Magnetic fluid and nonoparticle applications to nanotechnology, Journal of Nanoparticle Research, 3 (2001), 73-78.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(63) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return