Influence of a magnetic field on the Soret-effect-dominated thermal convection in ferrofluids

Andrey Ryskin; Harald Pleiner

doi:10.1103/PhysRevE.69.046301

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Influence of a magnetic field on the Soret-effect-dominated thermal convection in ferrofluids

Andrey Ryskin and Harald Pleiner

Max-Planck-Institut für Polymerforschung, D 55021 Mainz, Germany

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Phys. Rev. E 69, 046301 – Published 14 April, 2004

DOI: https://doi.org/10.1103/PhysRevE.69.046301

Abstract

We investigate theoretically the influence of a magnetic field on the growth of convective rolls in a slab of ferrofluid subject to a vertical temperature gradient. Due to the pronounced Soret effect of these materials in combination with a considerable solutal expansion, a dynamic description as a binary mixture is appropriate. We first derive a comprehensive set of magnetic field effects in the statics and dynamics of binary mixtures. Among those, the two prominent ones, the Kelvin force and magnetophoresis, are studied in detail with respect to their influence on the thermal convection behavior. The main difference from the case without an external field rests in the importance of the boundary layers, which influence the bulk problem through the magnetic boundary conditions. We discuss an analytical approximate solution and compare it with a numerical multimode expansion.

References (15)

R. E. Rosensweig, Ferrohydrodynamics (Cambridge University Press, Cambridge, England, 1985).
E. Blums, A. Mezulis, M. Maiorov, and G. Kronkalns, J. Magn. Magn. Mater. 169, 220 (1997).
A. Ryskin, H.-W. Müller, and H. Pleiner, Phys. Rev. E 67, 046302 (2003).
H. Pleiner and H. R. Brand, in Pattern Formation in Liquid Crystals, edited by A. Buka and L. Kramer (Springer, New York, 1996), Chap. 2.
E. Jarkova, H. Pleiner, H.-W. Müller, A. Fink, and H. R. Brand, Eur. Phys. J. E 5, 583 (2001).
A. Ryskin, H. Pleiner, and H.-W. Müller, Eur. Phys. J. E 11, 389 (2003).
We use rationalized Gaussian (or Heaviside-Lorentz) units. For a conversion to standard SI units, see J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).
With the usual heuristic definition $χ = \partial M / \partial H$ there is $χ = χ_{0} + χ_{H} H_{0}^{2}$ and $\bar{ε} = 1 + χ .$
A. Recktenwald and M. Lücke, J. Magn. Magn. Mater. 188, 326 (1998).
B. A. Finlayson, J. Fluid Mech. 40, 753 (1970).
G. K. Auernhammer and H. R. Brand, Eur. Phys. J. B 16, 157 (2000).
J. Lenglet, A. Bourdon, J.-C. Bacri, and G. Demouchy, Phys. Rev. E 65, 031408 (2002).
E. N. Lorenz, J. Atmos. Sci. 20, 130 (1963).
In the simplest case this is a five-mode model for $A_{1} {= A}_{0} (2 + k)$ and $A_{n} = 0$ for $n >~ 2.$
S. Hollinger, M. Lücke, and H.-W. Müller, Phys. Rev. E 57, 4250 (1998).

[c1] R. E. Rosensweig, Ferrohydrodynamics (Cambridge University Press, Cambridge, England, 1985).

[c2] E. Blums, A. Mezulis, M. Maiorov, and G. Kronkalns, J. Magn. Magn. Mater. 169, 220 (1997).

[c3] A. Ryskin, H.-W. Müller, and H. Pleiner, Phys. Rev. E 67, 046302 (2003).

[c4] H. Pleiner and H. R. Brand, in Pattern Formation in Liquid Crystals, edited by A. Buka and L. Kramer (Springer, New York, 1996), Chap. 2.

[c5] E. Jarkova, H. Pleiner, H.-W. Müller, A. Fink, and H. R. Brand, Eur. Phys. J. E 5, 583 (2001).

[c6] A. Ryskin, H. Pleiner, and H.-W. Müller, Eur. Phys. J. E 11, 389 (2003).

[c7] We use rationalized Gaussian (or Heaviside-Lorentz) units. For a conversion to standard SI units, see J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

[c8] With the usual heuristic definition $χ = \partial M / \partial H$ there is $χ = χ_{0} + χ_{H} H_{0}^{2}$ and $\bar{ε} = 1 + χ .$

[c9] A. Recktenwald and M. Lücke, J. Magn. Magn. Mater. 188, 326 (1998).

[c10] B. A. Finlayson, J. Fluid Mech. 40, 753 (1970).

[c11] G. K. Auernhammer and H. R. Brand, Eur. Phys. J. B 16, 157 (2000).

[c12] J. Lenglet, A. Bourdon, J.-C. Bacri, and G. Demouchy, Phys. Rev. E 65, 031408 (2002).

[c13] E. N. Lorenz, J. Atmos. Sci. 20, 130 (1963).

[c14] In the simplest case this is a five-mode model for $A_{1} {= A}_{0} (2 + k)$ and $A_{n} = 0$ for $n >~ 2.$

[c15] S. Hollinger, M. Lücke, and H.-W. Müller, Phys. Rev. E 57, 4250 (1998).

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Influence of a magnetic field on the Soret-effect-dominated thermal convection in ferrofluids

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