FTW / ITP, Otto von Guericke Universitat, Postfach 4120, D 39016 Magdeburg

The free surface of a ferrofluid, subjected to a vertically oriented and uniform magnetic field, becomes unstable above a critical value of the field [1]. This Rosenszweig or normal field instability gives rise to a hexagonal or square pattern of peaks [2]. In contrast, a magnetic field field tangential to the undisturbed free interface tends to stabilizes the surface in the direction of the field.

Extending previous studies [3], we inves tigate theoretically the pattern formation in the presence of an arbitrary oriented magnetic field. The stability of hexagonal, square, ridge and rhombic planforms is analyzed by means of an energy variational method. Moreover, adding the wavenumber modulus k to the set of variations parameters, the wavenumber selection problem is addressed. The possible minimization in k is a special advantage of our approach, since it allows an in vestigation of the non linear wavenumber of the patterns decreases with increasing super criticality of the magnetic field. Figure 1 display this dependence of the wavenumber k and the corresponding surface de flexion z on the magnetic field.

Furthermore in the general case of a tilted magnetic field we also include the mutual orientation of the corresponding wave vectors into the set of variations parameters to analyze the relative stability of various patterns. Thereby the anisotropy induced by the tangential magnetic field is taken into account in a proper way, Such an extension is not easily accessible to other approaches used in previous theoretical investigations [5, 6].

The nonlinear analyze reveals that in the case of a tilted magnetic field its tangential component can stabilize parallel ridges of ferrofluid. On increasing the normal field these ridges become unstable at first with respect to surface waves with a modulus of the wave vector k differing from the critical wavenumber of the Rosenszweig in stability. Due to this perturbations the ridges split up into peaks, which can align in various stretched patterns. For example an array of elongated hexagons forms a stable pattern due to the breaking of the rotational symmetry by the external field.

Figure 1: Surface deflection z (0,0) at the cusp of the hexagonal (H), square (S) or ridge (R) planform and emerging wavenumber k as function of the super criticality parameter e = H₀²/H₀² – 1 for a vertical magnetic field H₀ = e_zH₀.

Acknowledgments.

This work was supported by the Deutsche Forschungsgemeinschaft under the project En 278 / 2 1.

References.

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2. B. Abou, J. E. Wesfreid and S. Roux // J. Fluid Mech. 416 (2000) 217 – 237.

3. A. Gailitus // J. Fluid Mech // J. Fluid Mech. 82 (1977) 401 – 413.

4. R. Friedrichs and A. Engel, «Pattern and wavenumber selection in ferrofluids», submitted to Phys. Rev. E.

5. M. Silber and E. Knobloch // Physica D 30 (1988) 83 – 98.

6. R. Bajaj and S. K. Malik // J. Magn. Magn. Mater. 149 (1995) 158 – 161.