ROTATING FERROFLUID DROPS
A.
Engel 1
, A. V. Lebedev 2 , K. I. Morozov 2 .
1.
Otto-von-Guericke Universitat
Magdeburg, PSF 4120, 39016 Magdeburg, Germany.
2.
Institute of Continuos Media
Mechanics, 1 Korolev Street, 614013, Perm, Russia.
The shapes of rotating fluid drops have al ways
fascinade scientists and artists alike. Ferrofluid drops floating in a non
magnetic liquid of the same density can be brought into rotation by a rotating
magnetic field set up by two perpendicular pairs of Helmholtz coils. Depending
on the magnetic permeability of the ferrofluid these drops show a variety of
shape transformations when frequency and strength of the magnetic field are
varied.
In [1] the shape of rotating microdrops with a typical radius of 10 mm has been
studied both experimentally and theoretically. In the present paper we
investigate the shape transformations of drops with radius of several
millimeters, i.e. a hundred times larger.
It turns out that new shape transformations may show up which were missed in
[1], probably due to the smallness of the drops.
We will only consider the case of a fast rotating
magnetic field, i.e. with a frequency large as compared with the in verse of
the typical time for shape relaxation of the droplet. The shape is then to a
very good approximation determined by the balance between the surface energy
and the magnetic energy averaged over
one period of the field. Assuming an elliptical shape of the deformed droplet
both energies can be determined analytically for a linear relation between
magnetization and magnetic field [2]. Minimizing the resulting expression in
eccentricities of the ellipsoid the various shape transformations can be
determined.
For small susceptibilities, m < 5, we find the initially spherical droplet deforms with increasing
field strength H into an oblate
spheroid of increasing eccentricity. For larger values of m there is at intermediate values of H
a transition to a three-axis ellipsoid which transforms back to a flat oblate
spheroid at large H values. The
transition to and from the three axis
ellipsoid occurs via backward bifurcation for m > 10 (see figures).
|
Figure 1. Ratio a / b
between the semiaxes of the rotating drop in the plane perpendicular to the
external field as function of the magnetic Bond number B0 = H2R / s for a fluid of
permeability m = 20,4. The upper curve results from the theory using a
linear magnetization curve, the lower one uses the Langevine law. Symbols
represent experimental results. |
Figure 2. Same as Figure 1 for the ratio a / с between the largest and the smallest semiaxes of the
droplet. |
At the moment where a transition to a three axis
ellipsoid takes place the rotation frequency of the drop becomes easily
accessible to an experimental investigation. The rotation frequency can be
determined theoretically from the balance between viscous and magnetic
torque’s. The former can be calculated by using Jef fray’s solution [3] for an
ellipsoidal solid moving in a viscous fluid whereas the de termination of the
latter requires the calculation of the phase
lag of the magnetization with respect to the external magnetic field.
Our theoretical results are compared with experimental
findings. Quite generally it turns out that although a theory using a linear
relation M (H) is able to explain the
experimental results qualitatively to obtain also quantitative agreement
requires to take into account saturation effects in the magnetization curve.
This is not surprising since the experimentally relevant fields are of the
order to the saturation magnetization. We present theoretical investigation using Langevin or more sophisticated
[4,5] magnetization curves and find considerably better agreement between
theory and experiment, both for the shape and the rotation frequency of the
drop.
References.
1.
J. C. Bacri, A. Cebers, R.
Perzynski // Phys. Rev. Lett., 72 (1994) 2705.
2.
K. I. Morozov // Journal Theoretical and experimental
physics, 85 (1997) 728.
3.
G. B. Jeffrey // Proc. R. Soc. London, Ser. A, 102 (1922) 161.
4.
A. F. Pshenichnikov, V. V.
Mekhonoshin, A. V. Lebedev // J. Magn.
Magn. Mater., 161 (1996) 94.
5.
B. Huke, M. Lucke // Phys. Rev., E62 (2000) 6875.