  We analyse small-amplitude oscillations of a weakly viscous electrically
conducting liquid drop in a strong uniform DC magnetic field. An asymptotic
solution is obtained showing that the magnetic field does not affect the shape
eigenmodes, which remain the spherical harmonics as in the non-magnetic case.
Strong magnetic field, however, constrains the liquid flow associated with the
oscillations and, thus, reduces the oscillation frequencies by increasing
effective inertia of the liquid. In such a field, liquid oscillates in a
two-dimensional (2D) way as solid columns aligned with the field. Two types of
oscillations are possible: longitudinal and transversal to the field. Such
oscillations are weakly damped by a strong magnetic field - the stronger the
field, the weaker the damping, except for the axisymmetric transversal and
inherently 2D modes. The former are overdamped because of being incompatible
with the incompressibility constraint, whereas the latter are not affected at
all because of being naturally invariant along the field. Since the magnetic
damping for all other modes decreases inversely with the square of the field
strength, viscous damping may become important in a sufficiently strong
magnetic field. The viscous damping is found analytically by a simple energy
dissipation approach which is shown for the longitudinal modes to be equivalent
to a much more complicated eigenvalue perturbation technique. This study
provides a theoretical basis for the development of new measurement methods of
surface tension, viscosity and the electrical conductivity of liquid metals
using the oscillating drop technique in a strong superimposed DC magnetic
field.
