We investigate the stability of interfacial waves in
conducting fluids under the influence
of a vertical current density, paying particular attention to aluminium
reduction cells
in which such instabilities are commonly observed. We develop a wave equation
for the interface in which the Lorentz force is expressed explicitly in
terms of the
fluid motion. Our wave equation differs from previous models, most notably
that
developed by Urata (1985), in that earlier formulations rested on a more
complex,
implicit coupling between the fluid motion and the Lorentz force. Our formulation
furnishes a number of quite general stability results without the need
to resort to
Fourier analysis. (The need for Fourier analysis typifies previous studies.)
Moreover,
our equation supports both travelling and standing waves. We investigate
each in turn.We obtain three new results. First, we show that travelling waves may
become
unstable in the presence of a uniform, vertical magnetic field. (Our travelling
waves are
quite different to those discovered by previous investigators (Sneyd 1985
and Moreau
& Ziegler 1986) which require more complex magnetic fields to become
unstable.)
Second, in line with previous studies we confirm that standing waves may
also become
unstable. In this context we derive a simple energy criterion which shows
which types
of motion may extract energy from the background magnetic field. This indicates
that a rotating, tilted interface is particularly prone to instability,
and indeed such a
motion is often seen in practice. Finally, we use Gershgorin's theorem
to produce a
sufficient condition for the stability of standing waves in a finite domain.
This allows
us to place a lower bound on the critical value of the background magnetic
field at
which an instability first appears, without solving the governing equations
of motion.
Stability and dynamics of magnetocapillary interactions