Linear stability analysis of an alternating magnetic field driven flow in a spinning container

We present a numerical analysis of the free surface liquid metal flow driven by an alternating (AC) magnetic field in a spinning cylindrical container. The axysimmetric flow structure is analyzed for various values of the magnetohydrodynamic interaction parameter N and the Ekman number E. The governing hydrodynamic equations are solved by a spectral collocation method. The alternating magnetic field distribution is found by a boundary-integral method. The electromagnetic and hydrodynamic fields are fully coupled via the shape of the liquid free surface. The upper free boundary was found simultaneously with the flow by a Newton method. It is found that in all considered parameter ranges the flow contains four main toroidal eddies. This is caused by the non-uniformity of the magnetic field near the edges of the liquid volume. The interaction parameter N controls the intensity of the flow. The additional container spinning leads to a deformation of the flow structure. At Ekman number E < 1 ´ 10-2 the meridional flow is reduced. The secondary azimuthal flow has its maximum in the Ekman number range of E ~ 10-3 - 10-2, at smaller Ekman number the azimuthal flow is suppressed too.
The three-dimensional stability analysis of the flow showed that the spinning leads mainly to a destabilization of the base flow. Only at very small Ekman numbers E the flow in the spinning container is more stable than in the non-spinning case. The instability at large Ekman numbers is of oscillatory type and the most unstable azimuthal wave number is m = 3. At smaller Ekman numbers the azimuthal wave number increased to m = 5, m = 6, etc. At E < 2.1 ´ 10-4 the most unstable wave number is m = 16. Except the narrow Ekman number range of 1.935 ´ 10-2 < E < 2.376 ´ 10-2 where the instability is of oscillatory type, at all other values of Ekman number E < 4.6136 ´ 10-2 the instability is of steady type.