Linear stability analysis of a ferrofluid in a radially heated concentric cylindrical annulus with an applied magnetic field

A study of linear stability analysis (LSA) was conducted on a ferrofluid confined in an an infinitely long cylindrical annulus, with differential radial heating. A stack of magnets inside the inner cylinder providing a magnetic field. In addition, the cylinders can rotate rigidly with an angular frequency. Besides the Archimedean buoyancy and the centrifugal buoyancy, the ferrofluid is subjected to the magnetic Kelvin force. The later consists of a conservative part which does not create flow and non-conservative force which can be seen as a magnetic buoyancy with an effective radial magnetic gravity $g_m$. To test the stability of the base state in the annulus, we add small perturbations to the base state and linearize the resulting equations. The small perturbations are expanded the perturbations in form of normal modes and we search the marginal states with a zero temporal growth rate. Two cases are investigated. In the first one, microgravity conditions are considered in which either the cylindrical annulus is at rest or it is rigidly rotated for Ta=10, Ta=20 and Ta=40. We found that the threshold increases with the radius ratio $\eta$ and that the solid-body rotation delays the threshold of instability. In all cases, the threshold $Ra_{mc}$ is independent of the Prandtl number (Pr). In the second case, the Earth gravity acts on the ferrofluid confined in a stationary vertical cylindrical annulus. For weak value of Gr, i.e. $Gr Gr_c$ and $Ra_m < Ra_{mc}$, critical modes are either oscillatory hydrodynamic modes for Pr < 12.45 and oscillatory thermal modes for Pr > 12.45. An energetic analysis will be performed to get a better insight into the powers associated to each buoyancy force. The obtained results are analogous to those obtained in the same cylindrical annulus subject to a dielectrophoretic force and to a solid-body rotation.