Investigated first is the linear stability of a Newtonian ferrofluid centred on a rigid wire, surrounded by another ferrofluid with a different magnetic susceptibility. An electric current runs through the wire, generating an azimuthal magnetic field which produces a magnetic stress at the interface of the fluids. Three dimensional disturbances to the system are considered and the linearised Navier-Stokes equations are solved analytically in terms of
an implicit expression for the growth rate of the disturbance. The growth rate is found numerically for arbitrary Reynolds number and given explicitly in the inviscid and Stokes regimes. Investigated next is a ferrofluid, whose magnetic susceptibility varies radially, centred on a rigid wire, subject to an azimuthal, non-uniform field. It is proven that if the gradient of the susceptibility is positive anywhere in the fluid, the system is linearly
unstable. Moreover, it is proven that applying an axial field can stabilise disturbances for both continuous and discontinuous susceptibilities.