We study the stability of a ferrofluid cylinder placed in an ambient fluid and immersed in a static asymmetric magnetic field under isothermal conditions. After formulating the problem of ferrohydrodynamics, we conduct stability analysis by successively considering the viscosity of the ferrofluid, the properties of the ambient fluid, and the first nonlinear effects for the admissible magnetic field shapes. The three developed models are then compared with experimental data taken from the literature for the case of a stabilizing magnetic field. Given the excellent agreement with the data, we intend to conduct a new campaign of experiments in order to further explore this comparison and experimentally identify the destabilizing magnetic field shapes that are theoretically predicted and are of particular interest for applications in the printing domain and medical field.