Equilibrium shape of ferrofluid in the uniform external field

It is well-known that morphology of magnetic or electric ferrofluids is extremely sensitive to the applied external electromagnetic fields. Even in the static regimes, they demonstrate a variety of qualitative and quantitative transformations often mentioned as free surface instabilities. That makes their computational modeling rather challenging. Therefore, for the sake of validation and verification, there is a need for having exact solutions. Modelers noticed that quite often isolated ferrofluids take on the shape of a prolate spheroid. So, there were several claims that a consistent theory allows to prove that isolated drop of ferrofluid exposed to a uniform field, indeed, has a shape of a spheroid. In this paper we analyze this problem one more time using a special technique of the analysis of the equilibrium shape, which traces back to the papers of Eshelby.