Analysis of a ferrofluid in a radial magnetic field

Dedicated to Professor Gheorghe Constantin on the occasion of his 70th anniversary Abstract. We study the dynamics of an unit volume of ferrofluid lo-cated in a vertical Hele-Shaw cell. The fluid is exposed to the radial magnetic field produced by a steady current through a wire parallel to the impermeable bottom of the cell. In an analytic frame work we prove well-posedness of the volume pre-serving flow. We also find a critical value of current’s intensity with the property that the flat interface is exponentially stable for intensities less than this value. When the current’s intensity passes certain values, the flat interface may change its shape and stationary fingering patterns occur