Thermal Marangoni Instability and Magnetic Pressure for a Thin Ferrofluid Layer

We study the linear coupling between the Marangoni and Cowley-Rosensweig instabilities for a thin layer of ferrofluid subjected to a temperature gradient and a magnetic field. Both are perpendicular to the reference horizontal boundaries, one of which is a rigid plate, while the other is a free surface remaining flat as long as the magnetic field is smaller than the critical value of the onset of the static isothermal Cowley-Rosensweig instability. Our study considers at first a ferrofluid layer resting on the rigid border. In the stationary case, when heating is directed from the rigid side, a magnetic field, smaller than the Cowley-Rosensweig critical one, can induce a new pattern: the critical Marangoni number is much lower than in the nonmagnetic undeformable case, for a dimensionless wavenumber of O(root Bo) less than 1.992, its Newtonian classical value. When heating from the gaseous phase, an oscillatory marginal case exists theoretically, but for unphysical conditions. We consider also the case when the ferrofluid is hanging down from the rigid side. Only the wavelength critical value of the Rayleigh-Taylor instability that separates a stable region from an unstable one changes.