On the Hill Equation Describing Oscillations of a Ferrofluid Free Surface in a Vertical Magnetic Field

摘要

We consider an isothermal ferrofluid layer which is nearly inviscid and semi-infinite. It is subjected to a magnetic field that consists of two parts. The first one is a vertical constant component. The second part is oscillating with time in a vertical plane. The linearized Laplace law describing the deformation of the free surface reduces to the study of a Hill equation, thus generalizing the Mathieu equation studied for a purely oscillating magnetic field. Using a classical method, we derive the marginal oscillating conditions and compare them with experimental situations.