Computation of Magnetic Liquid-Free Surface Shape in a Quasi-Homogeneous Magnetic Field With Differential Evolution

This paper presents a method to compute the magnetic liquid-free surface shape of a single spike in a quasi-homogeneous magnetic field produced by the Helmholtz coil. In the Rosensweig instability, the magnetic liquid-free surface deforms in a spike-like shape pattern as a magnetic field above some critical value is applied to the liquid. The free surface of the magnetic liquid is described as a polynomial function in cylindrical coordinates with the applied cylindrical symmetry. To obtain the shape deformation, the system of nonlinear magnetically augmented Young-Laplace equations is solved iteratively. The approach to the solution is executed in two steps. In the first step, magnetic field distribution along the surface is computed by the finite-element method. When magnetic field distribution is known, the second step occurs in which the system of Young-Laplace equations, defined as an optimization problem, is solved by differential evolution.