Mechanical Equilibrium of a Nonmagnetic Body Immersed in a Cylindrical Container with a Magnetic Fluid Magnetized by an External Homogeneous Magnetic Field

Abstract

Purpose. Analytical and numerical description of the magnetohydrodynamic forces acting on a small nonmagnetic spherical body in a cylindrical container with magnetic fluid (magnetofluid dispenser and separator approximation) that determine the hydrostatic mechanical equilibrium in the system.Methods. The numerical study solves the magnetostatic problem by the finite element method in the FEMM program package using the Lua script language. The system of Maxwell’s equations is solved by the standard method in the vector potential formulation. The analytical solution of the magnetostatic problem is obtained by the mirror image method using a simplifying model representation of the linear law of magnetization of a magnetic fluid. The ponderomotive force acting on a body immersed in a magnetic fluid is calculated using the Rosensweig formula and the energy approach.Results. A refined expression for the magnetic ponderomotive force acting on a nonmagnetic sphere immersed in a cylindrical container with magnetized magnetic fluid is obtained. Direct numerical simulation of the laboratory experiment is performed, which allows us to compare the accuracy of the numerical and analytical solutions with the experimental data. Despite violating the limits of applicability of the analytical theory, the new expression correctly describes the nonmonotone coordinate dependence of the force, and the error in determining the coordinate extremums does not exceed 6 % and 26 % in absolute value. The physical justification for the condition of mechanical equilibrium in the model system under study is given.Conclusion. The competition of two oppositely directed magnetic forces leads to the fact that a nonmagnetic sphere in a cylindrical container with magnetized magnetic fluid has one unstable mechanical equilibrium position in the center of the container, so that the body is pressed against the wall, or (additionally) two stable equilibrium positions that allow the body to levitate near the container wall without touching it.