Modeling the Magnetic Relaxation Behavior of Micropolar Ferrofluids by Means of Homogenization

Abstract

In this paper a simple particle population homogenization approach is used in order to estimate the magnetic relaxation time of a ferrofluid by means of a microscopic analysis. At a macroscopic level the ferrofluid is modeled as a micropolar fluid with rotational degrees of freedom. The governing equations for these degrees of freedom are the spin balance and the magnetic relaxation equation. They are solved analytically for a simple unidirectional magnetic setup. On a microscopic level the ferrofluid is considered to consist of rigid spherical permanent magnets suspended in a non-magnetic carrier fluid. Due to both, the friction of the micro magnets with the carrier fluid and their own inertia, the alignment of the magnets with an applied external field is retarded. By neglecting thermal effects and therefore the Brownian motion, it is possible to reduce the equations of motion to a nonlinear pendulum equation, which is readily solved using numerical methods for ordinary differential equations. By averaging over all possible initial configurations of the micro magnets, a pseudo homogenization is obtained, which can then be compared to the macroscopic solution. From this comparison the relaxation time at a continuum level can be estimated.