| Magnetorheological (MR) fluids constitute an important class of controllable fluids, whose rheological properties can be dramatically altered with the application of a magnetic field. These fluids typically consist of micron-sized, magnetizable particles dispersed in a non-magnetic carrier fluid. The essential characteristic of MR fluids is that they may be varied from a free-flowing fluid in the absence of an applied magnetic field to that of a Bingham solid in a moderate field. Moreover, this transformation is nearly instantaneous and reversible. Understanding the overall magnetic properties of MR fluids is an integral part of the design process of MR fluid-based devices. It also provides valuable insight into the main characteristics of the microstructure responsible for their field-dependent rheology. Prediction of their overall magnetic properties, however, is a challenging task due to the size, density, and nonlinear magnetic properties of the particles. To deal with this challenge, we propose a model for magnetic response of MR fluids that is based on the mathematical theory of homogenization. We begin with the derivation of Maxwell's equations and describe the periodic microstructure for the model geometry. We derive effective constitutive laws that govern the magnetic response of (periodically arranged) particle-chains through magnetic saturation. Comparison of numerical results for these equations with experimental data show good agreement which suggests that our. approach could be useful in the design of improved MR fluids. With this in mind, we use this model to study the effects of different microstructure on the magnetic properties of these fluids. In particular, we consider fluids that contain spherical inclusions and ferrite particles which are different sizes. To obtain numerical results for the general case of nonlinear fluids, it is necessary to solve a system of nonlinear equations. We use the method of successive iterates to do this and we prove its convergence. Moreover, we discuss an alternative approach to that found in the literature for estimating the linear permeability and present its extension for more realistic geometries. Finally, we present a homogenization result that characterizes the asymptotic behavior of composite systems which are governed by the nonlinear vector potential equation in magnetostatics or electrostatics. |
