Flow of electro-rheological materials

Abstract

Summary. An electro-rheological fluid is a material in which a particulate solid is suspended in an electrically non-conducting fluid such as oil. On the application of an electric field, the viscosity and other material properties undergo dramatic and significant changes. In this paper, the particulate imbedded fluid is considered as a homogeneous continuum. It is assumed that the Cauchy stress depends on the velocity gradient and the electric field vector. A representation for the constitutive equation is developed using standard methods of continuum mechanics. The stress components are calculated for a shear flow in which the electric field vector is normal to the velocity vector. The model predicts (i) a viscosity which depends on the shear rate and electric field and (ii) normal stresses due to the interaction between the shear flow and the electric field. These expressions are used to study several fundamental shear flows: the flow between parallel plates, Couette flow, and flow in an eccentric rotating disc device. Detailed solutions are presented when the shear response is that of a Bingham fluid whose yield stress and viscosity depends on the electric field. During the past few years, there has been a great deal of interest in the manufacture and use of a class of materials which can be classified as field dependent theological materials. These materials are essentially fluids which are imbedded with particulate solids which react to an electrical field in that on the application of a field the viscosity and other material properties undergo dramatic and significant changes. Such materials are being touted as agents for enhancing the performance and efficiency of a variety of engineering devices in very diverse fields. Much of the activity in this area is devoted to producing this material and performing experiments in order to understand the scientific basis for their behavior. Little, if any effort has been devoted to mathematically modeling these materials. The need for understanding the mechanics of such materials and mathematically modeling their behavior is made all the more important as these materials are already finding day-to-day applications in the design of ubiquitous devices like clutches and brakes in cars, vibration dampers and absorbers, lubricating fluids in bearings to name some. In this paper we shall present a mathematical model for field dependent materials which is consistent with the phenomena which have been observed. We shall solve a series of boundary value problems the results of which can be compared with future experiments, as these boundary value problems are in domains which are amenable to experimentation. Unlike the field of magnetohydrodynamies, we do not have an equation like Maxwell's equation which governs the applied field, as the fluid which forms the base for the particulate media is non-conducting. The presence of the field alters the basic material properties of the particulate imbedded fluid, which is considered as a homogeneous continuum. Thus, for instance, the Cauchy stress is dependent on the gradient of the