Petgamoo Int. J. EngngSci. Vol. 32, No. 3. pp. 481-500, 1994 Copyright @ 1994 ElsevicrScience Ltd Printed zyxwvutsrqponmlkjihgfedcbaZ in G re a t Britain. All rightsreserved 0020-7225/% $6.00 + 0.00 zyxwvut MODELING ELECTRO-RHEOLOGICAL MATERIALS THROUGH MIXTURE THEORY K. R. RAJAGOPAL and R. C. YALAMANCHILI Department of Mechanical Engineering, University of Pittsburgh, Pittsburgh, PA 15261, U.S.A. A. S. WINEMAN Department of Mechanical Engineering and Applied Mechanics, University of Michigan, Ann Arbor, MI48109, U.S.A. (Communicated by C. G. SPEZIALE) Abstract-Electra-rheological materials are suspensions of particles in non-conducting fluids, and all models that have been developed to date describe their behavior by treating them as a homogenized single continuum, and ignoring the multicomponent structure of the material. The theory of interacting continua is ideally suited for modeling such mixtures and in this paper we present a simple theory which takes into account the distribution of the particles in the fluid, the applied electric field, and the relative motion of the two constituents. To illustrate the utility of such a theory we study the flow of an electro-rheological material between two parallel plates under the application of an electrical field normal to the plates. 1. INTRODUCTION Electra-rheological materials are suspensions of non-conducting particulate media in non- conducting fluids. Properties like viscosity of the suspension change significantly on the application of an electric field. This phenomenon was observed over three decades ago by Winslow [l]. Such behavior can be gainfully employed in a wide range of technological applications from the design of clutches, brakes, shock absorbers and journal bearings to a plethora of applications in hydraulics. Much of the effort that has been expended in recent years in the field of electro-rheology is in designing and tailor-making these materials. There has also been a reasonable amount of experimental work on electro-rheological materials. However, little if any effort has been directed towards providing a comprehensive theory to describe the behavior of these materials. Recently, Rajagopal and Wineman [2] developed a mathematical model for field dependent materials based on the basic principles of continuum mechanics, which predicts behavior in keeping with experimentally observed phenomena. The theory of Rajagopal and Wineman [2] assumes that the electro-rheological suspension can be regarded as a single continuum. A good case can be made for such an approach on the basis of homogenization which yields average properties for the suspensions. However, it would be remiss not to try to model such a suspension via the theory of interacting continua (cf. Truesdell [3,4]). I n such an approach, balance laws are postulated for each constituent which allows for interaction between the constituents including the possibility of generation of the individual species, chemical reactions, electro-mechanical, electro-chemical and other effects. The theory also allows us to account for the fact that we have particles moving through the fluid by including interactions such as drag, virtual mass effect, magnus effect, spin-lift, density gradient effects, buoyancy effects amongst others. Mixtures of fluids and solid particulate media within the context of a purely mechanical theory of interacting continua have been studied by Massoudi [5], and Johnson et zyxwvutsrqponml al. [6, 71. These studies were primarily aimed at problems of fluidization and the transport of mixtures of fluids and granular solids. The mixture was assumed to be made up of a classical linearly viscous fluid and a granular solid. Thus, the partial stresses associated with the fluid and the granular material depend on the stretching tensors associated with both the fluid and solid motion in addition to the way in which the solid is distributed, which is described by a volume distribution function u and the gradient of the volume distribution function. ES 32:3-6 481