International Journal of Non-Linear Mechanics 42 (2007) 381 – 390 www.elsevier.com/locate/nlm A three-dimensional non-linear constitutive law for magnetorheological fluids, with applications A. Dorfmann a , ∗ , R.W. Ogden b , A.S. Wineman c a Department of Civil and Environmental Engineering, Tufts University, Medford, MA 02155, USA b Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK c Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109, USA Received 29 September 2006; received in revised form 2 March 2007; accepted 2 March 2007 Abstract In this paper we first summarize the magnetic and mechanical balance equations for magnetorheological fluids undergoing steady motion in the presence of a magnetic field. A general three-dimensional non-linear constitutive law for such a fluid is given for the case in which the magnetic induction vector is used as the independent magnetic variable. The equations are needed for the analysis of boundary-value problems involving fluids with dispersed micron-sized ferrous particles subjected to a time-independent magnetic field. For illustration, the equations are applied, in the case of an incompressible fluid, to the solution of some basic problems. We consider unidirectional flow in a region confined by two infinite parallel plates with a magnetic field applied perpendicular to the plates. Next, we examine two problems involving a circular cylindrical geometry with the fluid occupying the region between two concentric cylinders: axial flow subjected to an axial magnetic field and circumferential flow with a circumferential field. After making some simplifying assumptions on the constitutive law and choosing material parameters, numerical solutions for the velocity profiles are illustrated.  2007 Elsevier Ltd. All rights reserved. Keywords: Magnetorheological fluids; Constitutive laws; Steady magnetorheological flow 1. Introduction Magnetorheological (MR) fluids respond to an applied mag- netic field with a rapid change in their rheological properties. The MR response results from the polarization of suspended micron-sized ferrous particles by the application of an ex- ternal magnetic field. The interaction between the result- ing induced dipoles causes the particles to form columnar (chain-like) structures, parallel to the applied magnetic field. These structures hinder the motion of the fluid, thereby increas- ing the viscosity characteristics of the suspension. The mechan- ical energy needed to produce yielding of the material increases with the applied magnetic field, resulting in a field-dependent yield stress. A detailed review of the interparticle interaction and of the changes of viscous properties in MR fluids due to the ∗ Corresponding author. Tel.: +1 617 627 6137; fax: +1 617 627 3994. E-mail address: Luis.Dorfmann@tufts.edu (A. Dorfmann). 0020-7462/$ - see front matter  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2007.03.002 application of a magnetic field is provided in the recent book by Odenbach [1]. Typically, the change in properties is manifested by the de- velopment of a yield stress that increases monotonically with the magnetic field. Materials with a yield stress are sometimes referred to as viscoplastic materials and their constitutive properties are frequently described by a generalized Bingham model (see, for example, [2–6]). In the review article by Bird et al. [2] the velocity profile of a material with a yield stress for different boundary conditions is determined and it is shown that a generalized Bingham viscoplastic model describes the mo- tion fairly well. The velocity profile develops a central region, known as a plug flow region, where the material moves at con- stant speed. Brodkey [5] and Bird et al. [2], for example, show that the velocity profile of a laminar flow through a circular conduit with magnitude of the shear stress greater than some critical value is flat in the central region where the material moves as an elastic solid. The velocity profile in the outer lay- ers corresponds to a non-Newtonian fluid and may be described