ABSTRACT
In Chapters 4 and 5 we studied the Þnite element models of Newtonian ßuids (i.e., ßuids whose constitutive behavior is linear). Fluids that are not described by the Newtonian constitutive relations are commonly encountered in a wide variety of industrial processes. For example, such materials include motor oils, high molecular weight liquids such as polymers, slurries, pastes, and other complex mixtures. The processing and transport of such ßuids are central problems in the chemical, food, plastics, petroleum, and polymer industries. Non-Newtonian behavior manifests itself in a number of different ways. Most
such ßuids exhibit a shear rate dependent viscosity, with “shear thinning” (i.e., decreasing viscosity with increasing shear rate) being the most prevalent behavior. Other phenomena associated with the elasticity and memory of the ßuid, such as recoil, are also observed in many situations. Differences in the normal stress components occur in many ßows and lead to such well-known effects as rod climbing or the Weissenberg effect, and the curvature of the free surface in an open channel ßow. A comprehensive list and discussion of these and other non-Newtonian effects is given in the book by Bird et al. [1]. For the present discussion non-Newtonian ßuids can conveniently be separated
into two distinct categories: (1) inelastic ßuids or ßuids without memory, and (2) viscoelastic ßuids, in which memory effects are signiÞcant. The distinction is an important one from both a physical and computational point of view. Basically, inelastic ßuids can be viewed as generalizations (in some sense) of the Newtonian ßuid. The viscosity function for such materials depends on the rate of deformation of the ßuid and thus allows “shear thinning” effects to be modeled. For numerical computations, inelastic ßuids can be treated using minor extensions to the standard Þnite element models developed for Newtonian ßuids. Viscoelastic ßuids, on the other hand, represent a signiÞcant departure from the Newtonian limit in terms of both physical behavior and computational complexity. The primary difficulty here is the “memory” of the ßuid; the motion of a material element depends not only on the present stress state, but also on the deformation history of the material element. This history dependence leads to very complex constitutive equations and the need for special computational procedures. It is the purpose of the present chapter to study some aspects of the Þnite element simulation of both inelastic and viscoelastic ßuids.
