ANALYSIS OF NON-NEWTONIAN FLOWS THROUGH CONTRACTIONS AND EXPANSIONS Luiz A. Reis Junior Department of Mechanical Engineering Pontif´ıcia Universidade Cat´ olica–Rio de Janeiro, RJ 22453-900, Brazil lareis@hotmail.com Mˆ onica F. Naccache Department of Mechanical Engineering Pontif´ıcia Universidade Cat´ olica–Rio de Janeiro, RJ 22453-900, Brazil naccache@mec.puc-rio.br Abstract. Flow of non-Newtonian fluids through contractions and expansions are found in several industrial processes. In this work, a numerical simulation of non-Newtonian fluid flows through an axysimetric expansion followed by a con- traction is performed. The numerical solution of conservation equations of mass and momentum is obtained via finite volume method. In order to model the non-Newtonian behavior of the fluid, it is used the Generalized Newtonian Fluid constitutive equation, with the Carreau viscosity function. The results obtained show the influence of rheological para- meters on flow patterns. keywords: Non-Newtonian fluids, contraction, expansion 1. Introduction In this work, the flow of non-Newtonian fluids through an abrupt axisymmetric expansion followed by an abrupt contraction is analyzed numerically. The mechanical behavior of the non-Newtonian fluid is modeled by the Generalized Newtonian Liquid constitutive equation (GNL) (Bird et al., 1987): τ = η(˙ γ )˙ γ˙ γ˙ γ˙ γ˙ γ (1) where τ is the extra-stress tensor, ˙ γ˙ γ˙ γ˙ γ˙ γ is the rate-of-deformation tensor, defined as grad v + (grad v) T , v is the velocity vector and η is the viscosity function, given by the Carreau-Yasuda model: η − η ∞ η 0 − η ∞ = [1 + (λ ˙ γ ) a ] (n−1)/a (2) In this equation, η 0 is the viscosity at low shear rates, η ∞ is the viscosity at high shear rates, λ is a time constant, n is the power-law exponent, and a is a dimensionless parameter that describes the transition region between the zero-shear- rate region and the power-law region. Depending on the values of these parameters, this equation can be used to model viscoplastic materials. Viscoplastic materials are used in many industrial processes, and their main characteristic is the presence of a yield stress. Above the yield stress the material behaves as a liquid, and, below it, as a solid. Barnes (1999a, 1999b) performed a comprehensive review about yield stress materials, reviving the argument that yield stress actually does not exist. He shows, for a large number of materials typically classified as viscoplastics, that when careful measurements are performed below the “yield stress,” it is found that flow actually takes place. Then, the viscosity function looks like a bi-viscosity model, with very high viscosity at small shear rates and lower viscosities for larger shear rates. However, an apparent yield stress can exist as a useful mathematical description of limited data, over a given range of flow conditions. Alexandrou et al. (2001) studied numerically the flow of Herschel-Bulkley fluids in a canonical three-dimensional expansion. The results were obtained for a 2:1 and a 4:1 expansion rate. The effects of Reynolds number and Bingham number on flow pattern and pressure distribution were investigated. It was observed that a strong interplay between the Reynolds and Bingham numbers occurs, and they influence the formation and break up of stagnant zones in the corner of the expansion. Souza Mendes et al. (2000) performed an experimental and numerical analysis of the flow of viscoplastic fluids through a converging-diverging channel. They observed experimentally a flow pattern transition, for a critical value of the ratio between the length and the diameter of the central region, indicating a possible material fracture. However, the viscosity model used in the numerical simulation was not capable to predict this behavior. The flow of Bingham materials through a 1 × 2 abrupt expansion was analyzed numerically by Vradis and ¨ Ot¨ ugen (1997). They observed that the reattachment length increases with Reynolds number and decreases with yield stress. Naccache and Souza Mendes (1997) studied numerically the flow of Bingham materials through abrupt expansions as a function of Reynolds number, yield stress and expansion ratio. It was noted that the reattachment length increases with the Reynolds number, decreases with yield stress and is essentially independent of the expansion ratio. An experimental study of the flow through axisymmetric