Numerical procedure for the computation of fluid flow with arbitrary stress-strain relationships

A finite-volume method is presented that allows for general stress-strain constitutive equations to be incorporated into a standard momentum± pressure-correction procedure. The method is sequential and segregated in nature, the various equations for mass and momentum conservation and for the evolution of the stress tensor are solved following a predefined order, and one of its features is the use of nonstaggered, and generally nonorthogonal, computational meshes. Two types of constitutive equations are used to test the method: the standard explicit and algebraic Newtonian model, and one of the simplest implicit differential equations, the upper-convected Maxwell model. In spite of its apparent simplicity, this latter model is known to pose the most severe numerical difficulties. However, the results in this article show the method to be effective in solving the equations for the flow of Newtonian and viscoelastic fluids through abrupt planar contractions with an area reduction of 4 to 1, one typical benchmark problem. The results are compared with available data and with solutions from a standard and validated code, and good agreement and consistency is found. A new formulation to evaluate stresses at cell faces is presented and shown to lead to improved results