Viscoelastic Fluids

Abstract

The theory of viscoelastic fluids developed in the second half of the 20th century with the industrial production of molten and dilute polymers and the growth of engineering that generated many new products. Building constitutive equations for these fluids requires new tensorial tools from continuum mechanics, such as objective tensor derivatives, that are introduced in this chapter. The Poiseuille and Couette flow problems are then solved explicitly: this computation points out some mathematical specificities of viscoelastic fluid models, as the possible non-uniqueness of the stationary solution, together with some mechanical ones, such as normal stresses and the spectacular Weissenberg effect. For solving more general flows, two numerical approaches are developed: a $\theta$-scheme algorithm, based on an operator splitting and a Newton method based on the log-conformation formulation. The discontinuous Galerkin finite element method is studied in details and then applied to the approximation of the constitutive equation. For the flow in an abrupt contraction and the flow in a driven cavity, examples of numerical computations are presented, illustrating the efficiency of these algorithm. Based on the log-conformation formulation, a global energy estimate is proved for this family of fluids.