Non-Newtonian and Viscoelastic Fluids

The theory of non-Newtonian and viscoelastic fluids flourished in the second half of last century with the developments of (molten and dilute) polymers and the growth of materials science and engineering that generated many new products and applications. The mathematical setting of the constitutive equations required new tools from tensor analysis and algebra that have already been exposed in previous chapters. Here the concentration will be on the various constitutive relations that became dominant over time because of their generality and/or their physical relevance. Despite the major effort carried out by numericists over the last four decades, numerical simulations at high Weissenberg number values (and zero Reynolds number) are not always feasible and this poses deep questions and concerns about getting the appropriate models for those fluids. However the log-formulation introduced by Fattal and Kupferman (J Non-Newton Fluid Mech 123:281–285, 2004) has eased that difficulty. Nonetheless in problems where experimental data are available, numerical results based on non-Newtonian models may sometimes be in error by an order of magnitude. This situation is detrimental in the long run to engineering and should be tackled by all means: theory, experiments and computations.