Abstract
It is evident that a major effort has been accomplished over the last ten years toward the development of numerical methods for solving viscoelatic flow. The problem was clearly much harder than expected. Several extensive reviews have been devoted to a detailed account of the difficulties encountered in reaching moderate values of the Weissenberg number. The numerical and analytical work undertaken in parallel by several research groups has led to some important conclusions which paved the way for recent promising development. First, numerical algorithms for solving highly nonlinear advective problems must be used with the greatest carefulness. When spurious solutions or unexpected effects such as limit points arise in numerical simulations, we have learned to question the validity of the numerical method as well as that of the constitutive equation. Typically, successive failures of numerical calculations with the Maxwell model at low values of We have often been attributed to its singular behavior in uniaxial elongational flow but, in the meantime, better adapted algorithms have led to solutions at ever increasing values of We. Secondly, the mathematical analysis of the partial differential equations governing the flow of viscoelastic fluids has revealed the possibility of changes of type of the vorticity equation under some circumstances, i.e., when the velocity of the fluid becomes comparable with the velocity of shear waves. The coexistence of hyperbolic and elliptic regions in a steady flow may be of great importance in explaining a number of experimental observations. Simultaneously, the analysis has led to the identification of artificial changes of type which partly explain some numerical failures, or at least give a pertinent diagnosis of numerical inaccuracy. Thirdly, it has been found that numerical algorithms must take into account the specific features of viscoelastic flow; among these, stress boundary layers, stress singularities, and advective (or memory) terms in the constitutive equations are prominent.
Localization of eigenvalues in elliptic regions
