On the use of characteristic variables in viscoelastic flow problems

The system of partial differential equations governing the how of an upper converted Maxwell fluid is known to be of mixed elliptic-hyperbolic type. The hyperbolic nature of the constitutive equation requires that, where appropriate, inflow conditions are prescribed in order to obtain a well-posed problem. Although there are three convective derivatives in the constitutive equation there are only two characteristic quantities which are transported along the streamlines. These characteristic quantities are identified. A spectral element method is described in which continuity of the characteristic variables is used to couple the extra stress components between contiguous elements. The continuity of the characteristic variables is treated as a constraint on the constitutive equation. These conditions do not necessarily impose continuity on the extra-stress components. The velocity and pressure follow from the doubly constrained weak formulation which enforces a divergence-free velocity field and irrotational polymeric stress forces. This means that both the pressure and the extra-stress tensor are discontinuous. Numerical results are presented to demonstrate this procedure. The theory is applied to the upper convected Maxwell model with vanishing Reynolds number. No regularization techniques such as streamline upwind Petrov Galerkin (SUPG), elastic viscous split stress (EVSS) or explicitly elliptic momentum equation (EEME) are used.