On the numerical stability of mixed finite-element methods for viscoelastic flows governed by differential constitutive equations

Mixed finite-element methods for computation of viscoelastic flows governed by differential constitutive equations vary by the polynomial approximations used for the velocity, pressure, and stress fields, and by the weighted residual methods used to discretize the momentum, continuity, and constitutive equations. This paper focuses on computation of the linear stability of the planar Couette flow as a test of the numerical stability for solution of the upper-convected Maxwell model. Previous theoretical results prove this inertialess flow to be always stable, but that accurate calculation is difficult at high De because eigenvalues with fine spatial structure and high temporal frequency approach neutral stability. Computations with the much used biquadratic finite-element approximations for velocity and deviatoric stress and bilinear interpolation for pressure demonstrate numerical instability beyond a critical value of De for either the explicitly elliptic momentum equation (EEME) or elastic-viscous split-stress (EVSS) formulations, applying Galerkin's method for solution of the momentum and continuity equations, and using streamline upwind Petrov-Galerkin (SUPG) method for solution of the hyperbolic constitutive equation. The disturbance that causes the instability is concentrated near the stationary streamline of the base flow. The removal of this instability in a slightly modified form of the EEME formulation suggests that the instability results from coupling the approximations to the variables. A new mixed finite-element method, EVSS-G, is presented that includes smooth interpolation of the velocity gradients in the constitutive equation that is compatible with bilinear interpolation of the stress field. This formulation is tested with SUPG, streamline upwinding (SU), and Galerkin least squares (GLS) discretization of the constitutive equation. The EVSS-G/SUPG and EVSS-G/SU do not have the numerical instability described above; linear stability calculations for planar Couette flow are stable to values of De in excess of 50 and converge with mesh and time step. Calculations for the steady-state flow and its linear stability for a sphere falling in a tube demonstrate the appearance of linear instability to a time-periodic instability simultaneously with the apparent loss of existence of the steady-state solution. The instability appears as finely structured secondary cells that move from the front to the back of the sphere.Financial support for this research was given by the National Science Foundation, the Office of Naval Research, and the Defense Research Projects Agency. Computational resources were supplied by a grant from the Pittsburgh National Supercomputer Center and by the MIT Supercomputer Facility.