Steady, two-dimensional flows of viscoelastic liquid with memory are analyzed by means of a nonlinear integral constitutive equation, novel streamlined finite elements, Galerkin's method of weighted residuals, and Newton iteration. By their relation to streamlines the new elements allow, for the first time, full Newton iteration of the algebraic equation set to which the governing integrodifferential system reduces. Streamlines are computed simultaneously with velocity components and pressure, the primary unknowns; to track the history of liquid particles the deformation equations are solved analytically in a Protean system of coordinates that conforms to the streamlines. In the cases studied the Newton iteration converges quadratically to a creeping flow state of Weissenberg number (product of upstream wall shear rate and relaxation time of the liquid) up to about 20 in channel flow, 10 in film flow, and 2 in die-swell flow. Shear-thinning shrinks the domain of convergence; it also reduces die-swell. A slip boundary condition in the vicinity of the contact line extends the domain of convergence.
Bibliographical noteFunding Information:
Financial support for this researchw as provided by grants-in-aid from the 3M Company and the Xerox Corporation, by the University of Minnesota Computer Center, and by the Supercomputer Institute at the University of Minnesota.
Part of the research was carried out at The University of Michigan and was supported by the Process and Reaction Engineering Program of the National Science Foundation-Grant CBT-8504607.