Stability of coextrusion through converging dies

Abstract

A linear stability analysis of multilayer flow of viscoelastic liquids through long, converging dies is performed by a rigorous two-dimensional analysis (2-D), as well as by a simplified one-dimensional analysis (1-D) on selected cross-sections along the die. The rheological behavior of liquids is represented by a modified Oldroyd-B model with a shear-rate dependent viscosity, described by the Carreau viscosity function. In the 2-D analysis, a streamlined Galerkin/finite-element method is used to discretize the domain. The resulting asymmetric generalized eigenvalue problem is large (of the order of 3000-8000), sparse, and banded, with a singular mass matrix. The leading eigenvalues of this complex problem are computed by using an iterative Arnoldi's algorithm, modified by Schur-Weilandt deflation, complex shift, and exponential preconditioning. With these series of modifications, the algorithm is now sufficiently flexible to solve any application that belongs to a generic class of large hydrodynamic stability problems. The effect of the die geometry on the neutral stability curves is investigated for various operating conditions and rheological parameters. In all the investigated cases, the critical flow-rate ratios in long converging channels are found to be independent of the shapes, and of the ratios of the thicknesses at the inlet to the outlet of the die. These results agree well with the approximate, simplified 1-D analysis, indicating that the most dangerous instability is at the inlet of the die. Thus, the analysis of the entire two-dimensional flow domain is unnecessary, at least for long channels, except for validating the 1-D analysis. The results also indicate that mesh-in-dependent eigensolutions cannot be obtained using the 2-D analysis when leading eigensolutions exist at large wavenumbers.