Stability analysis of polymer shear flows using the eXtended Pom–Pom constitutive equations

Abstract

The stability of polymer melt shear flows is explored using the recently proposed eXtended Pom–Pom (XPP) model of Verbeeten et al. [J. Rheol. 45 (2001) 823]. We show that both the planar Couette and the planar Poiseuille flow are stable for ‘small’ disturbances. From our one-dimensional eigenvalue analysis and two-dimensional finite element calculations, excellent agreement is obtained with respect to the rate of slowest decay. For a single mode of the XPP model, the maximum growth of a perturbation is fully controlled by the rightmost part of the continuous eigenspectrum. We show that the local fourth order ordinary differential equation has three regular singular points that appear in the eigenspectrum of the Couette flow. Two of these spectra are branch cuts and discrete modes move in and out of these continuous spectra as the streamwise wavelength is varied. An important issue that is addressed is the error that is contained in the rightmost set of continuous modes. We observe that the XPP model behaves much better as compared to, for example, the Upper Convected Maxwell (UCM) model in terms of approximating the eigenvalues associated with the singular eigenfunctions. We demonstrate that this feature is extended to the two-dimensional computations of a periodic channel which means that the requirements on the spatial grid are much less restrictive, resulting in the possibility to use multi-mode simulations to perform realistic stability analysis of polymer melts. Also, it is shown that inclusion of a nonzero second normal stress difference has a strong stabilizing effect on the linear stability of both planar shear flows.