Stability analysis of constitutive equations for polymer melts in viscometric flows

The stability of two representative constitutive equations for polymer melts, the exponential Phan-Thien-Tanner(PTT) model and the Giesekus model, are investigated in planar shear flows. For the PTT equation, instabilities arepredicted for both plane Couette and Poiseuille flows using transient finite-element calculations. A Chebyshev-Tauspectral method is used to confirm that these instabilities are not spurious or an artifact of the finite elementformulation. Mechanisms are proposed based on an energy analysis of the most unstable mode for each flow. Thestability of plane Couette flow of a Giesekus model is also probed using our spectral method and found to be stablefor the range of parameters investigated. However, in pressure driven flow, the Giesekus model is unstable overa critical local Weissenberg num! ber (Wi) based on the shear rate at the channel wall. We present the completeeigenspectrum for this model in both Couette and Poiseuille flows and note that ?ballooning? of the continuousspectrum, which can cause spurious instabilities, is significantly stabilized for this constitutive equation relative tothe upper convected Maxwell model or PTT model. Both Poiseuille flowinstabilities occur at moderate Weissenbergnumbers and, with more careful investigation, may be able to explain some of the unusual phenomena observed inslit flows of polymer melts