Stability analysis of constitutive equations for polymer melts in viscometric flows

Abstract

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 are predicted for both plane Couette and Poiseuille flows using transient finite-element calculations. A Chebyshev–Tau spectral method is used to confirm that these instabilities are not spurious or an artifact of the finite element formulation. Mechanisms are proposed based on an energy analysis of the most unstable mode for each flow. The stability of plane Couette flow of a Giesekus model is also probed using our spectral method and found to be stable for the range of parameters investigated. However, in pressure driven flow, the Giesekus model is unstable over a critical local Weissenberg number ( Wi) based on the shear rate at the channel wall. We present the complete eigenspectrum for this model in both Couette and Poiseuille flows and note that ‘ballooning’ of the continuous spectrum, which can cause spurious instabilities, is significantly stabilized for this constitutive equation relative to the upper convected Maxwell model or PTT model. Both Poiseuille flow instabilities occur at moderate Weissenberg numbers and, with more careful investigation, may be able to explain some of the unusual phenomena observed in slit flows of polymer melts [AIChE J. 22 (2) (1976) 209; J. Non-Newt. Fluid Mech. 77 (1998) 123; Adv. Polym. Sci. 138 (1999) 227].