Numerical approximation of the spectra of Phan-Thien Tanner liquids

The linear stability of the linear Phan-Thien Tanner (PTT) fluid model is investigated for plane Poiseuille flow. The PTT model involves parameters that can be used to fit shear and extensional data, which makes it suitable for describing both polymer solutions and melts. The base flow is determined using a Chebyshev-tau method. The linear stability equations are also discretized using Chebyshev approximations to furnish a generalized eigenvalue problem. The spectrum is shown to comprise a continuous part and a discrete part. The theoretical and numerical results are validated for the UCM and Oldroyd-B models, which are special cases of the PTT model, by comparing with results in the literature. It is demonstrated that the linear extensional and elasticity parameters considered. The computational efficiency and accuracy of the numerical method are also investigated.