Abstract:
The linear stability of the Giesekus and linear Phan-Thien Tanner (PTT) fluid models
is investigated for a number of planar Poiseuille flows in single, double and triple layered configurations. The Giesekus and PTT models involve parameters that can be used to fit shear and extensional data, thus making them 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 which is then solved using the QZ-algorithm.
The eigenspectra are shown to comprise of continuous parts and discrete parts. The
theoretical and numerical results are validated for the Oldroyd-B model, which is a simplified case of the Giesekus and PTT models, by comparing with results in the literature. The continuous and discrete parts of the eigenspectra are determined using a purely numerical
scheme to solve the discretized eigenvalue problem. The continuous spectra are then more
accurately determined using a semi-analytical scheme which uses an analytical solution of the Orr-Sommerfeld equation alongside a numerical solution for the base
flow.
A comprehensive survey of the effect of each shear thinning and extensional
fluid parameter is undertaken and an instability is found for particular parameter values for the Giesekus
fluid. A preliminary investigation of this instability is undertaken whereby the unstable discrete eigenvalue is investigated using an Orthonormal Runge-Kutta scheme within a shooting method which uses the results from the Chebyshev-QZ scheme as a starting point.
The linear PTT fluid is found to be stable to infinitesimal disturbances within the range of shear-thinning and extensional parameters considered. The computational e ciency and accuracy of the numerical methods are also investigated.
