The spectrum and Lyapunov linear instability of the stationary state for polymer fluid flows: the Vinogradov–Pokrovskii model

We study the Lyapunov linear stability of the stationary state for flows of an incompressible viscoelastic polymer fluid in an infinite planar channel. As a model we choose the Vinogradov–Pokrovskii rheological model well-suited for describing the flow characteristics of linear polymer melts. We find the spectrum of the mixed problem and prove that the solution to the linearized mixed problem in the class of periodic perturbations of the variable changing along the channel side grows faster in time than the exponential with a linear exponent. In other words, the stationary state is linearly unstable.