Spectrum of a problem about the flow of a polymeric viscoelastic fluid in a cylindrical channel (Vinogradov-Pokrovski model)

We study the linear stability of a resting state for flows of incompressible viscoelastic polymeric fluid in an infinite cylindrical channel in axisymmetric perturbation class. We use structurally-phenomenological Vinogradov-Pokrovski model as our mathematical model.
We formulate two equations that define the spectrum of the problem. Our numerical experiments show that with the growth of perturbations frequency along the channel axis there appear eigenvalues with positive real part for the radial velocity component of the first spectral equation. That guarantees linear Lyapunov instability of the resting state.