Spectrum of a linear problem about the MHD flows of a polymeric fluid in a cylindrical channel in case of an absolute conductivity (generalized Vinogradov-Pokrovski model)

We study the linear stability of a resting state for flows of incompressible viscoelastic polymeric fluid under the influence of homogenous magnetic field in an infinite cylindrical channel in axisymmetric perturbation class. The tension vector of the magnetic field is parallel to the cylinder axis. We use structurally-phenomenological Vinogradov-Pokrovski model as our mathematical model.
We formulate the equation 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. However for large Reynolds and Weissenberg numbers the exponential growth rate of the amplitude for high frequencies can be suppressed to quite low values by increasing the magnetic pressure.