Linear instability of a resting state of the magnetohydrodynamic flows of polymeric fluid in a cylindrical channel (generalized Vinogradov–Pokrovski model)

We study a linear stability of a resting state for flows of incompressible viscoelastic fluid in an infinite cylindrical channel under the influence of an external uniform magnetic field directed parallel to the cylinder axis (we use a generalized rheological Vinogradov–Pokrovski model as mathematical model) in a class of axisymmetric periodic along the axial variable flows. We establish that for some values of the parameters in the case of an absolute conductivity bm=0, the magnetic field can substantially lessen the real part of an exponent for perturbations of the radial velocity component, which is the main element of the instability development. For general case bm≠0, we justify the possibility of removing the instability based on the performed calculations.