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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 b m = 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 b m ≠ 0 , we justify the possibility of removing the instability based on the performed calculations. [ABSTRACT FROM AUTHOR]

We study the linear stability of a resting state for a generalization of the basic rheological Pokrovski–Vinogradov model for flows of solutions and melts of an incompressible viscoelastic polymeric medium to the nonisothermal case under the influence of magnetic field. We prove that the corresponding linearized problem describing magnetohydrodynamic flows of polymers in an infinite plane channel has the following property: for certain values of the conduction current which is given on the electrodes, i.e. on the channel boundaries, the problem has solutions whose amplitude grows exponentially (in the class of functions periodic along the channel). [ABSTRACT FROM AUTHOR]

We study a new rheological model describing flows of melts and solutions of incompressible viscoelastic polymeric media in an external uniform magnetic field in the presence of a temperature drop and a conduction current. We derive an asymptotic representation of the spectrum of the linear problem resulting from the linearization of the initial boundary value problem in an infinite plane channel about a Poiseuille-type flow. For this Poiseuille-type flow, we also determine the parameter domain of linear Lyapunov stability. [ABSTRACT FROM AUTHOR]