MHD model of incompressible polymeric fluid. Linear instability of the resting state

Abstract

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).

Keywords:

• Incompressible viscoelastic polymeric medium
• rheological relation
• magnetohydrodynamic flow
• resting state
• spectrum
• Lyapunov's stability

AMS SUBJECT CLASSIFICATIONS:

• 76E25
• 76A05

Acknowledgments

Authors are grateful to A.S. Rudometova for numerical experiments and to A.V. Yegitov for the help in formatting the work.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The statements of the problem and theorems were done by A.M. Blokhin. Their proofs are done by D.L. Tkachev. This work is supported by the Russian Scientific Fund, project number 20-11-20036.