TY - JOUR

T1 - Stability of the Poiseuille-type flow for a MHD model of an incompressible polymeric fluid

AU - Blokhin, A. M.

AU - Tkachev, D. L.

PY - 2020/3/1

Y1 - 2020/3/1

N2 - We study a generalization of the Pokrovski–Vinogradov model for flows of solutions and melts of an incompressible viscoelastic polymeric medium to nonisothermal flows in an infinite plane channel under the influence of magnetic field. For the linearized problem (when the basic solution is an analogue of the classical Poiseuille flow for a viscous fluid described by the Navier–Stokes equations) we find a formal asymptotic representation for the eigenvalues under the growth of their modulus. We obtain a necessary condition for the asymptotic stability of the Poiseuille-type shear flow. For analysis we use new result, that generalizes Birkhoff theorem on the case, when the coefficient matrix of the eigenvalue itself has zero with multiplicity greater than one as an eigenvalue. We also get the necessary condition for Lyapunov stability of the shear Poiseuille-type flow as a result of acquired representation.

AB - We study a generalization of the Pokrovski–Vinogradov model for flows of solutions and melts of an incompressible viscoelastic polymeric medium to nonisothermal flows in an infinite plane channel under the influence of magnetic field. For the linearized problem (when the basic solution is an analogue of the classical Poiseuille flow for a viscous fluid described by the Navier–Stokes equations) we find a formal asymptotic representation for the eigenvalues under the growth of their modulus. We obtain a necessary condition for the asymptotic stability of the Poiseuille-type shear flow. For analysis we use new result, that generalizes Birkhoff theorem on the case, when the coefficient matrix of the eigenvalue itself has zero with multiplicity greater than one as an eigenvalue. We also get the necessary condition for Lyapunov stability of the shear Poiseuille-type flow as a result of acquired representation.

KW - Incompressible viscoelastic polymeric fluid

KW - Magnetohydrodynamic flow

KW - Rheological relation

KW - INSTABILITY

KW - LINEARIZED PROBLEM

KW - MICROPOLAR FLUID

KW - ASYMPTOTICS

KW - SPECTRUM

UR - http://www.scopus.com/inward/record.url?scp=85076705348&partnerID=8YFLogxK

U2 - 10.1016/j.euromechflu.2019.12.006

DO - 10.1016/j.euromechflu.2019.12.006

M3 - Article

AN - SCOPUS:85076705348

VL - 80

SP - 112

EP - 121

JO - European Journal of Mechanics, B/Fluids

JF - European Journal of Mechanics, B/Fluids

SN - 0997-7546

ER -