A low-dimensional approach to nonlinear plane–Poiseuille flow of viscoelastic fluids

Abstract

The nonlinear stability and bifurcation of the one-dimensional plane–Poiseuille flow is examined for a Johnson–Segalman fluid. The methodology used is closely related to that of Ashrafi and Khayat [Phys. Fluids 12, 345 (2000)] for plane–Couette flow. The velocity and stress are represented by orthonormal functions in the transverse direction to the flow. The flow field is obtained from the conservation and constitutive equations using the Galerkin projection method. Both inertia and normal stress effects are included. The stability picture is dramatically influenced by the viscosity ratio, ε. The range of shear rate or Weissenberg number for which the base flow is unstable increases (from zero) as the fluid deviates from the Newtonian limit (as ε decreases). Typically, two turning points are observed near the critical Weissenberg numbers. The transient response is heavily influenced by the level of inertia. It is found that the flow responds oscillatorily when the Reynolds number is small, and monotonically at large Reynolds number (when elastic effects are dominated by inertia).