A low-dimensional approach to nonlinear plane-Couette flow of viscoelasticfluids

The nonlinear stability of the one-dimensional plane Couette flow is examin ed for a Johnson-Segalman fluid. The velocity and stress are represented by symmetric and antisymmetric Chandrasekhar functions in space. The flow fie ld is obtained from the conservation and constitutive equations using the G alerkin projection method. Both inertia and normal stress effects are inclu ded. For given Reynolds number and viscosity ratio, two critical Weissenber g numbers are found at which an exchange of stability occurs between the Co uette and other steady flows. The critical points coincide with the two ext rema of the stress/rate-of-strain curve. At low (high) Reynolds number, the flow decays monotonically (oscillatorily) toward the steady-state solution . The number and stability of the nontrivial branches around the critical p oints are examined using the method of multiple scales. Comparison between the approximate and the numerical branches leads to excellent agreement in the vicinity of the critical points. The influence of the higher-order mode s is assessed, showing low-order convergence and good accuracy when the flo w profiles are compared against existing finite-element results. (C) 2000 A merican Institute of Physics. [S1070-6631(00)02201-7].