Subcritical bifurcation of shear-thinning plane Poiseuille flows

International audienceIn a recent article (Nouar et al. 2007), a linear stability analysis of plane Poiseuille flow of shear-thinning fluids has been performed. The authors concluded that the viscosity stratification delays the transition and that is important to account for the viscosity perturbation. The current paper focuses on the first principles understanding of the influence of the viscosity stratification and the nonlinear variation of the effective viscosity with the shear rate on the flow stability with respect to a finite amplitude perturbation. A weakly nonlinear analysis, using the amplitude expansion method is adopted as a first approach to study nonlinear effects. The bifurcation to two-dimensional travelling waves is studied. For the numerical computations, the shear-thinning behavior is described by the Carreau model. The rheological parameters are varied in a wide range. The results indicate that (i) the nonlinearity of the viscous terms tends to reduce the viscous dissipation and to accelerate the flow, (ii) the harmonic generated by the nonlinearity of the viscosity is smaller and in opposite phase with that generated by the quadratic nonlinear inertial terms and (iii) with increasing shear-thinning effects, the bifurcation becomes highly subcritical. Consequently, the magnitude of the threshold amplitude of the perturbation, beyond which the flow is nonlinearly unstable, decreases. This result is confirmed by computing higher order Landau constants