Petrov–Galerkin computation of nonlinear waves in pipe flow of shear-thinning fluids: First theoretical evidences for a delayed transition

A pseudospectral Petrov–Galerkin code is developed in order to compute nonlinear traveling waves in pipe flow of shear-thinning fluids. The framework is continuum mechanics and the rheological model used is the purely viscous Carreau model. The code is validated, and a study of its convergence properties is made. It is shown that exponential convergence is obtained, despite the highly-nonlinear nature of the viscous diffusion terms. Physical computations show that, as compared with the case of a constant-viscosity fluid, i.e., a Newtonian fluid, in the case of shear-thinning fluids the critical Reynolds number of the saddle-node bifurcation where the waves with an azimuthal wavenumber m0 = 3 appear increases significantly when the non-Newtonian effects come into play.