This paper investigates the spatial evolution of a disturbance in an open-channel flow of a power-law fluid at non-uniform accelerated and decelerated initial profiles, up to the occurrence of roll-waves in mild and steep slope channels. Both theoretical and numerical analyses are applied to the depth-averaged continuity and momentum conservation equations, deduced from the von Kármán's integral method. For the theoretical investigation, the nonlinear near-front expansion technique was applied. Then, the full nonlinear problem in its conservative formulation was numerically solved. Independently of the rheology of the flowing medium, non-uniform initial conditions strongly influence the perturbation celerity, the disturbance evolution and the roll-wave development. For mild slope channels, an initially decelerated profile of shear-thinning fluids has a stabilizing effect, while the opposite is found for accelerated profiles. For shear-thickening fluids, only the stabilizing effect caused by a decelerated profile is observed. In steep slope channels, independently of the fluid rheology, decelerated initial conditions promote roll-wave occurrence, while accelerated conditions inhibit the perturbation growth. Although experimental verifications are needed, the present results have to be properly accounted for in defining roll-wave prediction methods and in assigning appropriate boundary conditions to enhance or to reduce their formation.
