Wave propagation in linearized shallow flows of power-law fluids

A depth-averaged model with a power-law rheology, usually adopted for debris flow of fine sediment-water mixtures, is analytically studied, considering the complete (Full Dynamic Model) and the simplified momentum equations, known as Kinematic, Diffusion, and Quasi-steady Models. Applying the Laplace transform to the linearized equations, the analytical first-order expressions of the upstream and downstream spatio-temporal channel response functions are derived for the full and the approximate models. Both shear-thinning and shear-thickening fluids have been considered in hypocritical and linearly stable conditions, to investigate the influence of the rheology on the wave propagation. The effect of the upstream and downstream boundary conditions is considered assuming a finite channel length. The analytic description of the various linearized approximations allows a multi-faceted comparison of the wave propagation. About the basic wave characteristics, in the entire investigated range of Froude numbers and for all the considered rheologies, the Kinematic and Diffusive models underestimate the primary wave celerity and the Quasi-steady model underestimates the wave round-trip time. However, the examination of the response function reveals that the approximate models may provide relatively accurate reproduction of the full dynamic response. Moreover, the analytical solutions in the Laplace variables permit a systematic analysis of some relevant shape factors of the response functions. Independently on the models and the rheology, the damping of the downstream response function is several orders of magnitude larger than that of the upstream one. The results of the shape factor analysis of the upstream response function indicate that the diffusive approximation is the most accurate one. The performed comparison among the wave characteristics of the different approximations provides indications on the applicability of the simplified models for predicting debris or mud flows.