Evaluation of shear rate formulations through steady uniform non-Newtonian fluid flows in the context of shallow-water equations

Non-Newtonian rheology effects, such as pseudoplasticity and viscoplasticity, are understood as shear stresses, incorporated to the energy slope term in the Shallow-Water Equations (SWE). However, non-Newtonian shear stresses are dependent of the shear rate, whose formulation is a function of the gradient of the velocity profile in the bottom. This study investigated two shear rate formulations that are commonly applied in the SWE literature: 1) a non-parameterized function; and 2) a function based on the Herschel-Bulkley rheological model. Their influence in steady uniform flows of non-Newtonian fluids was evaluated through numerical-theoretical comparisons. A Lax-Friedrichs scheme was implemented to solve the SWE system and allowed employing the shear rate formulations. Experimental tests were carried out and numerical simulations of hypothetical scenarios were performed. It was found that the non-parameterized formulation presented deviation in normal depth up to 14% in comparison with theoretical solution, while the formulation based on the Herschel-Bulkley model provided a good agreement, corroborated by punctual Computational Fluid Dynamics simulations (deviation less than 2%) and experimental data. The ratio of both shear rate formulations is strongly correlated to the deviation of normal depth, indicating that the non-parameterized shear rate function does not provide an acceptable result in the steady uniform flow.