On the propagation of viscous gravity currents of non-Newtonian fluids in channels with varying cross section and inclination

This paper presents a model for the laminar propagation of gravity currents of rheologically complex fluids over natural slopes. The study is motivated by the ubiquitous occurrence of gravity currents in environmental applications that are confined by channels that widen and have reduced slopes in the flow direction; typical examples are mud and lava flows. In these applications, many fluids exhibit nonlinear relationships between shear stress and shear rate, with or without the appearance of a yield stress. We consider Ostwald-de Waele and Herschel-Bulkley fluids. A power-law equation is used to capture the variations in the channel shape and slope in the flow direction. We study the motion of constant and time-dependent volumes of these fluids on smoothly varying topographies. Approximate similarity solutions are obtained for Ostwald-de Waele fluids, while for HB fluids, we use the methods of characteristics to compute front propagation. Constant volume and constant influx tests were conducted in a channel with a widening parabolic cross-section and an inclination decreasing downstream from $\approx 7^{\circ}$ to $\approx 3.2^{\circ}$. The front position was measured continuously over time, and the current thickness and the surface velocity were recorded for a subset of experiments in some cross sections. The experimental study confirms the theoretical formulation, with a better agreement for constant influx than constant volume currents.