Waves dynamics in a linearized mud-flow Shallow model

Abstract

The study of formation and evolution of waves in mud-flows is strongly motivated by their destructive power. The rheology of these flows, which involve massive solid transport, is often described using linear or non-linear viscoplastic models. The paper analyzes the wave dynamics of mud-flows, presenting the linearized response of the 1D uniform flow of a viscoplastic fluid with yield stress, i.e. Herschel & Bulkley fluid, to a pointwise impulsive forcing term. The solution in either linearly stable or unstable conditions is found, for both subcritical and supercritical flows, through the bilateral inverse Laplace transform. The analysis offers a unified description of the behaviour of different non-newtonian fluids, recovering as particular cases both the Bingham and the power-law models. The influence of the dimensionless governing numbers and of the rheological parameters on the shape and peak of waves is investigated.

Diffusive approximation for unsteady mud flows with backwater effect

“…1b suggests that the secondary wave attenuates more strongly than the primary one. Such a result is in agreement with the waves dynamics in linearized (laminar and turbulent) clear-water and mud flows investigated through the study of the Green's function [11,12,33]. Moreover, while for high values of wave periods the primary wave travels downstream without attenuating (Fig.…”

“…(9). Moreover, in order to consider only linearly stable conditions (r < 0), the Froude number of the corresponding uniform condition is always assumed below the following neutral stability value [9,12]:…”

“…[9,[12][13][14]38]), but rather limitedly to analyze the performance of simplified flood routing models. Di Cristo et al [15] and Di Cristo et al [16] analyzed the performance of three different approximations, i.e.…”

Gravity-driven flow of a shear-thinning power-law fluid over a permeable plane

“…As far as stable base flow is considered (Fig.4a), the instantaneous maximum perturbation height progressively moves from the downstream ray x′/t′=c 1 towards the one corresponding to the kinematic wave celerity x′/t′ = c * . Similarly to [15], it may be shown that the Green function tends to assume asymptotically in time a Gaussian shape with the centroid moving with c * . Consistently, the Green function tends to vanish at the two boundaries of the characteristic fan.…”

“…However, the complete solution of the corresponding linearized flow model has not been deduced. As far as the impermeable bed case is concerned, the solution of the linearized flow model allowed to enlighten the essential flow features of the waves for both Newtonian [17,18,38,34,13,14] and non-Newtonian [11,15,16] fluids.…”