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This paper proposes a computational approach to debris flow model. In recent years, the theoretical activity on the classical Herschel-Bulkley model (1926) has been very intense, but it was in the early 80s that the opportunity to explore the computational model has enabled considerable progress in identifying the subclasses of applicability of different sets of boundary conditions and their approximations. Here we investigate analytically the problem of the simulation of uniform motion for heterogeneous debris flow laterally confined taking into account mainly the geological data and methodological suggestions derived from simulation with cellular automata and grid systems, in order to propose a computational scheme able to operate a compromise between “global” predictive capacities and computing effort.

The mobility of granular clusters in the upper part of the mountain basins may cause a sliding similar to fluids currents. Obviously this happens in particular conditions of slope and if we have a particular solid-fluid concentration ratio. Inside the mixture that moves, there are several resistances. Despite the strong nonstationarity of currents, it is important to study the conditions of uniform or nearly uniform motion [

The observations in correspondence with the free surface as well as the first velocity measurements made within the mixture showed that the transverse velocity profile exhibits a maximum at the centerline of the section and minimum values in correspondence with the side walls. Concentration measurements have been obtained by some authors for the case of granular mixtures devoid of fine material, in correspondence with the side walls of the channel. These measurements have shown that the concentration increases with the depth of the current, with a roughly linear trend, reaching a maximum value equal to the concentration at the bottom. In [

The application of the model therefore allows simulating the different conditions that may occur for different combinations of the significant variables (bottom slope and concentration of the mixture) both in the presence of equilibrium with the erodible bed and in the presence of an interface with a nonerodible surface. The estimates of the average speed of the current obtained with the model described in [^{−8} sec^{−1}, on which depend the tangent tensions along the direction of flow. The physical reason is fairly straightforward: this value empirically fits a typical situation of the change in viscosity and the convective motions in the natural landslides. We wondered if it is possible to generalize this solution, eliminating the dependence on epsilon, so you can get the same useful evaluations also for situations more and more distant from those in which it developed the original model. We used a suggestion that comes from the simulation with cellular automata and “grid” methods [

In the study of debris flows the use of a numerical code is crucial, since the Navier-Stokes equation is nonlinear partial differential equation and it is often impossible to solve analytically. Therefore the goal is to find solutions computationally “complacent.” This departs substantially from the traditional mathematical physics approach. One of the key physical assumptions for the calculation code allows a numerical integration of the following conservation of momentum:

Since the tangent tension is not defined in the region where the gradient tensor is zero, we have used a relation proposed by Bercovier and Engleman [^{−8} sec^{−1}, empirical value calibrated on geophysical data of landslides,

As first step, the idea is to consider the following relations:

We set

In practice we have modified the numerical code eliminating the parameter

In [

The fact that the concentration grows with the depth has been observed experimentally but it is very difficult to find a relation

Global reasonable assumption is that the concentration is constant along the edges of the river bed and equal to maximum packing concentration and, instead, in the points of coordinates

These conditions allow finding the links among the coefficients

By uniting the two systems, we obtain

By resolving the previous system we get

This relation has been modified introducing two new variables

Initially we have used the new distribution of concentration to integrate the problem through (^{−8} sec^{−1} and instead, for such value, the linear distribution does not allow finding solutions. Let us notice that, in the simulation of the experiments, with widths of the river bed of 0.203 m we have observed that the variation of

We have proposed a numerical code in the context of uniform motion for heterogeneous debris flow laterally confined. This computational scheme reflects a global description whose value critically depends on (

The authors declare that there is no conflict of interests regarding the publication of this paper.