Unlike most fluids, granular materials include coexisting solid, liquid or gaseous regions, which produce a rich variety of complex flows. Dense flows of grains driven by gravity down inclines occur in nature and in industrialprocesses. To describe the granular flow on an inclined surface, several studies were carried out. We can cite in particular the description of Saint-Venant which considers a dry granular flow, without cohesion and it only takes into account the substance-substrate friction, this model proposes a simplified form of the granular flow, which depends on the one side on the angle of inclination of the substrate with respect to the horizontal plane and on the other side on the thickness of the substance H. The numerical simulation we have developed is first based on the Saint-Venant model, it allowed us to visualize the variation of the speed according to the thickness of the substance (from 0 to H) and to deduce the average speed of the substance on an inclined plane. However, this restrictive model does not take into account the effect of particle friction on the flow and considers that the thickness H is constant. To make our simulation more realistic, we opted for the Savage-Hatter model. We took into account the variation of the thickness on the particles speed, in addition we have studied the effect of the variation of many parameters on the granular flow, namely the temperature and the roughness of the substrate, the density and the compactness of the substance, we found that the speed of the particles increases and the treatment time decreases with an increase in temperature.

Granular materials have a multiphase behavior as they can behave like solids, ﬂuids or gases. They appear in many industrial applications such as food processing or pharmaceutics and also in geotechnical and physical phenomena like avalanches.

Despite a large amount of dedicated research [

In our numerical study and in order to enlarge the specter of parameters and factors that intervenes in the flow, we have opted for the Savage-Hatter model [

We consider the flow of a granular mass (we denote “m” and “d” for the average mass and the diameter of the grains) along a slope of inclination θ, under the effect of gravity g.

For simplicity, we describe the two-dimensional situation, where the position along the slope is marked by the direction ‘x’. We denote ‘y’ the direction perpendicular to the slope (y = 0 corresponds to substance-substrate contact surface), and u(y) the velocity profile (

At the base of the flow, there is a pressure P and a shear stress τ in the material. We call

The effective description of the flow, known as of Saint-Venant model, is obtained by integrating the mass conservation equations (

The K parameter determined by the ratio of the normal stresses within the flow. We can see in the second equation the crucial importance of the effective coefficient of friction

The granular flow is similar to the dense flow regime; thus, we have the following friction law [

With:

I: Inertial Nombrer

_{g}: Surface mass,

P: Pressure,

b: Parameter of grain type.

For a uniform stationary flow, the inclination θ is equal to the effective coefficient of friction

We then used the friction law just identified in homogeneous shear to discuss the shear rate profile in dense stationary flow on a rough inclined plane. The shear rate allowed to determine the variation of the speed _{x}(

To determine the shear rate, we consider the density

The latter (shear rate) is used to determine the speed profile:

We thus find the law of flow:

The combined measurement of the thickness H of the flowing layer, the inclination θ and the average speed

We have developed a computational program that aims to visualize the velocity profile and the effect of substrate roughness on flow blockage.

A study is carried out on an elementary volume of a rectangular substance 10 m long and 6 m thick placed on a rough substrate. The substance is characterized by a density of

Our first simulation consists in visualizing the profile of the speed along the thickness

In our second simulation, we have used the effective description of Saint-Venant flow,

Several simulations were performed to visualize the effect of height, density, compactness and temperature of the substance on the flow velocity. From

Density and compactness are crucial parameters that affect the speed of the flow. In order to visualize the effect of density, we have considered three different values as (_{1} = 1000, _{2} = 1600, _{3} = 2000 ^{3}). It can be seen from

Furthermore, to study the variation of the flow velocity as a function of the substance temperature, it was necessary to add to the momentum (

η: viscosity (η = cste T)

In our simulation, we have chosen the water viscosity for three different temperatures _{1} = 1, 787_{2} = 1, 002_{3} = 0.2818

The results presented in

In this second part, we have studied the effect of roughness on the blocking of the flow. We mean by blocking the flow when the grains are trapped in the roughness and can only start rolling from an inclination _{start}.

Accelerated by gravity and slowed down by dissipative shocks on the substrate, the grains reach a stationary speed _{stop}. A simulation was performed in order to show the effect of roughness.

For a coefficient of friction

The average layer speed will be slowed down to blockage for all _{i} < 30°-.

The mean velocity of the layer will be uniform for _{i} = 30°-.

The average speed of the layer will be accelerated for all _{i} > 30°-.

This simulation proves that in order that our substance does not block, it is necessary that (

In this work we have opted for the Savage-Hatter model to study a granular flow in an inclined plane. We have provided new numerical results that highlight the effect of substrate temperature, the roughness, the compactness, the density of the substance on the flow behavior, which was neglected in previous models. We demonstrated that, the more the density and the compactness are important and the temperature of the substance is low, the more we will have a significant flow velocity. Furthermore, the inclination angle plays an important role in the speed of the granular flow, to have a flow; the angle of inclination must be greater than the start angle. Despite, the advantages of the Savage-Hatter model, but this latter, do not take into account the effects of forces that appear between particles during the flow. For this purpose, we aimed in the future work to highlight the effect of these forces (Forces of Hertz, Visco-elastic force…) on the flow velocity of particles on a plane with variable angles of inclination.