# A Free Interface Model for Static/Flowing Dynamics in Thin-Layer Flows of Granular Materials with Yield: Simple Shear Simulations and Comparison with Experiments

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## Abstract

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## 1. Introduction

## 2. The Free Interface Model

#### 2.1. Origin of the Model: Viscoplastic Rheology with Yield Stress

#### 2.2. Free Interface Model with Source Term

- If $\nu >0$ then$${\partial}_{Z}\left(\nu {\partial}_{Z}U\right)(t,b(t))=\nu {\partial}_{ZZ}^{2}U(t,b(t))=S(t,b(t)),$$$$\dot{b}(t)=\left({\displaystyle \frac{{\partial}_{Z}S(t,b(t))-{\partial}_{ZZ}^{2}(\nu {\partial}_{Z}U)(t,b(t))}{S(t,b(t))}}\right)\nu .$$
- If $\nu =0$ and ${\partial}_{Z}U(t,b(t))\ne 0$, then$$\dot{b}(t)={\displaystyle \frac{S(t,b(t))}{{\partial}_{Z}U(t,b(t))}}.$$

#### 2.3. Free Interface Model with Constant Source

## 3. Analytical Solution for the Inviscid Model with Constant Source Term

#### 3.1. Analytical Solution

#### 3.2. Choice of the Parameters and Initial Conditions

- linear profile ${U}^{0}(Z)={\alpha}_{1}(Z-{b}^{0})$, with ${\alpha}_{1}=70$ s${}^{-1}$,
- exponential profile ${U}^{0}(Z)={\alpha}_{2}({e}^{\beta Z}-{e}^{\beta {b}^{0}})$, with ${\alpha}_{2}=0.1$ ms${}^{-1}$ and $\beta =130$ m${}^{-1}$,
- Bagnold profile ${U}^{0}(Z)={\alpha}_{3}({(h-{b}^{0})}^{\frac{3}{2}}-{(h-Z)}^{\frac{3}{2}})$, with ${\alpha}_{3}=545$ m${}^{-1/2}$s${}^{-1}$.

- $b(t)={\displaystyle \frac{S}{{\alpha}_{1}}}t+{b}^{0}$,
- $b(t)={\displaystyle \frac{1}{\beta}}log\left({\displaystyle \frac{S}{{\alpha}_{2}}}t+{e}^{\beta {b}^{0}}\right)$,
- $b(t)=h-{\left({(h-{b}^{0})}^{3/2}-\frac{S}{{\alpha}_{3}}t\right)}^{2/3}$.

#### 3.3. Results and Comparison with Experiments

## 4. Numerical Solution to the Viscous Free Interface Model

#### 4.1. Discretization by Moving the Interface

#### 4.2. Discretization by an Optimality Condition

#### 4.3. Scales in the Transient Regime

#### 4.4. Results and Comparison with Experiments for Constant Viscosity

#### 4.5. Variable Viscosity

## 5. Discussion and Conclusions

- If $\nu >0$ and $S(t,X)\ne 0$, then$$\dot{b}(t,X)={\displaystyle \frac{-{\partial}_{ZZ}^{2}(\nu {\partial}_{Z}U)(t,X,b(t))}{S(t,X)}}\nu .$$
- If $\nu =0$ and ${\partial}_{Z}U(t,X,b(t))\ne 0$, then$$\dot{b}(t,X)={\displaystyle \frac{S(t,X)}{{\partial}_{Z}U(t,X,b(t))}}.$$

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Experimental setup from [9,12] to study granular column collapse over inclined planes covered by an initially static layer made of the same grains as those released in the column. For slope angles $\theta $ a few degrees smaller than the typical friction angle $\delta $ of the involved material, a quasi-uniform flow develops behind the front.

**Figure 2.**Position of the static/flowing interface b as a function of time t until the granular mass stops, measured at $X=90$ cm from the gate, from experiments of granular collapse over an initially static granular layer of thickness ${b}^{0}=5$ mm on an inclined channel of slope angle $\theta ={19}^{\circ}$ (green squares), $\theta ={22}^{\circ}$ (blue stars), $\theta ={23}^{\circ}$ (red crosses), and $\theta ={24}^{\circ}$ (black vertical crosses). Time $t=0$ s corresponds to the time when the front of the flowing layer reaches the position $X=90$ cm. The approximate upward velocity $\dot{b}$ of the static/flowing interface is indicated in m/s for each slope angle. These new results have been extracted from the experiments performed by [12,13] for granular columns of initial down-slope length ${r}_{0}=20$ cm, initial thickness ${h}_{0}=14$ cm, and width $W=20$ cm (i.e., volume $V=5600$ cm${}^{3}$). Note that the position of the free surface when the mass stops, i.e., when $b=h$, represented by the upper point for each angle, slightly depends on the slope angle and decreases as the slope angle increases, from about 0.025 m at $\theta ={19}^{\circ}$ to about 0.018 m at $\theta ={24}^{\circ}$.

**Figure 3.**Velocity profiles $U(Z)$ at different times until the granular mass stops, measured at $X=90$ cm from the gate, in experiments of granular collapse over an initially static granular layer of thickness ${b}^{0}=5$ mm on an inclined channel of slope angle (

**a**) $\theta ={19}^{\circ}$; (

**b**) $\theta ={22}^{\circ}$; and (

**c**) $\theta ={24}^{\circ}$. Time $t=0$ s corresponds to the time when the front of the flowing layer reaches the position $X=90$ cm. These new results have been extracted from the experiments performed by [12,13] for granular columns of initial down-slope length ${r}_{0}=20$ cm, initial thickness ${h}_{0}=14$ cm, and width $W=20$ cm (i.e., volume $V=5600$ cm${}^{3}$).

**Figure 4.**Simplified flow configuration consisting of a uniform flowing layer over a uniform static layer, both parallel to the rigid bed of slope angle $\theta $. The static/flowing interface position is $b(t)$ and the total thickness of the mass h is constant.

**Figure 5.**Static/flowing interface position b as a function of time t for the inviscid model, different slope angles and an initially static granular layer of thickness ${b}^{0}=5$ mm, using (

**a**) a linear, (

**b**) an exponential, and (

**c**) a Bagnold initial velocity profile. The experimental results from Figure 2 are shown for comparison.

**Figure 6.**Velocity profiles $U(Z)$ at different times for the inviscid model, with a linear initial velocity profile and an initially static granular layer of thickness ${b}^{0}=5$ mm over an inclined plane of slope (

**a**) $\theta ={19}^{\circ}$ and (

**b**) $\theta ={24}^{\circ}$. The experimental results from Figure 3 are shown for comparison.

**Figure 7.**Velocity profiles $U(Z)$ at different times for the inviscid model, with an initially static granular layer of thickness ${b}^{0}=5$ mm over an inclined plane of slope (

**a**,

**c**) $\theta ={19}^{\circ}$ and (

**b**,

**d**) $\theta ={24}^{\circ}$, using (

**a**,

**b**) an exponential and (

**c**,

**d**) a Bagnold initial velocity profile. The experimental results from Figure 3 are shown for comparison.

**Figure 8.**Velocity ${U}_{1}^{n+1}$ versus ${b}^{n+1}$. The chosen value for ${b}^{n+1}$ is determined by the intersection of the curve with the horizontal axis.

**Figure 9.**Schematic evolution of the thickness of the static/flowing interface as a function of time.

**Figure 10.**Static/flowing interface position b as a function of time t for different slope angles using a linear velocity profile and different viscosities (

**a**) $\nu ={10}^{-5}$ m${}^{2}$s${}^{-1}$; (

**b**) $\nu =5\times {10}^{-5}$ m${}^{2}$s${}^{-1}$; and (

**c**) $\nu ={10}^{-4}$ m${}^{2}$s${}^{-1}$. The experimental results from Figure 2 are shown for comparison.

**Figure 11.**Velocity profiles $U(Z)$ at different times, with a linear velocity profile, for two different viscosities (

**a**,

**b**) $\nu =5\times {10}^{-5}$ m${}^{2}$s${}^{-1}$ and (

**c**,

**d**) $\nu ={10}^{-4}$ m${}^{2}$s${}^{-1}$, and for the slope angles (

**a**,

**c**) $\theta ={19}^{\circ}$; and (

**b**,

**d**) $\theta ={24}^{\circ}$. The experimental results from Figure 3 are shown for comparison.

**Figure 12.**(left) Velocity profiles $U(Z)$ for an inclined plane of slope $\theta ={24}^{\circ}$ and (right) evolution of the thickness of the static/flowing interface position b for different slope angles, with a viscosity $\nu =5\times {10}^{-5}$ m${}^{2}$s${}^{-1}$, using an exponential (

**a**,

**b**) and a Bagnold (

**c**,

**d**) initial velocity profile. The experimental results from Figure 2 and Figure 3 are shown for comparison.

**Figure 13.**${t}_{\nu =0}^{\mathrm{stop}}-{t}_{\nu}^{\mathrm{stop}}$ with respect to viscosity in log scale.

**Figure 14.**Static/flowing interface position b as a function of time t, for variable viscosity associated with the $\mu (I)$ law, with linear initial velocity profile and slope angles $\theta ={19}^{\circ}$, $\theta ={22}^{\circ}$, $\theta ={24}^{\circ}$. The experimental results from Figure 2 are shown for comparison.

**Figure 15.**Velocity profiles $U(Z)$ at different times, for variable viscosity associated with the $\mu (I)$ law, with linear initial velocity profile and slope angles $\theta ={19}^{\circ}$ and $\theta ={24}^{\circ}$. The experimental results from Figure 3 are shown for comparison.

**Figure 16.**Static/flowing interface position b as a function of time t (left) and velocity profiles $U(Z)$ at time $t=0.5$ s (right), with linear initial velocity profile and slope angle $\theta ={22}^{\circ}$, for respectively our model without viscosity, with constant viscosity $\nu =5\times {10}^{-5}$ m${}^{2}$s${}^{-1}$, variable viscosity associated with the $\mu (I)$ law, and experimental measurements.

**Figure 17.**Thickness profile h as a function of down-slope position X in the experiments performed by [12,13], at times $t=0.66$ s, $0.78$ s, $1.02$ s. The value $X=0$ corresponds to the position of the gate. The slope angle is $\theta ={22}^{\circ}$. Note that the vertical scale differs from the horizontal one, since the thickness of the flow represents only 7% of its horizontal extension.

**Table 1.**Stopping time ${t}^{\mathrm{stop}}$ for the inviscid model ${t}_{\nu =0}^{\mathrm{stop}}={U}^{0}(h)/S$ (for linear initial velocity profile) and in the experiments (from Figure 2), for different slope angles.

${\mathit{t}}^{\mathbf{stop}}$ (s) | $\mathit{\theta}={\mathbf{19}}^{\circ}$ | $\mathit{\theta}={\mathbf{22}}^{\circ}$ | $\mathit{\theta}={\mathbf{24}}^{\circ}$ |
---|---|---|---|

inviscid model | $0.79$ | $1.4$ | $2.8$ |

experiments | $0.94$ | $1.3$ | $3.3$ |

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**MDPI and ACS Style**

Lusso, C.; Bouchut, F.; Ern, A.; Mangeney, A.
A Free Interface Model for Static/Flowing Dynamics in Thin-Layer Flows of Granular Materials with Yield: Simple Shear Simulations and Comparison with Experiments. *Appl. Sci.* **2017**, *7*, 386.
https://doi.org/10.3390/app7040386

**AMA Style**

Lusso C, Bouchut F, Ern A, Mangeney A.
A Free Interface Model for Static/Flowing Dynamics in Thin-Layer Flows of Granular Materials with Yield: Simple Shear Simulations and Comparison with Experiments. *Applied Sciences*. 2017; 7(4):386.
https://doi.org/10.3390/app7040386

**Chicago/Turabian Style**

Lusso, Christelle, François Bouchut, Alexandre Ern, and Anne Mangeney.
2017. "A Free Interface Model for Static/Flowing Dynamics in Thin-Layer Flows of Granular Materials with Yield: Simple Shear Simulations and Comparison with Experiments" *Applied Sciences* 7, no. 4: 386.
https://doi.org/10.3390/app7040386