Particle impact on a cohesive granular media

We investigate numerically the impact process of a particle of diameter $d$ and velocity $V_i$ onto a cohesive granular packing made of similar particles via two-dimensional discrete element method simulations. The cohesion is ensured by liquid bridges between neighboring particles and described by short range attraction force based on capillary modeling. The outcome of the impact is analyzed through the production of ejected particles from the packing, referred to as the splash process. We quantify this production as a function of the impact velocity for various capillary strength $\mathrm{\Gamma }$ and liquid content $\mathrm{\Omega }$. The numerical data indicate that the splash process is modified when the dimensionless cohesion number $\text{Co}=6\mathrm{\Gamma }/{\rho }_{p}g{d}^2$ (where ${\rho }_p$ is the particle density, $d$ its diameter, and $g$ the gravitational acceleration) exceeds a critical value of the order of the unity. Above this value, we highlight that the ejection process is triggered above a threshold impact Froude number, $\text{Fr}=V_i/\sqrt{gd}$, which depends both on $\mathrm{\Gamma }$ and $\mathrm{\Omega }$ and scales as ${\mathrm{\Gamma }}^{\beta }{\mathrm{\Omega }}^{\delta }$, where the values of the exponents are found close to $1/2$ and $1/6$, respectively, and can be derived from rational physical arguments. Importantly, we show that, above the threshold, the number of splashed particles follows a linear law with the impact Froude number as in the cohesionless case.

Figure 1

2D packing built with $16000$ particles of mean diameter $d$ using pluviation method in a square box with dimensions $40d\times 40d$. To avoid crystallization, the particles of the packing have a polydispersity of $20\%$ and the walls of the box consist of spherical particles of diameter $2d$. The mean volume fraction of the packing is $\varphi =0.8$.

Figure 2

Variation of the restitution coefficients $\overline{e}$ (a) and $\overline{e_z}$ (b) as a function of the incident angle ${\theta }_i$ for different Froude number $\text{Fr}=V_i/\sqrt{gd}$ ranging from 20 to 80. The solid lines represent empirical laws proposed by Beladjine et al. [13].

Figure 3

Variation of the mean number of ejected particles ${\overline{N}}_{ej}$ as a function of the impact Froude number $\text{Fr}=V_i/\sqrt{gd}$ for various incident angles ${\theta }_i$ ranging from ${10}^{\circ }$ to ${90}^{\circ }$. The dashed lines represent linear fits to the data using Eq. (10). Inset: mean number of ejected particles renormalized by the fraction of energy of the incident bead transferred to the packing [i.e., ${\overline{N}}_{ej}/(1-{\overline{e}}^2)$] as a function of the Froude number Fr. The dashed line represents the best linear fit to our data.

Figure 4

Variation of the restitution coefficients $\overline{e}$ and $\overline{e_z}$ as a function of the cohesion number for various incident angles. The impact Froude number and the water content are set to $\text{Fr}=60$ and $\mathrm{\Omega }=1\%$.

Figure 5

Mean number of ejected particles $\overline{N_{ej}}$ as a function of the Froude number Fr for various values of the cohesion number Co. The incident angle is kept constant and equal to ${\theta }_i={10}^{\circ }$ and $\mathrm{\Omega }=1\%$. Inset: mean number of ejected particles $\overline{N_{ej}}$ as a function of the modified Froude number ${\text{Fr}}^{*}$ [cf. Eq. (15)].

Figure 6

Variation of critical Froude number ${\text{Fr}}_c$ and the slope $N_0$ (see inset) as a function of the cohesion number. The horizontal solid lines represent the noncohesive limit where the dashed lines stand for the best fits using the following scaling laws: $\text{Fr}\propto {\text{Co}}^{{\beta }_1}$ and $N_0\propto {\text{Co}}^{-{\beta }_2}$, with ${\beta }_1\approx 0.4$ and ${\beta }_2\approx 0.45$.

Figure 7

Variation of the mean number of ejected particles rescaled by $(1-{\overline{e}}^2)$ as a function of the Froude number Fr for various impact angle ${\theta }_i$ and various cohesion number Co. Inset: variation of the mean number of ejected particles as a function of the Froude number Fr.

Figure 8

Mean number of ejected particles ${\overline{N}}_{ej}$ as a function of the Froude number Fr for various water content ranging from $0\%$ to $10\%$. ${\theta }_i={10}^{\circ }$ and $\text{Co}=100$. Dashed lines stand for the best fit to the data with linear laws. Inset: mean number of ejected particles as a function of the modified Froude number ${\text{Fr}}^{*}$.

Figure 9

Variation of the critical Froude number ${\text{Fr}}_c$ and coefficient $N_0$ (inset) as function of the water content $\mathrm{\Omega }$.

Figure 10

Variation of the mean number of ejected particles as a function of the modified Froude number ${\text{Fr}}^{*}$ for $\text{Co}=10$ and $\mathrm{\Omega }=5\%$ (the impact angle is kept fixed: ${\theta }_i={10}^{\circ }$). The solid line stands for Eq. (25). Inset: mean number of ejected particles as a function of the Froude number Fr. The solid line presents a linear fit to the data: ${\overline{N}}_{ej}=N_0(\text{Fr}-{\text{Fr}}_c)$, with ${\text{Fr}}_c\approx 20$ and $N_0\approx 0.2$.

Figure 11

Particle volume fraction $\varphi$ of a cohesive granular packing as a function of the dimensionless stiffness parameter $K=k_n/\mathrm{\Gamma }$. The dashed line represents the particle volume fraction of a noncohesive packing.

Figure 12

Illustration of 2D cohesive packings with different dimensionless stiffnesses $K=k_n/\mathrm{\Gamma }$: (a) $K=6250$ and (b) $K=62.5$. When $K$ is decreased below a critical value (i.e., $K\lesssim 5\times {10}^3$), strong heterogeneities appear within the bed.