Attraction and repulsion between objects in a granular flow

The objective of this work is the understanding of lift forces exerted on two side-by-side intruders within a granular flow that is uniform upstream. To this end, we have performed experiments that reveal long-range attraction and repulsion depending on flow velocity and the distance between intruders. We found an interesting correlation between the asymmetry of flow velocity around the obstacles and the lift force, although the understanding of this correlation rests as an open question for future work. Additionally, a hydrodynamic model based on kinetic theory for inelastic particles was solved numerically. The model reproduces remarkably well the general qualitative features of the experiments which shows that the system behaves similarly to a pair of hot intruders immersed in a liquid of temperature-dependent viscosity. However, quantitative discrepancies suggest that better constitutive relations are needed.

Figure 1

(a) Schematic representation of the experimental setup. Big, thick polycarbonate sheets placed face-to-face, 1.3 cm apart from each other form a quasi-two-dimensional container. The container is filled with tapioca seeds of diameter $d=0.33\pm 0.02\mathrm{cm}$ and density ${\rho }_t\approx 2.7{\text{g/cm}}^3$. (b) Two cylindrical obstacles of diameter $D=1.5\mathrm{cm}$ are placed side-by-side between the walls of the container without touching them, 55 cm above the bottom. The gap between the cylinders $X_{\text{sep}}$ can be varied, and the cylinders are attached to force gauges that allow measuring the vertical, $F_{\text{Drag}}$, and horizontal, $F_{\text{Lift}}$, forces on them. (c) The angle of tilted sticks placed at the bottom of the container controls mass flow out of the system. (d) Front view of the intruders immersed in the granular medium before the flow begins. See Supplemental Material [24] for more details.

Figure 2

(a) Mean value of the lift force as a function of the size of the gap between the intruders $X_{\text{sep}}$. $F_{\text{lift}}>0$ means repulsion and $F_{\text{lift}}<0$ attraction. Inset (a): Raw and smoothed data of the lift force as a function of time are plotted together for the left intruder for $U_{\infty }=9.37$ cm/s and $X_{\text{sep}}=3R$. The stationary state from which the mean force is calculated is clearly identified. (b) The same data as in panel (a) but presented as a function of the upstream flow velocity $U_{\infty }$. The main plots include error bars, but for most of the points they are of the order of the size of the markers.

Figure 3

(a) Attractive ($U_{\infty }=15.9\text{cm/s}$ and $X_{\text{sep}}=5R$) and (b) repulsive ($U_{\infty }=27.7\text{cm/s}$ and $X_{\text{sep}}=5R$) typical velocity fields obtained by applying particle image velocimetry (PIV) to the movies of the experiments (see Fig. S5 of the Supplemental Material [24] for an explanation of the procedure). (c) Schematic representations of the regions where the grains velocities were measured to compute the quantity $\mathrm{\Delta }{\langle v\rangle }^2$ defined in Eq. (1). (d) $\mathrm{\Delta }{\langle v\rangle }^2$ was computed from the velocity fields of the experiments and it was found that, when divided by $U_{\infty }^{3/2}$, its behavior was very similar to the behavior of the lift force shown in Fig. 2. (e) A good linear relation with a correlation coefficient of $r^2=0.9826$ is obtained when $\langle F_{\text{lift}}\rangle$ is plotted as a function of $\mathrm{\Delta }{\langle v\rangle }^2/U_{\infty }^{3/2}$.

Figure 4

Flow velocity in the middle point between obstacles, $v_{\text{mp}}$, plotted as a function of $U_{\infty }$. The dashed line is the bulk flow velocity at the level of the equator of intruders.

Figure 5

(a) Velocity field and (b) Velocity variance (granular temperature) of the experiment for a case of attractive lift. (c) Velocity variance along the horizontal axis, which passes through the equator of intruders for different flow velocities. Vertical lines signal the position of the intruders, which, in this case, are separated by a distance of $X_{\text{sep}}=1.5D$. We obtained these data from images of the experiments like that shown in panel (b). Note that the region of analysis does not correspond to the limits of the experiment: the total width of the container is 50 cm. (d) Mean bulk velocity variance as a function of $U_{\infty }$. The data was obtained from data like the one presented in panel (c) and averaged over all the experiments.

Figure 6

Temperature and velocity fields obtained from the hydrodynamic model (LBM). Comparing these fields with the experiments [Figs. 5 and 5] helped to find the best values of the parameters of the hydrodynamic model.

Figure 7

Lift coefficient $C_L$, Eq. (15), as a function of intruders separation from (a) the LBM, and (b) experiments. In the inset in panel (b), the experimental $C_L$ for all the flow velocities are plotted, in contrast with the main graph where only the fastest experiments are shown for clarity because the value of $C_L$ changes dramatically with $U_{\infty }$. (c) Lift coefficient from the LBM as a function of Re. (d) Experimental lift coefficient as a function of bulk velocity for $U_{\infty }>9$ cm/s. Inset: lift coefficient for the whole range of $U_{\infty }$.