Rising and sinking intruders in dense granular flows

We computationally determine the net bed force on single spherical intruder particles in dense granular flows as a function of particle size, particle density, shear rate, overburden pressure, and gravity. A simple buoyancy-like scaling law is recovered (analogous to that in fluids), but with a scale factor that depends on the particle size ratio due to discrete contacts. Comparing the bed force with the intruder weight results in predictions of whether an intruder rises or sinks that agree with data from various independent experiments of free surface granular flows.

Figure 1

(a) Intruder particle (red) in a controlled, constant-shear-rate flow. A virtual spring measures the net vertical bed force on the intruder. (b) Rheology for constant-shear-rate and chute flows ($\theta ={\{22,24,25,28\}}^{\circ }$). Data are depth-averaged values. Error bars indicate $\pm 1$ standard deviation.

Figure 2

Dependence of $F$ on various parameters. Data in (a)–(c) are from the constant-shear-rate system. (a) $F$ and $m_i{g}_z$ vs $R$ for ${\rho }_i=\rho$ ($\rho =2500{\mathrm{kg}/\mathrm{m}}^3$, $P_0=600\mathrm{Pa}$, $\dot{\gamma }=30{\mathrm{s}}^{-1}$, $g=9.81{\mathrm{m}/\mathrm{s}}^2$). Inset: varying ${\rho }_i$. (b) $F/m_i{g}_z$ vs $R$ for $R_{\rho }=\{0.5,1,3\}$ with varying $\rho$ and ${\rho }_i$. (c) $F/m_i{g}_z$ vs $R$ for $600\mathrm{Pa}⩽P_0⩽3000\mathrm{Pa}$, $10{\mathrm{s}}^{-1}⩽\dot{\gamma }⩽40{\mathrm{s}}^{-1}$, and $g=\{5,15\}{\mathrm{m}/\mathrm{s}}^2$. Inset: $F/m_i{g}_z$ vs $I$ for $1.4⩽R⩽1.6$ (selected for illustration). (d) Constant-shear-rate results ($P_0=600\mathrm{Pa}$, $\dot{\gamma }=30{\mathrm{s}}^{-1}$), chute flow results ($\theta ={\{22,24,25,28\}}^{\circ }$), and data from [14]. Shaded (unshaded) areas in (b)–(d) indicate that the intruder sinks (rises) from its initial position.

Figure 3

(a) $F/\varphi \rho g_z{V}_i$ vs $R$, with varying ${\rho }_i$, $\rho$, $P_0$, $\dot{\gamma }$, $\theta$, and $g$ in constant-shear-rate and chute flows. Data from [14] is also included (red stars). Solid curve is a fit to $F=f\left(R\right)\varphi \rho g_z{V}_i$, with dashed curves showing the two exponential terms defining $f\left(R\right)$; see text. Inset: extreme cases with $I=0.005$ ($P_0=3000\mathrm{Pa}$, $\dot{\gamma }=1{\mathrm{s}}^{-1}$) and ${\mu }_i=0$ ($P_0=1800\mathrm{Pa}$, $\dot{\gamma }=20{\mathrm{s}}^{-1}$), respectively. (b) $N_c/N_n$ vs $R$ ($P_0=2400\mathrm{Pa}$, $\dot{\gamma }=20{\mathrm{s}}^{-1}$). Inset: $xz$-plane view of contact network at various $R$, showing intruders (red), contacting particles (blue), and noncontacting neighbor particles (gray).

Figure 4

Predicted rise-sink transition compared to untethered intruders in (a) 88 constant-shear-rate simulations ($P_0=600\mathrm{Pa}$, $\dot{\gamma }=20{\mathrm{s}}^{-1}$) and (b) 189 experiments [9]. Curves show $R_{\rho }=\varphi f\left(R\right)$ with $\varphi =0.55$. Red, gray, and black symbols indicate rising, sinking, and neutral intruders, respectively. Data in (b) are from rotating drums ($\square$), chute flows ($\diamond$), and heap flows ($▹$).