Drag force in a cold or hot granular medium

We measure experimentally and analyze the resisting force exerted by a bidimensional packing of small disks on a larger intruder disk dragged horizontally at constant velocity $V_0$. Depending on the vibration level of the packing that leads to a granular “cold” or “hot” packing, two force regimes are observed. At low vibration level (“cold” granular medium), the drag force $F$ does not depend on $V_0$, whereas for high vibrations (“hot” granular medium), the drag force increases linearly with $V_0$. Both regimes can be understood by the balance of two “granular temperatures” that can be defined in the system: a bulk temperature $T_b$ imposed by the external vibration to the overall packing and a local temperature $T_0$ induced by the own motion of the intruder disk in its vicinity. All experimental data obtained for different sizes and velocities of the intruder disk are shown to be governed by the temperature ratio $T_0/T_b$. A critical velocity $V_{0c}$, above which the system switches from “hot” to “cold,” can be obtained in this frame. Finally, we discuss how these two “viscous” regimes should be followed by an inertial regime where the drag force $F$ should increase as ${V_0}^2$ at high enough velocity values, for $V_0$ greater than a critical value $V_{0i}$ corresponding to high enough Reynolds or Froude number.

Figure 1

Sketch of the experimental setup.

Figure 2

(a) Evolution of the instantaneous drag force $F$ as a function of time $t$ and disk position $x$ for an intruder disk of diameter $d=8$ mm at velocity $V_0=0.1$ mm.${\mathrm{s}}^{-1}$ in the case of no external vibrations ($\omega =0$, - - -) or external vibrations ($\omega \simeq 63{\mathrm{s}}^{-1}$, —) of a granular packing of solid fraction $\varphi =0.80$. (b) Drag force $F$ as a function of velocity $V_0$ (a) without packing vibrations ($▴$, $\omega =0$) and (b) with packing vibration ($▵$, $\omega \simeq 63{\mathrm{s}}^{-1}$) for an intruder disk of diameter $d=8$ mm in a granular packing of solid fraction $\varphi =0.80$. (- - -) Mean value $F_0=0.25$ N for $\omega =0$. The error bars correspond to the standard deviation of the drag force signal $F\left(t\right)$.

Figure 3

Normalized drag force ${\tilde{F}}_0{({\varphi }_J-\varphi )}^{1/2}/\varphi$ without packing vibrations ($\omega =0$) as a function of the size ratio $d/d_g$. Measurements for an intruder disk of diameter $d=8$ mm ($▴$), 16 mm ($\blacksquare$), and 32 mm ($•$) in a granular medium of solid fraction $\varphi =0.80$, and for $d=8$ mm ($▾$) and $d=16$ mm ($⧫$) in a less dense packing $\varphi =0.76$ together with the model equation ${\tilde{F}}_0{({\varphi }_J-\varphi )}^{1/2}/\varphi =73{(d/d_g)}^{-1/2}$ (—). Inset : Same data for the normalized drag force ${\tilde{F}}_0=F_0/{\rho }_{s}g{h}^{2}d$ with the model equations ${\tilde{F}}_0=350{(d/d_g)}^{-1/2}$ (- - -) for $\varphi =0.80$ and ${\tilde{F}}_0=230{(d/d_g)}^{-1/2}$ (– $·$ –) for $\varphi =0.76$.

Figure 4

Normalized drag force $F^{★}=F/F_0$ as a function of the normalized velocity $V^{★}=V_0/V_{0c}$ in a granular medium of solid fraction $\varphi =0.80$ vibrated at $\omega \simeq 63{\mathrm{s}}^{-1}$ for an intruder diameter $d=8$ mm ($▵$), 16 mm ($\square$), or 32 mm ($\circ$). (- - -) Fitting curve of equation ${\displaystyle F^{★}=V^{★}/{[1+{V^{★}}^2]}^{1/2}}$ with $\alpha =0.02$ in the expression of $V_{0c}$.

Figure 5

Sketch of the evolution of the normalized drag force $F$ in a granular packing as a function of velocity $V_0$ in a log-log plot.