Compaction of liquid immersed granular packings by small upward flows

We study experimentally the compaction dynamics of a fully immersed granular packing subjected to a slow upward flow. We consider continuous flows as well as discontinuous flows featuring short flow bursts that act as flow “taps.” We monitor the compaction rate of initial packings prepared by settling, and we find that compaction is possible for a continuous flow with upward velocities below the fluidization threshold. Flow bursts are found strongly to enhance the compaction efficiency, especially when the bursts are above the fluidization threshold.

Figure 1

(a) Sketch of the experimental setup. (b) Continuous flow with the constant upward velocity $U_F\ne 0$. (c) Bursting flow with the time period $T=T_w+T_f$ made of successive flow time $T_f$ separated by nonflow time $T_w$.

Figure 2

Solid fraction of the granular bed $\varphi$ as a function of the upward fluid velocity $U_F$ normalized by the Stokes velocity $U_s$ (bottom axis) or the critical fluidization velocity $U_{Fc}$ (top axis). ($\circ$) Experimental data for fluidized beds above the fluidization threshold $U_{Fc}/U_s$ = 0.02 and (—) best fit by RZ law $\varphi =1-{(U_F/U_s)}^{1/n}$ with $n\simeq 4.2$ and $U_s\simeq 16\mathrm{mm}/\mathrm{s}$. ($\times$) Final value ${\varphi }_{\infty }$ from best fit by Eq. (1) of $\varphi \left(t\right)$ data of Fig. 3 corresponding to compaction experiments at $U_F<U_{Fc}$. Inset: zoom on flow range $U_F/U_{Fc}⩽2$.

Figure 3

Time evolution of the normalized solid fraction $\varphi /{\varphi }_0$ of the packing submitted to an upward continuous flow at three different velocities $U_F/U_{Fc}=0.4$ ($\square$), $0.6$ ($◊$), and $0.7$ ($○$) from an initial packing fraction ${\varphi }_0=0.606$ ($\square$), $0.607$ ($◊$), and $0.603$ ($○$). (—) Best fit by Eq. (1) with ${\varphi }_{\infty }=0.74$ and ($\square$) $B=0.019$ and $\tau =55\mathrm{s}$, ($◊$) $B=0.027$ and $\tau =25\mathrm{s}$, ($○$) $B=0.037$ and $\tau =7\mathrm{s}$. The horizontal gray stripe corresponds to RCP.

Figure 4

(a) Amplitude $B$ and (b) dimensionless characteristic time $\tau U_{Fc}/a$ of the compaction dynamics by upward continuous flow as a function of the dimensionless flow velocity $U_F/U_{Fc}$. ($\circ$) Experimental data and (—) best linear fits.

Figure 5

Normalized solid fraction $\varphi /{\varphi }_0$ as a function of time $t$ for upward flow bursts of dimensionless velocity ($*$) $U_F/U_{Fc}=0.4$ ($T=15$ s) and ($\times$) $U_F/U_{Fc}=2.6$ ($T=11,6$ s) corresponding to the same dimensionless flow displacement $T_f{U}_F/a=0.54$ with accordingly varying burst time duration $T_f$, and the same waiting time $T_w=T-T_f=11$ s. (- - -) Best fits by Eq. (1) with ${\varphi }_{\infty }=0.74$ and with ($*$) $B=0.07$ and $\tau =945\mathrm{s}$, and ($\times$) $B=0.21$ and $\tau =37\mathrm{s}$. ($\square$) Same data and fit as in Fig. 3 for continuous flow at $U_F/U_{Fc}=0.4$. The horizontal thick gray line corresponds to RCP.

Figure 6

Time evolution of the top position $h$ of the packing relative to its initial value $h_0$ made dimensionless by the bead radius $a$ during a few first flow bursts for a burst flow velocity $U_F$ below ($---$), $U_F/U_{Fc}=0.6, T_f/T=0.26$ and above ($————$), $U_F/U_{Fc}=2.6, T_f/T=0.08$ the fluidization threshold $U_{Fc}$. The waiting time $T_w=11$ s has been kept constant, whereas the flow time $T_f$ has been varied to keep constant the fluid displacement $T_f{U}_F$.

Figure 7

(a) Compaction amplitude $B$ and (b) dimensionless characteristic time $\tau U_{Fc}/a$ of the compaction dynamics as a function of the dimensionless flow velocity $U_F/U_{Fc}$ with the same fluid displacement during flow burst $T_f{U}_F/a=0.5$. Inset: Corresponding characteristic number of bursts $N_c=\tau U_{Fc}/a$ as a function of $U_F/U_{Fc}$.