Phase transition in peristaltic transport of frictionless granular particles

Flows of dissipative particles driven by the peristaltic motion of a tube are numerically studied. A transition from a slow “unjammed” flow to a fast “jammed” flow is found through the observation of the flow rate at a critical width of the bottleneck of a peristaltic tube. It is also found that the average and fluctuation of the transition time, and the peak value of the second moment of the flow rate exhibit power-law divergence near the critical point and that these variables satisfy scaling relationships near the critical point. The dependence of the critical width and exponents on the peristaltic speed and the density is also discussed.

Figure 1

(a) Snapshot of our simulation showing some of the parameters in our model. See the text for details. (b) Sequential snapshots of the simulation for $w=2.0$, $\dot{\epsilon }=1.83\times {10}^{-1}$, and $\overline{\rho }=2.96\times {10}^{-1}$. Time increases from left to right, and the peristaltic wave of the wall progresses upward. The system remains in the unjammed flow regime when $t<\tau$. (c) Snapshot of stationary state for the parameters used in (b) showing the jammed flow regime when $t_{}^{}>\tau$. Dark (red) spheres and light (green) spheres are transported particles and particles embedded in the wall, respectively. Wall particles facing the reader are not visualized to show the transported particles clearly.

Figure 2

(a) Time evolutions of the normalized flow rate $J_0(t_{}^{},\Delta t_{}^{})/\tilde{J}$ with $\tilde{J}\equiv Nc/L$ for $w=2.2448$, $\dot{\epsilon }=1.83\times {10}^{-1}$, and $\overline{\rho }=2.96\times {10}^{-1}$. The definition of $J_0(t_{}^{},\Delta t_{}^{})$ is given by Eq. (9). The observation times $\Delta t_{}^{}$ for $J_0(t_{}^{},\Delta t_{}^{})$ are set to $T$, $2T$, and $5T$, where $T=5.48$ is the period of the peristaltic wave. (b) Time evolutions of the normalized flow rates $J\left(t_{}^{}\right)/\tilde{J}$ and $J_0(t_{}^{},5T)/\tilde{J}$. The parameters are the same as those in (a). (c) Typical time evolution of normalized flow rate $J/\tilde{J}$ with $\dot{\epsilon }=1.83\times {10}^{-1}$ and $\overline{\rho }=2.96\times {10}^{-1}$. (d) Same plot as (c) for $w=3.24$ to show the definition of the transition time $\tau$. See text for details.

Figure 3

(a) Transition time $\tau$ as a function of $w_{\mathrm{c}}-w$. (b) Scaling function $f\left(x\right)$ with $x=(w_{\mathrm{c}}-w)/\dot{\epsilon }{}^{\nu }$ and $\nu =3/2$; see Eq. (10). (c) Fluctuation of transition time ${\chi }_{\tau }$ as a function of $w_{\mathrm{c}}-w$. (d) Scaling function $g\left(x\right)$ with $x=(w_{\mathrm{c}}-w)/\dot{\epsilon }{}^{\nu }$; see Eq. (11). For all figures, we use the density $\overline{\rho }=2.96\times {10}^{-1}$, and circles, squares, triangles, and inverted triangles correspond to the data for $\dot{\epsilon }=3.65\times {10}^{-1},1.83\times {10}^{-1},9.13\times {10}^{-2}$, and $4.56\times {10}^{-2}$, respectively.

Figure 4

(a) Normalized stationary flow rate $J_{\mathrm{st}}/\tilde{J}$ as a function of $w$ for $\overline{\rho }=2.96\times {10}^{-1}$ and various $\dot{\epsilon }$, where the dashed line is given by Eq. (13). The symbols are the same as those in Fig. 3. (b) Plot of $w_{\mathrm{c}}$ as a function of $\dot{\epsilon }$ for $\overline{\rho }=2.96\times {10}^{-1}$. (c) Hysteresis loop of flow rate for $\overline{\rho }=2.96\times {10}^{-1}$ and $\dot{\epsilon }=1.83\times {10}^{-1}$. Circles and squares correspond to $J_{\mathrm{st}}/\tilde{J}=0$ and 1, respectively.

Figure 5

(a) Normalized stationary flow rate $J_{\mathrm{st}}/\tilde{J}$ as a function of $w$ for various $\overline{\rho }$. The dashed line shows Eq. (13). Circles, squares, triangles, inverted triangles, and diamonds correspond to $\overline{\rho }=1.48\times {10}^{-1},2.96\times {10}^{-1},4.44\times {10}^{-1},5.92\times {10}^{-1}$, and $7.40\times {10}^{-1}$, respectively. (b) Unnormalized stationary flow rate. (c) Plot of $w_{\mathrm{c}}$ as a function of $\overline{\rho }$. (d) Plot of $\alpha$ as a function of $\overline{\rho }$. All figures are obtained for $\dot{\epsilon }=1.83\times {10}^{-1}$.

Figure 6

(a) Snapshot of stationary state for $w=2.0$, $\dot{\epsilon }=1.83\times {10}^{-1}$, and $\overline{\rho }=7.40\times {10}^{-2}$. Although particles do not become stuck at bottlenecks, the stationary flow rate $J_{\mathrm{st}}$ reaches $\tilde{J}$. (b) Snapshot of stationary state for $w=2.0$, $\dot{\epsilon }=1.83\times {10}^{-1}$, and $\overline{\rho }=7.40\times {10}^{-1}$.

Figure 7

(a) Second moment of the flow rate ${\chi }_J^{}$ as a function of $t_{}^{}$. We set $\dot{\epsilon }=1.8\times {10}^{-1}$, and curves from left to right correspond to $w=2.0,2.12,2.2,2.232,2.24$, and $2.244$. (b) Scaling function $h\left(x\right)$ of peak values of ${\chi }_J^{}$, ${\chi }_J^{\max }$, with $x=(w_{\mathrm{c}}-w)/\dot{\epsilon }{}^{\nu }$; see Eq. (14). Circles, squares, and triangles correspond to $\dot{\epsilon }=1.83\times {10}^{-1},9.13\times {10}^{-2}$, and $4.56\times {10}^{-2}$, respectively. Both figures are obtained for $\overline{\rho }=2.96\times {10}^{-1}$.

Figure 8

Schematic diagram of the configuration of wall particles in the ${\theta }_{}^{}$-$z_{}^{}$ plane. Here, $R=a+d_{}^{}/2$.