Liquid transport produced by a cluster of peristaltic contractions in a circular channel

Clustered contractions are observed in the human small intestine after fatty meals or in patients with gastrointestinal diseases. We investigate flow produced by a cluster of peristaltic contractions propagating along a circular channel for a finite Reynolds number $\mathrm{Re}\sim {10}^2$ and a finite contraction width (wavelength) $\lambda /D\sim 1$. We perform a computational fluid dynamics analysis of a Newtonian liquid using the cumulant lattice Boltzmann method where zero pressure is given to both ends of the channel. We show that clustered contractions cause retrograde flow over the cluster and fluid trapping in the upstream and downstream regions of the cluster. Flow rate is not proportional to the number of peristaltic waves in a cluster, in particular, for large contraction ratio (amplitude). To understand the underlying mechanism, we simulate flow produced by an isolated contraction in channels with different lengths. The magnitude of pressure gradient produced by the isolated contraction does not decrease largely with increasing the channel length, minimizing a decrease in flow rate. When flow rates of clustered contractions with different numbers of waves are plotted as a function of the channel length per wave, the flow rates collapse onto a single curve of each contraction ratio. In addition, the flow rates of the clustered contractions converge to the flow rates of infinitely periodic contractions. We also compare the characteristics of the flow rate for $\mathrm{Re}\sim {10}^2$ and $\lambda /D\sim 1$ to those for $\mathrm{Re}\to 0$ and $\lambda /D\to \infty$.

Figure 1

Wall geometry for $N_p=2$, where $L$ is the channel length, $D$ is the channel diameter, $D_p$ is the contraction amplitude, and $\lambda$ is the contraction width. The contraction ratio is given by $D_p/D$.

Figure 2

(a) Effect of the lattice spacing $\mathrm{\Delta }x$ and the relaxation time $\tau$ on numerical result. Flow rate $Q$ is shown for $N_p=1,\mathrm{Re}={10}^2,D_p/D=0.75,\lambda /D=1$, and $L/D=5$. (b) Flow rate of periodic waves for $\mathrm{Re}={10}^{-1}$ as a function of $\lambda /D$. Dashed lines show analytical solutions for $\mathrm{Re}\to 0$ and $\lambda /D\to \infty$ [10].

Figure 3

Effect of Re on $Q/D^{2}V$ for $N_p=2,L/D=15$, and $\lambda /D=1$.

Figure 4

Stationary frame streamlines with velocity magnitude at the channel center ($x\text{−}y$ cross section) for $N_p=2,L/D=15$, and $\lambda /D=1$: (a) $D_p/D=0.25$; (b) $D_p/D=0.50$; (c) $D_p/D=0.75$. Contracted regions are magnified.

Figure 5

The $x$-directional velocity profiles at (a) the most contracted region of the first wave ($x=x_{p,1}+Vt$), and (b) the widest region between the first and second waves $[x=(x_{p,1}+x_{p,2})/2+Vt]$ for $N_p=2,L/D=15$, and $\lambda /D=1$. Corresponding regions are shown in (c).

Figure 6

Wave frame streamlines at the channel center ($x\text{−}y$ cross section) for $N_p=2,L/D=15$, and $\lambda /D=1$: (a) $D_p/D=0.25$; (b) $D_p/D=0.50$; (c) $D_p/D=0.75$. Contracted regions are magnified.

Figure 7

Relationship between $N_p$ and $Q/Q_{N_p=1}$, where $Q_{N_p=1}$ is the flow rate for $N_p=1.L/D=15$, and $\lambda /D=1$. The dotted line represents the proportional line.

Figure 8

Effect of channel length on flow rate for isolated contraction ($N_p=1$), where $Q_{L/D=15}$ is the flow rate for $L/D=15$.

Figure 9

Cross-sectional mean pressure along the channel for (a) $D_p/D=0.25$ and (b) $D_p/D=0.75$. Peristaltic wave is located at $x=0$. Arrows indicate pressure gradient produced by contractions.

Figure 10

Comparison of the cross-sectional mean pressure distribution for three cases: $N_p=1$ with $L/D=5,N_p=2$ with $L/D=10$ and $N_p=3$ with $L/D=15$ where the position and pressure are normalized as $x/L$ and $p{D}^2/\mu VL$, respectively. Arrow indicates pressure gradient.

Figure 11

Flow rate $Q/Q_{\max }$ as a function of the channel length per wave. Black square symbols represent flow rates of periodic contractions.

Figure 12

Effect of the contraction ratio of the second wave on flow rate, where $D_{p1}$ and $D_{p2}$ denote the contraction ratios of the first and second waves, respectively.

Figure 13

Analytical solution for $\mathrm{Re}\to 0$ and $\lambda /D\to \infty$. (a) Relationship between $N_p$ and $Q/Q_{N_p=1}$. (b) Flow rate $Q/Q_{\max }$ as a function of $L/N_p\lambda$. (c) Comparison of the cross-sectional mean pressure distribution for three cases: $N_p=1\text{–}3$ with $L/N_p\lambda =5$.

Figure 14

Trajectories of ten tracer particles for five wave periods, where $\lambda /D=1$ and $D_p/D=0.5$. (a) isolated and (b) periodic contraction waves at $\mathrm{Re}={10}^{-1}$, (c) isolated, and (d) periodic contraction waves at $\mathrm{Re}={10}^2$. The initial position of particles are shown by circles. The particles are initially located at both the sides of the isolated wave (a) and (c) and at the most contracted and the widest regions for the clustered waves (b) and (d). The initial positions of the waves are also shown.

Figure 15

Particle travel distance for five wave periods, where $\lambda /D=1$ and $D_p/D=0.5$. (a) isolated, (b) periodic contraction waves at $\mathrm{Re}={10}^{-1}$, (c) isolated, and (d) periodic contraction waves at $\mathrm{Re}={10}^2$. Some 1000 particles are initially located at random positions within $-1.5D<x<1.5D$ on the $x\text{−}y$ plane.