Particle motion in unsteady two-dimensional peristaltic flow with application to the ureter

Particle motion in an unsteady peristaltic fluid flow is analyzed. The fluid is incompressible and Newtonian in a two-dimensional planar geometry. A perturbation method based on a small ratio of wave height to wavelength is used to obtain a closed-form solution for the fluid velocity field. This analytical solution is used in conjunction with an equation of motion for a small rigid sphere in nonuniform flow taking Stokes drag, virtual mass, Faxén, Basset, and gravity forces into account. Fluid streamlines and velocity profiles are calculated. Theoretical values for pumping rates are compared with available experimental data. An application to ureteral peristaltic flow is considered since fluid flow in the ureter is sometimes accompanied by particles such as stones or bacteriuria. Particle trajectories for parameters that correspond to calcium oxalates for calculosis and Escherichia coli type for bacteria are analyzed. The findings show that retrograde or reflux motion of the particles is possible and bacterial transport can occur in the upper urinary tract when there is a partial occlusion of the wave. Dilute particle mixing is also investigated, and it is found that some of the particles participate in the formation of a recirculating bolus, and some of them are delayed in transit and eventually reach the walls. This can explain the failure of clearing residuals from the upper urinary tract calculi after successful extracorporeal shock wave lithotripsy. The results may also be relevant to the transport of other physiological fluids and industrial applications in which peristaltic pumping is used.

Figure 1

Geometry of the sinusoidal periodic wave and an immersed particle in the medium.

Figure 2

[(a), (c), and (e)] fluid instantaneous streamlines and [(b), (d), and (f)] axial velocity profiles in a fixed frame at $t=0$ for different values of $Q$ with parameters $\left\{\text{Re}=0\left({\text{Re}}_m=0\right);\varphi =0.6;ϵ=0.1\right\}$. The scaling $U_{\text{rel}}=U/\left[0.75{⟨U\left(X=0\right)⟩}_{\text{mean}}\right]$.

Figure 3

(a) $\mathrm{Re}=0({\mathrm{Re}}_m=0)$, (b) $\mathrm{Re}=0({\mathrm{Re}}_m=0)$, (c) $\mathrm{Re}=50({\mathrm{Re}}_m=5)$, (d) $\mathrm{Re}=50({\mathrm{Re}}_m=5)$, (e) $\mathrm{Re}=100({\mathrm{Re}}_m=10)$, and (f) $\mathrm{Re}=100({\mathrm{Re}}_m=10)$ Fluid instantaneous streamlines and [(b), (d), and (f)] axial velocity profiles for a fixed frame at $t=0$ for different values of Re with parameters $\left\{\varphi =0.6;ϵ=0.1;Q=0\right\}$. The scaling $U_{\text{rel}}=U/\left[0.75{⟨U\left(X=0\right)⟩}_{\text{mean}}\right]$.

Figure 4

(a) $\varphi =0.6$ and (b) $\varphi =0.9$ Pumping curves for different Reynolds numbers with geometrical parameters $ϵ=0.1$ and $\varphi$.

Figure 5

Effect of the Reynolds number on the maximum pressure rise with $ϵ=0.014$ and $\varphi =0.7$. Data points are from the experiments of Weinberg et al. [13]. Dashed curve is theory based on second-order expansion in ${\text{Re}}_m$ of Weinberg et al. [13]. Solid line is present solution up to a second-order expansion in $ϵ$. Viscosities in centipoise: ○, 1; ◻, 6; ▽, 12; △, 14; and ●, 150.

Figure 6

Effect of the Reynolds number on the time–mean flow rate for zero pressure change. Dashed curve is theory based on second-order expansion in ${\text{Re}}_m$ of Weinberg et al. [13]. Solid line is present solution up to a second-order expansion in $ϵ$. Viscosities in centipoise: ○, 1; ◻, 6; ▽, 12; △, 14; and ●, 150.

Figure 7

Schematic of the urinary system and cross section of the kidney.

Figure 8

Regions of trapping or separated flows with parameters $ϵ=0.0654$ and $\text{Re}=84.27$.

Figure 9

(a) $Q=-0.2$, retrograde flow; (b) $Q=0$, zero-pumping; (c) $Q_0=0.327$, free pumping; (d) $Q=0.45$, augmented flow. Particle trajectories with the fluid parameters $\left\{ϵ=0.0654;\varphi =0.8;\text{Re}=84.27\right\}$ for different values of the time–mean flow rate. Particle parameters: for bacteria, $\left\{\alpha =0.0004;S=1.16;\text{St}=5.687\times {10}^{-8}\right\}$; for CaOx, $\left\{\alpha =0.0041;S=1.96;\text{St}=9.997\times {10}^{-6}\right\}$. Particle initial positions are at $x=0$ on $y=\left\{0,\pm 0.45,\pm 0.9\right\}$, with zero initial velocities, integration carried up to $T=3$. Peristalsis acts in the positive $X$ direction and gravity in the negative $Y$ direction. Particle trajectory: solid line, bacteria; dashed line, CaOx.

Figure 10

Effect of gravity on particle motion for bacteria with flow rate $Q_0=0.327$. Fluid and particle parameters $\left\{ϵ=0.0654;\varphi =0.8;\text{Re}=84.27;\alpha =0.0004;S=1.16;\text{St}=5.687\times {10}^{-8}\right\}$. Particle initial positions are $x=0$, $y=\left\{0,\pm 0.45,\pm 0.9\right\}$ with zero initial velocities; integration carried to $T=3$. Solid line is gravity in positive $X$ direction; dashed line is gravity in negative $Y$ direction.

Figure 11

Effect of gravity on particle motion for CaOx with flow rate $Q_0=0.327$. Fluid and particle parameters $\left\{ϵ=0.0654;\varphi =0.8;\text{Re}=84.27;\alpha =0.0041;S=1.96;\text{St}=9.997\times {10}^{-6}\right\}$. Particle initial positions are $x=0$, $y=\left\{0,\pm 0.45,\pm 0.9\right\}$ with zero initial velocities; integration carried to $T=3$. Solid line is gravity in positive $X$ direction; dashed line is gravity in negative $Y$ direction.

Figure 12

Particle trajectory for bacteria with $Q=0$ and gravity in positive $X$ direction. Fluid and particle parameters $\left\{ϵ=0.0654;\text{Re}=84.27;\alpha =0.0004;S=1.16;\text{St}=5.687\times {10}^{-8}\right\}$. Initial position (0,0.9) and zero initial velocity, integration to $T=10$. Solid line is $\varphi =0.8$; dashed line is $\varphi =0.99$.

Figure 13

Particle longitudinal path for bacteria with $Q=0$ and gravity in positive $X$ direction. Fluid and particle parameters $\left\{ϵ=0.0654;\text{Re}=84.27;\alpha =0.0004;S=1.16;\text{St}=5.687\times {10}^{-8}\right\}$. Initial position (0,0.9) and zero initial velocity, integration to $T=10$. Solid line is $\varphi =0.8$; dashed line is $\varphi =0.99$. Plot is in dimensional units.

Figure 14

Comparison of the effects of forces on particle motion for bacteria with $Q=Q_0$. Fluid and particle parameters $\left\{ϵ=0.0654;\varphi =0.8;\text{Re}=84.27;\alpha =0.0004;S=1.16;\text{St}=5.687\times {10}^{-8}\right\}$. Solid line is all forces; dashed line is Stokes drag, gravity, and Basset; dotted line is Stokes drag and gravity; integration carried to $T=20$. Gravity acts in the negative $Y$ direction.

Figure 15

Comparison of the effects of forces on the particle motion for CaOx with $Q=Q_0$. Fluid and particle parameters $\left\{ϵ=0.0654;\varphi =0.8;\text{Re}=84.27;\alpha =0.0041;S=1.96;\text{St}=9.997\times {10}^{-6}\right\}$. Solid line is all forces; dashed line is Stokes drag, gravity, and Basset; dotted line is Stokes drag and gravity; integration carried to $T=20$. Gravity acts in the negative $Y$ direction.

Figure 16

(a) Initial, $T=0$, (b) $T=1$, (c) $T=3$, (d) $T=5$, (e) $T=7$. Position of 1280 bacteria particles initially evenly distributed within a rectangle defined by $X=\left\{-0.5,0.5\right\}$ and $Y=\left\{-0.8,0.8\right\}$. Fluid and particle parameters $\left\{ϵ=0.0654;\varphi =0.8;\text{Re}=84.27;\alpha =0.0004;S=1.16;\text{St}=5.68\times {10}^{-8}\right\}$. Gravity acts in the negative $Y$ direction.

Figure 17

(a) Initial, $T=0$, (b) $T=1$, (c) $T=3$, (d) $T=5$, (e) $T=7$.Position of 400 CaOx particles initially evenly distributed within a rectangle defined by $X=\left\{-0.5,0.5\right\}$ and $Y=\left\{-0.8,0.8\right\}$. Fluid and particle parameters $\left\{ϵ=0.0654;\varphi =0.8;\text{Re}=84.27;\alpha =0.0041;S=1.96;\text{St}=9.997\times {10}^{-6}\right\}$. Gravity acts in the negative $Y$ direction.

Figure 18

(a) $T=1$ (bacteria), (b) $T=1$ (CaOx), (c) $T=3$ (bacteria), (d) $T=3$ (CaOx), (e) $T=5$ (bacteria), (f) $T=5$ (CaOx), (g) $T=7$ (bacteria), (h) $T=7$ (CaOx). Comparison of recirculations inside the bolus corresponding to Figs. 16, 17.