Peristaltic pumping in thin non-axisymmetric annular tubes

Abstract

The two-dimensional laminar flow of a viscous fluid induced by peristalsis due to a moving wall wave has been studied previously for a rectangular channel, a circular tube and a concentric circular annulus. Here, we study peristaltic flow in a non-axisymmetric annular tube: in this case, the flow is three-dimensional, with motions in the azimuthal direction. This type of geometry is motivated by experimental observations of the pulsatile flow of cerebrospinal fluid along perivascular spaces surrounding arteries in the brain, which is at least partially driven by peristaltic pumping due to pulsations of the artery. These annular perivascular spaces are often eccentric and the outer boundary is seldom circular: their cross-sections can be well matched by a simple, adjustable model consisting of an inner circle (the outer wall of the artery) and an outer ellipse (the outer edge of the perivascular space), not necessarily concentric. We use this geometric model as a basis for numerical simulations of peristaltic flow: the adjustability of the model makes it suitable for other applications. We concentrate on the general effects of the non-axisymmetric configuration on the flow and do not attempt to specifically model perivascular pumping. We use a finite-element scheme to compute the flow in the annulus driven by a propagating sinusoidal radial displacement of the inner wall. Unlike the peristaltic flow in a concentric circular annulus, the flow is fully three-dimensional: azimuthal pressure variations drive an oscillatory flow in and out of the narrower gaps, inducing an azimuthal wiggle in the streamlines. We examine the dependence of the flow on the elongation of the outer elliptical wall and the eccentricity of the configuration. We find that the time-averaged volumetric flow is always in the same direction as the peristaltic wave and decreases with increasing ellipticity or eccentricity. The additional shearing motion in the azimuthal direction will increase mixing and enhance Taylor dispersion in these flows, effects that might have practical applications.

Keywords: peristaltic pumping.

Figure 1.

Cross-sections of the annular tube for the concentric elliptical annulus model (a) and the eccentric circular annulus model (b).

Figure 2.

The ‘Finer, Fine BL’ meshing scheme (bold in table 3) applied to the cross-sections of the most eccentric circular annulus (a), the concentric circular annulus (b) and the most elliptical concentric annulus (c).

Figure 3.

Hydraulic resistances (dimensionless) for steady Poiseuille flow calculated using our 3-D code and compared with the corresponding values calculated by Tithof et al. (2019). The cross-sectional area of the annulus is kept fixed. (a) Hydraulic resistances of concentric elliptical annuli of different ellipticity α/β. (b) hydraulic resistances of eccentric circular annuli of different eccentricity ϵ. Axial velocity profiles for the same elliptic annulus, computed by Tithof et al. (2019) (c) and our 3-D code (d). The maximum axial velocities agree within 1 %.

Figure 4.

Comparison of the instantaneous volumetric flow rate Q* given by the test simulation of peristaltic pumping in a concentric circular annulus with that of the analytical solution (in Appendix A), both plotted for one period of the peristaltic wave. (Parameter values for this simulation are given in the text.)

Figure 5.

Cross-sectional pressure distribution and velocity fields at the middle cross-section (one wavelength from either end) of the concentric elliptical annulus model with ellipticity α/β = 1.667. Values are plotted at three different phases of the peristaltic wave: beginning, middle and end (left to right). The displacement of the wall wave is plotted above at the top, with red vertical dashed lines marking the three times of the corresponding cross-sections. The colour scale indicates the magnitude of the dimensionless velocity magnitude, and the copper scale indicates the dimensionless pressure.

Figure 6.

Instantaneous streamlines for pumping in concentric annuli with ellipticity α/β equal to 1.00 (circular), 1.224 and 1.667. For the concentric circular annulus (a) the flow is axisymmetric and the streamlines wiggle only in the radial direction. When the outer wall of the annulus is slightly flattened into an ellipse (b), the flow becomes three-dimensional, with an oscillating azimuthal velocity component, and the streamlines also wiggle in the azimuthal direction. For substantial flattening (c), the azimuthal velocity is significant, as are the azimuthal wiggles in the streamlines. The colour and copper scales indicate values of the dimensionless velocity magnitude and pressure.

Figure 7.

As in figure 5, but for an eccentric circular annulus model with eccentricity ϵ = 0.349.

Figure 8.

As in figure 6, but for circular annuli with eccentricity ϵ equal to 0, 0.225 and 0.349. The instantaneous streamlines wiggle only in the radial direction in the concentric annulus (a), but also wiggle increasingly in the azimuthal direction with increasing eccentricity (b,c).

Figure 9.

Time-averaged volumetric flow rate (dimensionless) plotted as a function of the ellipticity of the elliptic annulus model (a), and as a function of eccentricity of the eccentric circular annulus model (b).