Simple analytic model for peristaltic flow and mixing

Peristaltic flows occur when fluid in a channel is driven by periodic, traveling wall deformations, as in industrial peristaltic pumps, urethras, stomachs, and cochleae. Peristaltic flows often vary periodically at every point in space but nonetheless cause net transport and mixing of solutes because of Lagrangian (Stokes) drift. Direct numerical simulation can predict peristaltic flows but is computationally expensive, particularly for determining functional relationships between drive parameters and transport or mixing. We present a simple analytic model of peristaltic flow that expresses flow velocity and drift velocity in terms of deformation speed and amplitude. The model extends beyond prior studies by allowing arbitrary wave forms via Fourier series. To validate our analytic model, we present experiments and simulations; both closely match the analytic model over a range of deformation speeds and amplitudes. We demonstrate the applicability of the model by quantifying variations in the thickness of the reflux region (where fluid drifts opposite the direction of travel of deformations) and by modeling mixing in the cochlea, which is promoted by peristaltic flow.

Figure 1

Sketch of the analytic model's domain. The domain is an infinite two-dimensional channel of mean width $L$ which is periodic in the $x$ direction, with a spatial period $\lambda$. Fluid flows because one boundary is deformed in a periodic shape $\eta (x,t)$ moving with constant speed $c$ in the $x$ direction and characteristic amplitude $A$.

Figure 2

Sketch of the experimental apparatus. Not shown are the tubing and water reservoirs that allow for the system to be filled and to produce different pressure boundary conditions if desired. Flow is driven via the actuation of the stepper motors, which deform the rubber sheet to produce a shape that closely approximates a periodic wave traveling with constant speed in the $x$ direction.

Figure 3

Comparison of the root-mean-square velocity in the bulk flow region in experiments, modeling, and simulations, with $\epsilon =0.01$ (top panel) and $\epsilon =0.05$ (bottom panel). The wavelength was held constant at $\lambda =30$ cm.

Figure 4

Comparison of $x$-direction velocity profiles in experiments, modeling, and simulations. Each black line indicates the location of the corresponding profiles. Profiles are plotted on two separate panels only to allow enlargement for the sake of visibility. Parameters: $c$ = 20 cm/s, $\lambda$ = 30 cm, $\epsilon =$ 0.01.

Figure 5

Comparison of vorticity fields from the analytic model (a) and simulation (b), with the difference plotted in (c). In all three, vorticity is normalized by the absolute maximum vorticity of the model, and $\lambda =500\mu \mathrm{m}$, $L=16\mu \mathrm{m}$, $\epsilon =0.13\%$, and $c=60\mathrm{m}$/s.

Figure 6

Examples of (a) simulated, (b) analytic, and (c) experimentally measured particle path lines, with $c=1$ m/s, $\lambda =30$ cm, and $\epsilon =3\%$. Black points mark particle locations a period apart.

Figure 7

(a) Comparison of the normalized drift velocity profile $u_{\text{drift}}/\text{max}\left(u_{\text{drift}}\right)$ at different Reynolds numbers Re. The black line marks the zero velocity point. (b) Location of the edge of the reflux region, as predicted by the analytic model. The red line is obtained through numerical integration of the Eulerian velocity field ($u_{\mathrm{drift}}$). The blue line is the analytic approximation ($U_{\mathrm{Lag}}$). The dots are the edge location for the curves on (a), and the color matches the relevant Reynolds number. As the Reynolds number increases, the reflux region shrinks.

Figure 8

Variation of the mean drift velocity $U_{\mathrm{drift}}$ with wave speed $c$ and normalized deformation amplitude $\epsilon$, and the calculated Reynolds number. The wavelength was held constant at $\lambda =30$ cm.

Figure 9

Change of the concentration of a passive scalar subjected to both diffusion and the modeled Lagrangian mean flow or subjected to diffusion alone. In the case with flow, $\lambda =200\mu \mathrm{m}$, $L=50\mu \mathrm{m}$, $\epsilon =0.2\%$, and $c=50$ m/s. In both cases, $D=7\times {10}^{-10}{\mathrm{m}}^2$/s. Peristaltic flow causes more mixing than diffusion alone.