Volumetric effects in viscous flows in circular and annular tubes with wavy walls

Abstract

We point out that, in the usual way of specifying a sinusoidal waviness of the wall of a tube of circular cross section, in which the mean radius is kept constant, the interior volume of the tube increases with increasing wave amplitude. We compare this case with the case where the interior volume is kept constant by reducing the mean radius as the wave amplitude increases. We present and compare numerical results of these two cases for steady, pressure driven, laminar viscous flow in a tube with a stationary wavy wall, for both circular and annular tubes. The volume flow rate and the hydraulic resistance can differ in the two cases by as much as 10% for wave amplitudes as small as 20% of the mean radius and as much as 50% for larger wave amplitudes. For a circular tube, we derive a scaling law that relates the two cases based on dimensional analysis, allowing the behavior in the constant-volume case to be determined from that in the constant-mean-radius case. Additionally, we consider peristaltic pumping due to a moving sinusoidal wall wave and show that the volume-change effect is significant even at small wave amplitudes, and that the volume flow rates in the two cases can differ significantly, by as much as 50% as the wave amplitude approaches its maximum value.

Figures (7)

• Figure 1: Schematic diagram of an circular tube with a wavy wall (cf. equation \ref{['eq:sinusoid']}).
• Figure 2: Hydraulic resistance of a wavy circular tube. (a-b) Plots of scaled hydraulic resistance $\mathcal{R}/\mathcal{R}_0$ versus scaled wave amplitude $b/r_0$ for both the constant mean radius and constant volume cases, for two different reference Reynolds numbers and two different wavelengths. The driving axial pressure drop $\Delta p$ is kept constant for each plot. The hydraulic resistance is always greater in the constant-volume case, and it goes asymptotically to infinity as the scaled wave amplitude approaches its limiting value $b/r_0 = r_{*\mathrm{min}}/r_0 = \sqrt{2/3} \doteq 0.8165$ (when the tube is pinched off into a sequence of separate compartments).
• Figure 3: Examples of the velocity field for steady Poiseuille flow in tubes with wavy walls, for the cases with constant mean radius and constant volume, for two different wavelengths ($\lambda/r_0 =$ 1 and 10), two different reference Reynolds numbers ($Re_0 =$ 1 and 100), and wave amplitude $b/r_0 = 0.6$. The left column shows the tube wall profiles, and the color plots show the streamlines, flow speed, and values of the scaled hydraulic resistance $\mathcal{R}/\mathcal{R}_0$. Note especially the case $Re_0 =$ 100, $\lambda/r_0 =$ 10, where there is a recirculating eddy in the constant-mean-radius case but not in the constant-volume case.
• Figure 4: Numerical simulations with equal scaled volume flow rates $\hat{Q}$ and $\hat{Q_*}$ ($Re_{0}$ = $Re_{*0}$ = 1, $\frac{Q_0}{r_0} = \frac{Q_{*0}}{r_*}$, $\frac{\lambda}{r_0} = \frac{\lambda_*}{r_*}$=1, and $\frac{b}{r_0} = \frac{b_*}{r_*}$). (a) To maintain dynamic similarity with the constant-mean-radius case, the pressure drop $\Delta p_*$ in the constant-volume case increases by a factor of $(r_0/r_*)^2$ as the mean radius $r_*$ decreases. (b) For $\frac{b}{r_0} = \frac{b_*}{r_*}$ = 0.6, the constant-mean-radius and constant-volume cases show similar flow fields.
• Figure 5: Annular tube with sinusoidal outer boundary. (a) Scaled hydraulic resistance $\mathcal{R}/\mathcal{R}_0$ versus scaled wave amplitude $b/(r_2-r_1)$ for $Re_0 = 100$, $r_2 = 2r_1$, and $\lambda/r_2 = 1$ and $10$. (b-c) Plots of flow speeds for the constant-mean-radius and constant-volume cases, for $b/(r_2-r_1) = 0.6$.