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

TL;DR

This paper investigates how sinusoidal wall waviness affects volumetric flow and hydraulic resistance in viscous flows within circular and annular tubes, highlighting significant differences based on volume constraints and wall motion.

Contribution

It introduces a comparative analysis of constant-volume and constant-mean-radius conditions, deriving a scaling law and examining peristaltic pumping effects in wavy-walled tubes.

Findings

Flow rate and resistance differ by up to 10-50% depending on wave amplitude and volume constraint.

A scaling law relates constant-volume and constant-mean-radius cases for circular tubes.

Volume change effects are significant in peristaltic pumping, affecting flow rates substantially.

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.