Non-steady peristaltic transport in finite-length tubes

Abstract

The classical lubrication-theory model of steady peristaltic transport of periodic sinusoidal waves in infinite-length tubes (Shapiro et al. 1969) is generalized to arbitrary wave shape and wavenumber in tubes of finite length. Whereas the classical model is steady in a frame of reference moving with the peristaltic waves, peristaltic transport in a finite-length tube is inherently non-steady. It may be shown, however, that pumping performance is independent of tube length if there exists an integral number of peristaltic waves in the tube. Three particularly interesting characteristics of non-steady peristalsis are described: (i) fluctuations in pressure and shear stress arise due to a non-integral number of waves in the finite-length tube; (ii) retrograde motion of fluid particles during peristaltic transport (reflux) has inherently different behaviour with single peristaltic waves as compared to multiple ‘train waves’, and (iii) finite tube length, the number of peristaltic waves and the degree of tube occlusion affect global pumping performance. We find that, whereas significant increases in pressure and shear stress result from the tube-to-wave length ratio being non-integral, global pumping performance is only slightly degraded by the existence of a non-integral number of waves in the tube during peristaltic transport. Furthermore, the extent of retrograde motion of fluid particles is much greater with single waves than with train waves. These results suggest that in the design and analysis of peristaltic pumps attention should be paid to the unsteady effects of finite tube length and to the differences between single and multiple peristaltic waves.