Rheological fluid motion in tube by metachronal waves of cilia

Abstract

This paper presents a theoretical study of a non-linear rheological fluid transport in an axisymmetric tube by cilia. An attempt has been made to explain the role of cilia motion in the transport of fluid through the ductus efferent of the male reproductive tract. The Ostwald-de Waele power-law viscous fluid is considered to represent the rheological fluid. We analyze pumping by means of a sequence of cilia beats from row-to-row of cilia in a given row of cells and from one row of cells to the next (metachronal wave movement). For this purpose, we consider the conditions that the corresponding Reynolds number is small enough for inertial effects to be negligible, and the wavelength-to-diameter ratio is large enough so that the pressure can be considered uniform over the cross section. Analyses and computations of the fluid motion reveal that the time-average flow rate depends on ϵ, a non-dimensional measure involving the mean radius a of the tube and the cilia length. Thus, the flow rate significantly varies with the cilia length. Moreover, the flow rate has been reported to be close to the estimated value 6×10−3 ml/h for human efferent ducts if ϵ is near 0.4. The estimated value was suggested by Lardner and Shack (Lardner, T. J. and Shack, W. J. Cilia transport. Bulletin of Mathematical Biology, 34, 325–335 (1972)) for human based on the experimental observations of flow rates in efferent ducts of other animals, e.g., rat, ram, and bull. In addition, the nature of the rheological fluid, i.e., the value of the fluid index n strongly influences various flow-governed characteristics. An interesting feature of this paper is that the pumping improves the thickening behavior for small values of ϵ or in free pumping (ΔP = 0) and pumping (ΔP > 0) regions.