A Mathematical Model for the Flow of a Casson Fluid due to Metachronal Beating of Cilia in a Tube

A mathematical model is developed to study the transport mechanism of a Casson fluid flow inspired by the metachronal coordination between the beating cilia in a cylindrical tube. A two-dimensional system of nonlinear equations governing the flow problem is formulated by using axisymmetric cylindrical coordinates and then simplified by employing the long wavelength and low Reynolds number assumptions. Exact solutions are derived for the velocity components, the axial pressure gradient, and the stream function. However, the expressions for the pressure rise and the volume flow rate are evaluated numerically. The features of the flow characteristics such as pumping and trapping are illustrated and discussed with the help of graphs. It is observed that the volume flow rate is influenced significantly by the width of plug flow region H (p) as well as the cilia length parameter ε. The analysis is also applied and compared with the estimated value of the volume flow rate of epididymal fluid in the ductus efferentes of the human male reproductive tract.