A theoretical approach to mathematical modeling of sperm swimming in viscoelastic Ellis fluid in a passive canal

This study aimed at investigating the biomechanics of sperm swimming through a passive channel. The spermatozoa are modeled as a Taylor's swimming sheet which is immersed in a viscoelastic Ellis fluid. A typical small size of microorganisms and their swimming protocol lead to negligible internal forces (or dominant viscous forces). The constitutive equations of sperm-driven flow are derived by employing the creeping flow and long-wavelength approximation. For some initial guesses of a flow rate and a swimming speed, the reduced differential equation is numerically solved via the MATLAB routine bvp5c for large values of the pertinent parameters. To find the unknowns (flow rate and sperm speed), Broyden's algorithm is employed in such a way that refines values of flow rate and swimming speed satisfying the dynamic equilibrium conditions. Using these refined values, one gets the energy diminution. The swimming speed, flow rate, power expended by the swimmer, streamlines pattern and velocity profiles are displayed in graphical figures.