Analytical and numerical study of creeping flow generated by active spermatozoa bounded within a declined passive tract
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The fluid-swimmer interaction is effective in two ways, i.e. the rheology of the surrounding fluid assists/resists the swimming mechanism and an efficient swimming generates an enhanced fluid flow. In particular the swimming of spermatozoa consists in biorheological hydrodynamic propulsion at low Reynolds numbers, where the viscous damping by far dominates over inertia. In this paper, we study the self-propulsion of a spermatozoon in the human cervix. The organism is modeled as a self-propelling infinite sheet and the cervix is simulated as a two-dimensional channel composed of two rigid walls. We assumed that the cervical walls are declined at a certain angle to the horizontal. The cervical fluid is characterized by the non-Newtonian Carreau model which exhibits shear rate-dependent viscosity, a characteristic of cervical medium. The amplitude of the waves traveling down the declined sheet surface is arbitrary while their wavelength is assumed larger than the declined channel spacing. The flow equations are formulated under this assumption. The analytic expressions of pressure gradient and velocity of the fluid are obtained by utilizing the perturbation technique. These expressions (effective for small values of Carreau parameters) are used in dynamics equilibrium conditions to calculate cell speed and power losses. The propulsive speed and power delivered by the spermatozoa at larger values of the rheological parameters are computed by employing the implicit finite deference technique combined with the Broyden method (modified secant method). The most striking result of the present study is that the optimum speed can be achieved by suitably adjusting the rheological parameters of the cervical fluid and likewise a fast moving organism can generate a better fluid flow.
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