Abstract
Very low Reynolds number propulsion is a topic of enduring interest due to its importance in biological systems such as sperm migration in the female reproductive tract. Motivated by the fibrous nature of cervical mucus, several recent studies have considered the effect of anisotropic rheology; these studies have generally employed the classical swimming sheet model of G. I. Taylor. The models of Cupples et al. (J. Fluid Mech. vol. 812, 2017, pp. 501–524) and Shi & Powers (Phys. Rev. Fluids vol. 2, 2017, 123102) consider related problems which in a common limit (passive, slightly anisotropic) make different predictions regarding how swimming speed depends on alignment angle. In the present paper we find that this discrepancy is due to missing terms in the analysis of Cupples et al., and that when these terms are correctly included, the models agree in their common limit. We further explore the predictions of the corrected model for both passive and active cases; it is found that for certain combinations of alignment angle and activity parameter, propulsion is halted; in other cases the small amplitude asymptotic expansion is no longer valid, motivating future numerical study.
Highlights
Small organisms swimming at very low Reynolds numbers, for example spermatozoa in cervical mucus, cannot propel themselves by utilising the inertia of the surrounding fluid; time-reversible kinematics result in zero net displacement for the small body
It is immediately seen that the inclusion of the extra term has altered the mean swimming velocity and introduced a dependence on the initial orientation angle φ
Increasing μ2 dominates the impact of the initial orientation angle on the mean swimming velocity, and the anisotropic shear viscosity works to collapse the results back towards the Newtonian value; when both parameters are zero we return to the Newtonian solution as expected
Summary
Small organisms swimming at very low Reynolds numbers, for example spermatozoa in cervical mucus, cannot propel themselves by utilising the inertia of the surrounding fluid; time-reversible kinematics result in zero net displacement for the small body. Taylor’s pioneering study presented the first model of zero-Reynolds-number swimming where time-reversal symmetry is broken by the wave direction (Taylor 1951). This model was formulated as the far-field Stokes flow produced by a swimming motion given by a small amplitude sinusoidal wave, and the mean rate of working was calculated as a measure of the energetic cost of swimming. A solution to the active case is considered in § 3, where a spatially averaged swimming velocity is calculated and discussed
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