On viscous propulsion in active transversely isotropic media

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angle and activity parameter, propulsion is halted; in other cases the small amplitude asymptotic expansion is no longer valid, motivating future numerical study.