Waving transport and propulsion in a generalized Newtonian fluid

Cilia and flagella are hair-like appendages that protrude from the surface of a variety of eukaryotic cells and deform in a wavelike fashion to transport fluids and propel cells. Motivated by the ubiquity of non-Newtonian fluids in biology, we address mathematically the role of shear-dependent viscosities on both the waving flagellar locomotion and ciliary transport by metachronal waves. Using a two-dimensional waving sheet as model for the kinematics of a flagellum or an array of cilia, and allowing for both normal and tangential deformation of the sheet, we calculate the flow field induced by a small-amplitude deformation of the sheet in a generalized Newtonian Carreau fluid up to order four in the dimensionless waving amplitude. Shear-thinning and shear-thickening fluids are seen to always induce opposite effects. When the fluid is shear-thinning, the rate of working of the sheet against the fluid is always smaller than in the Newtonian fluid, and the largest gain is obtained for antiplectic metachronal waves. Considering a variety of deformation kinematics for the sheet, we further show that in all cases transport by the sheet is more efficiency in a shear-thinning fluid, and in most cases the transport speed in the fluid is also increased. Comparing the order of magnitude of the shear-thinning contributions with past work on elastic effects as well as the magnitude of the Newtonian contributions, our theoretical results suggest that the impact of shear-dependent viscosities on transport could play a major biological role.