Waving transport and propulsion in a generalized Newtonian fluid

Abstract

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. The net flow induced far from the sheet can be interpreted either as a net pumping flow or, in the frame moving with the sheet, as a swimming velocity. At leading order (square in the waving amplitude), the net flow induced by the waving sheet and the rate of viscous dissipation is the same as the Newtonian case, but is different at the next nontrivial order (four in the waving amplitude). If the sheet deforms both in the directions perpendicular and parallel to the wave progression, the shear-dependence of the viscosity leads to a nonzero flow induced in the far field while if the sheet is inextensible, the non-Newtonian influence is exactly zero. 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, which beyond the Carreau model are valid for a wide class of generalized Newtonian fluids, suggest that the impact of shear-dependent viscosities on transport could play a major biological role.