  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.
