  Micro-organisms expend energy moving through complex media. While propulsion
speed is an important property of locomotion, efficiency is another factor that
may determine the swimming gait adopted by a micro-organism in order to
locomote in an energetically favorable manner. The efficiency of swimming in a
Newtonian fluid is well characterized for different biological and artificial
swimmers. However, these swimmers often encounter biological fluids displaying
shear-thinning viscosities. Little is known about how this nonlinear rheology
influences the efficiency of locomotion. Does the shear-thinning rheology
render swimming more efficient or less? How does the swimming efficiency depend
on the propulsion mechanism of a swimmer and rheological properties of the
surrounding shear-thinning fluid? In this work, we address these fundamental
questions on the efficiency of locomotion in a shear-thinning fluid by
considering the squirmer model as a general locomotion model to represent
different types of swimmers. Our analysis reveals how the choice of surface
velocity distribution on a squirmer may reduce or enhance the swimming
efficiency. We determine optimal shear rates at which the swimming efficiency
can be substantially enhanced com- pared with the Newtonian case. The
non-trivial variations of swimming efficiency prompt questions on how
micro-organisms may tune their swimming gaits to exploit the shear-thinning
rheology. The findings also provide insights into how artificial swimmers
should be designed to move through complex media efficiently.
