  Many microorganisms find themselves immersed in fluids displaying
non-Newtonian rheological properties such as viscoelasticity and shear-thinning
viscosity. The effects of viscoelasticity on swimming at low Reynolds numbers
have already received considerable attention, but much less is known about
swimming in shear-thinning fluids. A general understanding of the fundamental
question of how shear-thinning rheology influences swimming still remains
elusive. To probe this question further, we study a spherical squirmer in a
shear-thinning fluid using a combination of asymptotic analysis and numerical
simulations. Shear-thinning rheology is found to affect a squirming swimmer in
nontrivial and surprising ways; we predict and show instances of both faster
and slower swimming depending on the surface actuation of the squirmer. We also
illustrate that while a drag and thrust decomposition can provide insights into
swimming in Newtonian fluids, extending this intuition to problems in complex
media can prove problematic.
