  It is widely believed that the swimming speed, $v$, of many flagellated
bacteria is a non-monotonic function of the concentration, $c$, of
high-molecular-weight linear polymers in aqueous solution, showing peaked
$v(c)$ curves. Pores in the polymer solution were suggested as the explanation.
Quantifying this picture led to a theory that predicted peaked $v(c)$ curves.
Using new, high-throughput methods for characterising motility, we have
measured $v$, and the angular frequency of cell-body rotation, $\Omega$, of
motile Escherichia coli as a function of polymer concentration in
polyvinylpyrrolidone (PVP) and Ficoll solutions of different molecular weights.
We find that non-monotonic $v(c)$ curves are typically due to low-molecular
weight impurities. After purification by dialysis, the measured $v(c)$ and
$\Omega(c)$ relations for all but the highest molecular weight PVP can be
described in detail by Newtonian hydrodynamics. There is clear evidence for
non-Newtonian effects in the highest molecular weight PVP solution.
Calculations suggest that this is due to the fast-rotating flagella `seeing' a
lower viscosity than the cell body, so that flagella can be seen as
nano-rheometers for probing the non-Newtonian behavior of high polymer
solutions on a molecular scale.
