  It has long been known that some microswimmers seem to swim
counter-intuitively faster when the viscosity of the surrounding fluid is
increased, whereas others slow down. This conflicting dependence of the
swimming velocity on the viscosity is poorly understood theoretically. Here we
explain that any mechanical microswimmer with an elastic degree of freedom in a
simple Newtonian fluid can exhibit both kinds of response to an increase in the
fluid viscosity for different viscosity ranges, if the driving is weak. The
velocity response is controlled by a single parameter $\varGamma$, the ratio of
the relaxation time of the elastic component of the swimmer in the viscous
fluid and the swimming stroke period. This defines two velocity-viscosity
regimes, which we characterize using the bead-spring microswimmer model and
analyzing the different forces acting on the parts of this swimmer. The
analytical calculations are supported by lattice-Boltzmann simulations, which
accurately reproduce the two velocity regimes for the predicted values of
$\varGamma$.
