  Topological chaos has emerged as a powerful tool to investigate fluid mixing.
While this theory can guarantee a lower bound on the stretching rate of certain
material lines, it does not indicate what fraction of the fluid actually
participates in this minimally mandated mixing. Indeed, the area in which
effective mixing takes place depends on physical parameters such as the
Reynolds number. To help clarify this dependency, we numerically simulate the
effects of a batch stirring device on a 2D incompressible Newtonian fluid in
the laminar regime. In particular, we calculate the finite time Lyapunov
exponent (FTLE) field for three different stirring protocols, one topologically
complex (pseudo-Anosov) and two simple (finite-order), over a range of
viscosities. After extracting appropriate measures indicative of both the
amount of mixing and the area of effective mixing from the FTLE field, we see a
clearly defined Reynolds number range in which the relative efficacy of the
pseudo-Anosov protocol over the finite-order protocols justifies the
application of topological chaos. More unexpectedly, we see that while the
measures of effective mixing area increase with increasing Reynolds number for
the finite-order protocols, they actually exhibit non-monotonic behavior for
the pseudo-Anosov protocol.
