  This paper describes topological kinematics associated with the stirring by
rods of a two-dimensional fluid. The main tool is the Thurston-Nielsen (TN)
theory which implies that depending on the stirring protocol the essential
topological length of material lines grows either exponentially or linearly. We
give an application to the growth of the gradient of a passively advected
scalar, the Helmholtz-Kelvin Theorem then yields applications to Euler flows.
The main theorem shows that there are periodic stirring protocols for which
generic initial vorticity yields a solution to Euler's equations which is not
periodic and further, the $L^\infty$ and $L^1$-norms of the gradient of its
vorticity grow exponentially in time.
