Explicit invariant manifolds and specialised trajectories in a class of unsteady flows

A class of unsteady two- and three-dimensional velocity fields for which the associated stable and unstable manifolds of the Lagrangian trajectories are explicitly known is introduced. These invariant manifolds form the important time-varying flow barriers which demarcate coherent fluids structures, and are associated with hyperbolic trajectories. Explicit expressions are provided for time-evolving hyperbolic trajectories (the unsteady analogue of saddle stagnation points), which are proven to be hyperbolic in the sense of exponential dichotomies. Elliptic trajectories (the unsteady analogue of stagnation points around which there is rotation, i.e., the "centre of a vortex") are similarly explicitly expressed. While this class of models possesses integrable Lagrangian motion since formed by applying time-dependent spatially invertible transformations to steady flows, their hyperbolic/elliptic trajectories can be made to follow any user-specified path. The models are exemplified through two classical flows: the two-dimensional two-gyre Duffing flow and the three-dimensional Hill's spherical vortex. Extensions of the models to finite-time and nonhyperbolic manifolds are also presented. Given the paucity of explicit unsteady examples available, these models are expected to be useful testbeds for researchers developing and improving diagnostic methods for tracking flow structures in genuinely time-dependent flows. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4769979]