Weak finite-time Melnikov theory and 3D viscous perturbations of Euler flows

The ordinary differential equations related to fluid particle trajectories are examined through a 3D Melnikov approach. This theory assesses the destruction of 2D heteroclinic manifolds (such as that present in Hill’s spherical vortex) under a perturbation which is neither differentiable in the perturbation parameter ε, nor defined for all times. The rationale for this theory is to analyse viscous flows that are close to steady Euler flows; such closeness in ε can only reasonably be expected in a weak sense for finite times. An expression characterising the splitting of the two-dimensional separating manifold is derived.