A two layer shear flow model with piecewise uniform potential vorticity is used to show that circulation or free exchange of parcels in an isopycnal layer of uniform potential vorticity can be greatly inhibited by a strong potential vorticity front in another isopycnal layer. On the other hand, a net mass transfer across the edge of a shear flow can be produced by the initial presence of a strong mesoscale eddy. The entrainment resulting from this eddy-shear flow interaction is defined and quantified for an ensemble of initial realizations. It is suggested that dynamically similar entrainment processes, occurring in more realistic potential vorticity distributions, are important in coupling the recirculation gyres to the Gulf Stream, thereby providing the observed downstream increase in transport.