This paper is concerned with the investigation of stratified flows of viscoelastic fluids by three-dimensional numerical simulation. The aim is to elucidate the mechanism behind the phenomenon of encapsulation in the coextrusion of polymer fluids in which one component progressively encapsulates the other along the coextrusion die. It is commonly accepted that the main reason for fluid encapsulation is the viscosity difference between the two components, since in most of the experiments, the less viscous fluid is observed to encapsulate the more viscous one. The occasional occurrence of the opposite case is usually explained as an effect of the elasticity difference between the fluids, which is responsible for a so-called "elastic encapsulation" counteracting the effects of the "viscous encapsulation." Three-dimensional simulations showed that the encapsulation sense does not always agree with the criteria of elastic or viscous encapsulations but can only be correctly predicted by the sign of the second normal stress difference jump across the interface separating the two fluids. Moreover, the onset of encapsulation, which is sometimes improperly attributed to either a viscous or elastic flow instability, is shown to be a reaction to either a pressure or a second normal stress difference unbalance on both sides of the initially flat interface. Due to this nonequilibrium condition, the interface can react either by shifting in the normal direction or by increasing its curvature. These two mechanisms were examined separately and characterized by their own time and length scales. (C) 2014 The Society of Rheology.