Instability of coextruded elastic liquids at high Weissenberg number

We reconsider the linear stability of pressure-driven flow of two Oldroyd-B or UCM fluids symmetrically placed in a channel or a pipe (a “coextrusion flow”). The fluids have matched viscosities but different relaxation times. Inertia and surface tension are neglected. We focus on the high Weissenberg number limit, where the distance travelled by a fluid particle in a relaxation time is large compared to the width of the channel. Given a prescribed location of the unperturbed interface, the growth rate at sufficiently high wavenumber can be deduced from previous analysis of Couette flow [Y. Renardy, Stability of the interface in two-layer Couette flow of upper convected Maxwell liquids, J. Non-Newtonian Fluid Mech. 28 (1988) 99–115; K. Chen, D.D. Joseph, Elastic short-wave instability in extrusion flows of viscoelastic liquids, J. Non-Newtonian Fluid Mech. 42 (1992) 189–211; J.C. Miller, J.M. Rallison, Interfacial instability between sheared elastic liquids in a channel, J. Non-Newtonian Fluid Mech. 143 (2007) 71–87; however, we find that the structure of the perturbation flow differs from the Couette case. A new regime with different stability properties was first observed in a companion paper [J.C. Miller, J.M. Rallison, Interfacial instability between sheared elastic liquids in a channel, J. Non-Newtonian Fluid Mech. 143 (2007) 71–87]. In this narrow-core regime the fraction of the channel occupied by the inner fluid is small and the wavenumber is large. We examine the corresponding distinguished limit and demonstrate that instability can occur at intermediate wavenumbers in flow geometries for which the long-wave and short-wave limits are both stable, contradicting some literature claims [P. Laure, H.L. Meur, Y. Demay, J.C. Saut, S. Scotto, Linear stability of multilayer plane Poiseuille flows of Oldroyd-B fluids, J. Non-Newtonian Fluid Mech. 71 (1997) 1–23; S. Scotto, P. Laure, Linear stability of three-layer Poiseuille flow for Oldroyd-B fluids, J. Non-Newtonian Fluid Mech. 83 (1999) 71–92]. The wavespeed of this mode can be substantially higher than the speed of the base flow. The perturbation grows on the time-scale of the shorter relaxation time of the two fluids. Although most of the analysis is performed for asymptotically large Weissenberg numbers, the instability is found to persist for values which should be experimentally accessible. For core-annular pipe flow the curvature of the unperturbed interface plays a role, and the stability characte