Interfacial instability between sheared elastic liquids in a channel

We consider the linear stability of the interface between two sheared elastic liquids at large Weissenberg number (Wi) with negligible inertia. The liquids are of Oldroyd-B or UCM type and have matched viscosity. In UCM liquids, Renardy [Y. Renardy, Stability of the interface in two-layer Couette flow of upper convected Maxwell liquids, J. Non-Newton. Fluid Mech. 28 (1988) 99-115] found a purely elastic instability for short-waves in the absence of surface tension for which the perturbation flow decays exponentially away from the interface. For UCM liquids at large Wi we show that this instability persists even though the wavelength is larger than the channel width and the disturbance occupies the entire channel. Surprisingly, the growth rate is not affected by the location of the walls, even though the mode structure is altered. This analysis suggests a reappraisal of the appropriateness of the short-wave and long-wave classifications for instabilities of viscoelastic liquids in order to accommodate the additional length scale introduced by fluid velocity and relaxation. The instability persists for Oldroyd-B liquids even as the elastic contribution to viscosity approaches zero. Surprisingly too, the inclusion of surface tension does not affect the asymptotic growth rate at large wavenumber. When more modest values of Wi are considered, we find parameter values for which arbitrarily large surface tension reduces the growth rate but does not stabilize the flow; previously proposed mechanisms based on the interface displacement are therefore inadequate to explain the instability. Because the instability is locally generated, it appears in other high Wi flows with interfaces, both in channels and in pipes.