3-DIMENSIONAL INSTABILITY OF VISCOELASTIC ELLIPTIC VORTICES

An analysis of the three-dimensional instability of two-dimensional vi scoelastic elliptical flows is presented, extending the inviscid analy sis of Bayly (1986) to include both viscous and elastic effects. The p roblem is governed by three parameters: E, a geometric parameter relat ed to the ellipticity; Re, a wavenumber-based Reynolds number; and De, the Deborah number based on the period of the base flow. New modes an d mechanisms of instability are discovered. The flow is generally susc eptible to instabilities in the form of propagating plane waves with a rotating wavevector, the tip of which traces an ellipse of the same e ccentricity as the flow, but with the major and minor axes interchange d. Whereas a necessary condition for purely inertial instability is th at the wavevector has a non-vanishing component along the vortex axis, the viscoelastic modes of instability are most prominent when their w avevectors do vanish along this axis. Our analytical and numerical res ults delineate the region of parameter space of (E, Re, De) for which the new instability exists. A simple model oscillator equation of the Mathieu type is developed and shown to embody the essential qualitativ e and quantitative features of the secular viscoelastic instability. T he cause of the instability is a buckling of the 'compressed' polymers as they are perturbed transversely during a particular phase of the p assage of the rotating plane wave.