Session Cc - General Instabilities.
ORAL session, Sunday, November 23
303, Moscone Center
An analysis of the three dimensional instability of two-dimensional viscoelastic elliptical flows is presented, extending the inviscid analysis of Bayly^1 to include both viscous and elastic effects. The problem is governed by three parameters: E, a geometric parameter related to the ellipticity; R\!e, a wavenumber-based Reynolds number; and De, the Deborah number based on the period of the base flow. New modes and mechanisms of instability are discovered. The flow is generally susceptible to instabilities in the form of propagating plane waves with a rotating wavevector, the tip of which traces an ellipse of the same eccentricity as the flow, but with the major and minor axes interchanged. Whereas a necessary condition for purely-inertial instability is that the wavevector has a nonvanishing component along the vortex axis, the viscoelastic modes of instability are most prominent when their wavevectors do vanish along this axis. Our analysis delineates the region of parameter space of (E,R\!e,De) for which the new instability exists. An oscillator equation of the Mathieu type is shown to embody the essential features of the secular viscoelastic instability. The cause of the instability is a buckling of the `compressed' polymers as they are perturbed transversely during the correct phase of the passage of the rotating plane wave. \small ^1~Bayly, B. J.,Phys.\ Rev.\ Lett., 57(17):2160--2163, 1986