zbMATH — the first resource for mathematics

Three-dimensional instability of viscoelastic elliptic vortices. (English) Zbl 0905.76037
An analysis of the three-dimensional instability of two-dimensional viscoelastic elliptical flows is presented which includes both viscous and elastic effects. 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 non-vanishing component along the vortex axis, the viscoelastic modes of instability are most prominent when their wavevectors vanish along this axis. A simple model oscillator equation of the Mathieu type is derived and shown to embody the essential qualitative and quantitative features of the secular viscoelastic instability. The cause of the instability is a buckling of the ‘compressed’ polymers as they are perturbed transversely during a particular phase of the passage of the rotating plane wave.

76E99 Hydrodynamic stability
76A10 Viscoelastic fluids
Full Text: DOI