Analysis for inertial and elastic instabilities in extensional flow and comparisons with cross-slot flow

We theoretically investigate the instabilities of a steady planar extensional flow of viscoelastic fluids with the flow vorticity equation. The results of this linear stability analysis indicate two distinct instabilities depending on the values of Reynolds number ($Re$) and Weissenberg number ($Wi$). One instability is an inertia-dominated one occurring at a critical $Re$, in which the vorticity component $\omega_x$ becomes unstable, suggesting an emerging axial vortex in the extensional direction $x$. The other instability is an elasticity-dominated one at high $Wi$, where the vorticity component $\omega_z$ in the direction normal to elongational plane becomes unstable, indicating a symmetry breaking on the elongational $xy$-plane. The predicted critical $Re$ and $Wi$ numbers of these two instabilities by the linear stability theory are critically compared with experimental and numerical results in the cross-slot channel flows.