Inertio-elastic instability of a vortex column

We analyse the instability of a vortex column in a dilute polymer solution at large Re and De with El = De/Re, the elasticity number being finite. The stability of small-amplitude two-dimensional perturbations in this distinguished limit is governed by the elastic Rayleigh equation whose spectrum is parameterized by E = El(1-β), β being the ratio of the solvent to the solution viscosity. In the said limit, neglecting the relaxation terms implies that the polymer solution supports undamped elastic shear waves propagating relative to the base-state flow, leading to a pair of elastic continuous spectra besides the familiar inviscid one. Further, unlike the neutrally stable inviscid case, an instability of the vortex column arises for finite E due to a pair of elastic shear waves being driven into a resonant interaction under the differential convection by the irrotational shearing flow outside the core. An asymptotic analysis for the Rankine profile shows the absence of an elastic threshold for this instability. The growth rate is O(Ω₀) for order unity E, although it becomes transcendentally small for E«1, being O(Ω₀E²e^-1/√E). An accompanying numerical investigation shows that the instability persists for smooth monotonically decreasing vorticity profiles, provided the radial extent of the transition region (from the rotational core to the irrotational exterior) is less than a certain E-dependent threshold.