Inertio-elastic instability of a vortex column

We analyze the instability of a vortex column in a dilute polymer solution at large $\textit{Re}$ and $\textit{De}$ with $\textit{El} = \textit{De}/\textit{Re}$, the elasticity number, being finite. Here, $\textit{Re} = Ω_0 a^2/ν_s$ and $\textit{De} = Ω_0 τ$ are, respectively, the Reynolds and Deborah numbers based on the core angular velocity ($Ω_0$), the radius of the column ($a$), the solvent-based kinematic viscosity ($ν_s = μ_s/ρ$), and the polymeric relaxation time ($τ$). The stability of small-amplitude perturbations in this distinguished limit is governed by the elastic Rayleigh equation whose spectrum is parameterized by $\textrm{E} = \textit{El}(1-β)$, $β$ being the ratio of the solvent to the solution viscosity. The neglect of the relaxation terms, in the said limit, implies that the polymer solution supports undamped elastic shear waves propagating relative to the base-state flow. The existence of these shear waves leads to multiple (three) continuous spectra associated with the elastic Rayleigh equation in contrast to just one for the original Rayleigh equation. 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; although, for small E, the growth rate of the unstable discrete mode is transcendentally small, being O$(\textrm{E}^2e^{-1/\textrm{E}^{\frac{1}{2}}})$. An accompanying numerical investigation shows that the instability persists for smooth vorticity profiles, provided the radial extent of the transition region (from the rotational core to the irrotational exterior) is less than a certain $\textrm{E}$-dependent threshold.

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