Inertio-elastic instability of a vortex column

Abstract

We analyze the instability of a vortex column in a dilute polymer solution at large $\text{Re}$ and $\text{De}$ with $\text{El}=\text{De}/\text{Re}$, the elasticity number, being finite. Here, $\text{Re}={\Omega }_0{a}^2/{\nu }_s$ and $\text{De}={\Omega }_0\tau$ are, respectively, the Reynolds and Deborah numbers based on the core angular velocity (${\Omega }_0$), the radius of the column ($a$), the solvent-based kinematic viscosity (${\nu }_s={\mu }_s/\rho$), and the polymeric relaxation time ($\tau$). The stability of small-amplitude perturbations in this distinguished limit is governed by the elastic Rayleigh equation whose spectrum is parameterized by $\text{E}=\text{El}(1-\beta )$, $\beta$ 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$({\text{E}}^2{e}^{-1/{\text{E}}^{\frac{1}{2}}})$. 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 $\text{E}$-dependent threshold.

References in corpus (13)

• Elastic turbulence in a polymer solution flow
• Elasto-inertial turbulence
• Exceeding the asymptotic limit of polymer drag reduction
• An hydrodynamic shear instability in stratified disks
• Viscoelastic pipe flow is linearly unstable
• Critical-layer structures and mechanisms in elastoinertial turbulence
• Linear instability of viscoelastic pipe flow
• On the relation between viscoelastic and magnetohydrodynamic flows and their instabilities
• Unbalanced instabilities of rapidly rotating stratified shear flows
• Magnetorotational Instability in Electrically Driven Flow of Liquid Metal: Spectral Analysis
• Interacting vorticity waves as an instability mechanism for MHD shear instabilities
• Magnetorotational-type instability in Couette-Taylor flow of a viscoelastic polymer liquid
• Analog of Astrophysical Magnetorotational Instability in a Couette-Taylor Flow of Polymer Fluids

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