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Inertio–elastic instability of a vortex column

Published online by Cambridge University Press:  28 February 2022

Anubhab Roy Open the ORCID record for Anubhab Roy [Opens in a new window] ,
Piyush Garg Open the ORCID record for Piyush Garg [Opens in a new window] ,
Jumpal Shashikiran Reddy Open the ORCID record for Jumpal Shashikiran Reddy [Opens in a new window]  and
Ganesh Subramanian Open the ORCID record for Ganesh Subramanian [Opens in a new window]
Anubhab Roy*
Affiliation:
Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore560 064, India
Piyush Garg
Affiliation:
Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore560 064, India
Jumpal Shashikiran Reddy
Affiliation:
Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore560 064, India
Ganesh Subramanian*
Affiliation:
Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore560 064, India
*
‡Present address: Department of Applied Mechanics, Indian Institute of Technology Madras, Chennai600 036, India.
†Email address for correspondence: [email protected]
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Abstract

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. Here, ${{Re}} = \varOmega _0 a^2/\nu$ and ${{De}} = \varOmega _0 \tau$ are, respectively, the Reynolds and Deborah numbers based on the core angular velocity ($\varOmega _0$), the radius of the column ($a$), the total (solvent plus polymer) kinematic viscosity ($\nu = (\mu _s +\mu _p)/\rho$ with $\mu _s$ and $\mu _p$ being the solvent and polymer contributions to the viscosity) 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 ${E} = {{El}}(1-\beta )$, $\beta$ 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. 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 $O(\varOmega _0)$ for order unity $E$, although it becomes transcendentally small for ${E} \ll 1$, being $O(\varOmega _0 {E}^2{\rm e}^{-1/{E}^{{1}/{2}}})$. 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.

JFM classification

Geophysical and Geological Flows: Waves in rotating fluids Vortex Flows: Vortex instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 937 , 25 April 2022 , A27
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© The Author(s), 2022. Published by Cambridge University Press

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References

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