Taylor–Couette flow between rotating cylinders has served for over a century as a laboratory model for studying hydrodynamic instabilities. For Newtonian fluids, the picture is well understood: above a critical Reynolds number, axisymmetric Taylor vortices appear, while helical modes emerge only when the cylinders counter-rotate. Polymer additives change the rheology, and recent linear calculations have shown that for a small inner cylinder radius, the helical mode can become dominant. The present work confirms this finding with nonlinear simulations and adds the missing link between linear theory and experiment.
The study uses direct numerical simulation of three-dimensional viscoelastic flow in an annular gap. Geometry: dimensionless inner cylinder radius ξ = 0.049, computational domain length 21.3. The governing equations include an additional retarded term describing the elasticity of polymer molecules. For discretization, a spectral Fourier method is used along the cylinder axis, with a mesh of 545 elements in the cross-section; polynomial order p = 5, 50 Fourier modes, and a time step of 0.025. A finite-duration white noise was superimposed on the steady flow, and the evolution of perturbation energy was tracked.
For a Newtonian fluid (K = 0), the critical Reynolds number obtained from nonlinear calculations was about 265, matching linear theory. For a viscoelastic fluid with K = 0.1, the critical value dropped to 202. At Re = 190, the perturbation decayed; at Re = 210, the energy reached a constant level; and at Re = 250, it grew. Long-time evolution at supercritical numbers showed the formation of either Taylor vortices (for K = 0) or helical vortices (for K = 0.1). Thus, elasticity not only lowers the stability threshold but also changes the type of secondary flow.
This work stands apart from its neighbors in the archive: nearby studies include vortices in superfluid helium, electron vortices on nanostrings, and flows in porous media. The difference here is that it is the first to confirm, via direct nonlinear simulation, the linear theory prediction about mode competition in wide-gap viscoelastic Taylor–Couette flow. Neighboring works either use different fluid models or consider other geometries, so this study fills the niche of nonlinear verification for weakly elastic solutions.
A natural extension would be to broaden the parameter range: increasing elasticity K, changing the inner cylinder radius, and accounting for finite cylinder length. It would also be interesting to compare with experiments on real polymer solutions, where effects not captured by the Oskolkov model may appear. The transition to developed turbulence and the role of elastic stresses in forming secondary flows could also be investigated.
The results are important for industrial processes involving polymer solutions: mixing, coating, pipeline transport. Understanding the instability threshold and the type of vortices that arise will help predict and control the transition to turbulence.
Perform calculations at other values of ξ and K to construct a regime map. Compare with experimental data on polymer solution flow in a gap with a rotating inner cylinder.
In the hydrodynamics of viscoelastic fluids, the criteria for transition to turbulence and the role of elasticity in forming secondary flows remain open questions. This work links linear stability theory with nonlinear dynamics but does not solve the problem of turbulent transport in strongly viscoelastic media.
This section is addressed to the author of the paper and their colleagues. Neighbouring works were found by meaning-based search across our archive, not by word overlap. We do not review: we show the field around the work.
What the work does
This work numerically investigates the stability of a polymer solution flow between coaxial cylinders using the Oskolkov model. The author introduces white noise and tracks its evolution, obtaining critical Reynolds numbers consistent with linear theory, and shows that elasticity lowers the stability threshold and makes helical perturbations more dangerous.
Where its strength is
Direct nonlinear simulation confirms the linear theory predictions about mode competition and reveals a transition from Taylor vortices to helical vortices as elasticity increases, filling the gap between linear calculations and experiments.
Where it could go next
Worth trying
Why this may matter
Confirming the nonlinear dynamics of viscoelastic flows is important for developing technologies that use polymer additives to control the transition to turbulence, and could help create more accurate models for engineering calculations.
Nearby works in our archive
The archive is sparse around this work: the closest review is only 0.51, against a usual 0.64. It looks like few have drilled nearby.
Helical vortices that appear in a polymer solution occur in a Newtonian fluid only when both cylinders rotate in opposite directions. The elasticity of polymer molecules mimics this effect even though the outer cylinder remains stationary.