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Couette Taylor instabilities for counter-rotating cylinders in the small-gap regime

Dongfen Bian, Emmanuel Grenier, Gérard Iooss, Zhuolun Yang

Abstract

We study the Couette Taylor instabilities for an incompressible viscous fluid between two coaxial cylinders of nearly equal radii, allowing counter-rotation with the ratio of rotation rate $μ\in [-1,1]$. Working in a rotating frame and in a small-gap and small-viscosity regime, we derive the corresponding limiting Navier Stokes system and analyze the linear stability of the Couette flow. In particular, we numerically compute the critical Taylor number for general perturbations and identify a transition near $μ_c \approx -0.8$: for $μ> μ_c$ the most unstable mode is axisymmetric, whereas for $μ< μ_c$ the most unstable mode is non-axisymmetric. Near criticality, slowly varying traveling waves are governed by a time-independent Ginzburg Landau equation. The nonlinear coefficient changes sign near $\hatμ_c \approx -0.65$, yielding a supercritical regime for $μ> \hatμ_c$ and a subcritical regime for $μ_c < μ< \hatμ_c$. In the subcritical range, we classify small-amplitude steady states, including Taylor vortex flows, wavy vortices, a two-parameter family of quasi-periodic flows, and a localized traveling perturbation of the Couette flow.

Couette Taylor instabilities for counter-rotating cylinders in the small-gap regime

Abstract

We study the Couette Taylor instabilities for an incompressible viscous fluid between two coaxial cylinders of nearly equal radii, allowing counter-rotation with the ratio of rotation rate . Working in a rotating frame and in a small-gap and small-viscosity regime, we derive the corresponding limiting Navier Stokes system and analyze the linear stability of the Couette flow. In particular, we numerically compute the critical Taylor number for general perturbations and identify a transition near : for the most unstable mode is axisymmetric, whereas for the most unstable mode is non-axisymmetric. Near criticality, slowly varying traveling waves are governed by a time-independent Ginzburg Landau equation. The nonlinear coefficient changes sign near , yielding a supercritical regime for and a subcritical regime for . In the subcritical range, we classify small-amplitude steady states, including Taylor vortex flows, wavy vortices, a two-parameter family of quasi-periodic flows, and a localized traveling perturbation of the Couette flow.
Paper Structure (13 sections, 2 theorems, 155 equations, 12 figures)

This paper contains 13 sections, 2 theorems, 155 equations, 12 figures.

Table of Contents

  1. Introduction
  2. The small gap approximation
  3. Axi-symmetric instabilities
  4. Non axi-symmetric instabilities
  5. Study of the case $\mu_c < \mu < \hat{\mu}_c$
  6. Study near $\mu=\mu_c$
  7. Study in the region $\tau =\varepsilon \sigma ^{2},|\varepsilon |\ll 1,$ near the line $\tau =0$
  8. Case $\sigma >0,\tau =\varepsilon \sigma ^{2},0\leq |\varepsilon |\ll 1.$
  9. Case $\sigma <0,\tau =\varepsilon \sigma ^{2},0\leq |\varepsilon |\ll 1.$
  10. Study in the region $\sigma ^{2}+4\tau \ll \sigma ^{2}$ and $\sigma < 0$ (near the bottom curve $\Delta =0$)
  11. Appendix
  12. Computation of coefficient $c$
  13. Numerical computation of $c$

Key Result

Theorem 2

Let $\mu_c<\mu<\hat{\mu}_c$, and let $\alpha = \alpha_c(\mu)$. Then $c<0$ and $b_4>0$, and the $z$-periodic Taylor vortex flow with axial period $2\pi/\alpha_c$ bifurcates subcritically at $T = T_c$. Let $\tau = T - T_c$ and assume that $|\tau|$ is small enough. Then

Figures (12)

  • Figure 1: $T_c$ (left) and $\alpha_c$ (right) as functions of $\mu$ for axi-symmetric fields.
  • Figure 2: Left: $T_c(\mu,\mathfrak{B})$ as a function of $\mathfrak{B}$ for $\mu = -1, -0.8, ..., 1$. The lower curve corresponds to $\mu = 1$. Right: $T_c(\mu,\mathfrak{B}) - T_c(\mu,0)$ as a function of $\mathfrak{B}$ for $\mu = -1, -0.8, ..., 1$. The lower curve corresponds to $\mu=-1$.
  • Figure 3: Bifurcation diagram in the case $c<0$. The wavy vortices ($\beta\neq0$) are bifurcating branches parallel to the TVF branch, branching from the Couette flow at some $\tau>0$.
  • Figure 4: Phase portraits in the case $c<0$, for $K=0$.
  • Figure 5: $f(X)$ for $K\neq0$. Wavy vortices (WV) correspond to the limit curves.
  • ...and 7 more figures

Theorems & Definitions (6)

  • Remark 1
  • Theorem 2
  • Remark 3
  • Remark 4
  • Theorem 5
  • Remark 6