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Couette Taylor instabilities in the small-gap regime

Dongfen Bian, Emmanuel Grenier, Gérard Iooss, Zhuolun Yang

Abstract

The Couette-Taylor instability occurs in a viscous fluid confined between two coaxial rotating cylinders. When the Taylor number surpasses a critical value, the stable Couette flow destabilizes, giving way to steady Taylor vortices. As the Taylor number increases further, these vortices themselves become unstable, transitioning into wavy Taylor vortices. In this article, we focus on the small-gap limit, where the ratio of the cylinder radii approaches unity and the rotation rates of the cylinders are nearly identical. We provide a rigorous proof of the existence of a critical Taylor number $T_c$, at which the Couette flow loses stability. For Taylor numbers just above $T_c$, under fixed axial periodicity, the solutions to the limiting Navier-Stokes system are governed by a Ginzburg-Landau-type partial differential equation. Beyond the classical Taylor vortex flow, we demonstrate that a two-parameter family of solutions emerges at criticality for $T>T_c$. This family includes not only wavy vortices but also a variety of other exotic flow patterns, all of which remain steady in the frame rotating at the average angular velocity of the cylinders.

Couette Taylor instabilities in the small-gap regime

Abstract

The Couette-Taylor instability occurs in a viscous fluid confined between two coaxial rotating cylinders. When the Taylor number surpasses a critical value, the stable Couette flow destabilizes, giving way to steady Taylor vortices. As the Taylor number increases further, these vortices themselves become unstable, transitioning into wavy Taylor vortices. In this article, we focus on the small-gap limit, where the ratio of the cylinder radii approaches unity and the rotation rates of the cylinders are nearly identical. We provide a rigorous proof of the existence of a critical Taylor number , at which the Couette flow loses stability. For Taylor numbers just above , under fixed axial periodicity, the solutions to the limiting Navier-Stokes system are governed by a Ginzburg-Landau-type partial differential equation. Beyond the classical Taylor vortex flow, we demonstrate that a two-parameter family of solutions emerges at criticality for . This family includes not only wavy vortices but also a variety of other exotic flow patterns, all of which remain steady in the frame rotating at the average angular velocity of the cylinders.
Paper Structure (23 sections, 8 theorems, 228 equations, 8 figures)

This paper contains 23 sections, 8 theorems, 228 equations, 8 figures.

Table of Contents

  1. Introduction
  2. The Navier-Stokes equations in cylindrical coordinates
  3. Linear analysis in the axi-symmetric case
  4. Study of the linearized operator $\mathcal{L}_{(\eta ,\mu )}$
  5. Study of $\mathcal{L}_{1,\alpha}$
  6. A first bifurcation
  7. Linear analysis in the non axisymmetric case
  8. Set up
  9. Expansion of $\lambda$
  10. Study of the critical Taylor number
  11. The Ginzburg Landau equation
  12. Limit of Navier-Stokes system
  13. The amplitude equation
  14. Stability of bifurcating steady wavy vortices
  15. Stationary solutions of the amplitude equation
  16. ...and 8 more sections

Key Result

Theorem 1.1

For $\eta$ and $\mu$ close enough to $1$, there exists $T_c > 0$ such that the Couette flow (Couette) is linearly stable with respect to axisymmetric perturbations if $T < T_c$ and linearly unstable if $T > T_c$.

Figures (8)

  • Figure 1: TVF is given by $a_{3}\tau=c \rho^2$. Any point $(\tau,\rho)$ in the grey region corresponds to wavy vortices with $A=\rho e^{i\beta y}$ such that $b_{4}\mathfrak{R}^2 \beta^2=a_{3}\tau-c\rho^2$. These solutions correspond to the boundary of the bounded region described in Figure \ref{['caseKnon0']}.
  • Figure 2: Bounded solutions for K=0
  • Figure 3: $f(X)$ for $K\ne 0$.
  • Figure 4: Bounded solutions $\rho$ for $K>0$ and $\theta(y)$ for $H_{min}<H\leq H_{max}$.
  • Figure 5: Value of the smallest positive zero of $T \mapsto \mathfrak{F}_1(\alpha,T)$ as a function of $\alpha$
  • ...and 3 more figures

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Definition 3.4
  • ...and 5 more