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On spiral steady flows for the Couette-Taylor problem

Edoardo Bocchi, Filippo Gazzola, Antonio Hidalgo-Torné

Abstract

We investigate the Couette-Taylor problem for a steady incompressible viscous fluid in a 3D cylindrical annulus, where one of the two cylinders is still, under both Dirichlet and boundary conditions involving the vorticity that naturally appear in the weak formulation. The outcome of this study is twofold. First, we explicitly determine all the solutions with a specific geometric \emph{partial invariance}, which coincide with the so-called spiral Poiseuille or Poiseuille-Couette flows depending on the boundary conditions. Second, for small boundary data, we provide stability of such solutions, that is, no steady finite-energy perturbations are admissible. To achieve this result in presence of vorticity boundary conditions, we find a substantial analytical difference depending on which cylinder is still.

On spiral steady flows for the Couette-Taylor problem

Abstract

We investigate the Couette-Taylor problem for a steady incompressible viscous fluid in a 3D cylindrical annulus, where one of the two cylinders is still, under both Dirichlet and boundary conditions involving the vorticity that naturally appear in the weak formulation. The outcome of this study is twofold. First, we explicitly determine all the solutions with a specific geometric \emph{partial invariance}, which coincide with the so-called spiral Poiseuille or Poiseuille-Couette flows depending on the boundary conditions. Second, for small boundary data, we provide stability of such solutions, that is, no steady finite-energy perturbations are admissible. To achieve this result in presence of vorticity boundary conditions, we find a substantial analytical difference depending on which cylinder is still.
Paper Structure (11 sections, 14 theorems, 138 equations, 3 figures)

This paper contains 11 sections, 14 theorems, 138 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Boundary conditions and classification of solutions
  3. Dirichlet boundary conditions
  4. Explicit derivation of partially-invariant solutions with still outer cylinder.
  5. Stability with still outer cylinder.
  6. The case of still inner cylinder.
  7. Boundary conditions involving the vorticity
  8. Derivation and connection with Navier slip conditions.
  9. Explicit derivation of partially-invariant solutions.
  10. Stability.
  11. Analysis of bounds for stability

Key Result

Theorem 3.1

Let $({{\mathcal{U}}^C},{{\mathcal{P}}^C})$ be as in tc1, let ${\mathcal{U}}^P$ be as in Ualphabeta. Then, the spiral Poiseuille flows are the only partially-invariant solutions to SNSEcyl-nsstokes0$_1$ as $\beta$ varies in $\mathbb{R}$.

Figures (3)

  • Figure 1.1: The cylindrical annulus $\Omega$, with one rotating cylinder.
  • Figure 1.2: Classification of flow states, taken from Chossat.
  • Figure 3.1: The Poiseuille flow $\mathcal{U}_{0,-1}$ produced by the pressure gradient along the $z$-axis in the plane $(\rho, z)$ with $R_1=1$ and $R_2=10$ ($R_0\approx 4.64$).

Theorems & Definitions (28)

  • Definition 2.1
  • Theorem 3.1
  • proof
  • Corollary 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • Theorem 3.4
  • Theorem 4.1
  • ...and 18 more