On spiral steady flows for the Couette-Taylor problem

TL;DR

This paper analyzes steady viscous flows in a cylindrical annulus for the Couette-Taylor problem, explicitly characterizing spiral solutions and establishing their stability under small boundary perturbations.

Contribution

It explicitly determines all solutions with partial invariance and proves their stability for small boundary data, considering different boundary conditions involving vorticity.

Findings

Explicit solutions with partial invariance are identified.

Stability of these solutions is proven for small boundary data.

Analytical differences depend on which cylinder remains still.

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.