On spiral steady flows for the Couette-Taylor problem

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.

💡 Research Summary

The paper addresses the classical Couette‑Taylor configuration in a three‑dimensional cylindrical annulus Ω = { R₁ < ρ < R₂, z∈ℝ }, where one of the two coaxial cylinders is fixed and the other rotates. The fluid is assumed steady, incompressible, and viscous, governed by the Navier‑Stokes system
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