Potential flows away from stagnation in infinite cylinders
François Hamel, Aram Karakhanyan
TL;DR
The paper proves a Liouville-type rigidity for steady incompressible Euler or Navier-Stokes flows in infinite cylinders $\Omega=\mathbb{R}\times\omega$ with tangential boundary conditions: any potential flow $v=\nabla\psi$ that stays strictly away from stagnation ($\inf_{\overline{\Omega}}|v|>0$) must be constant and parallel to the cylinder. The key approach combines ODE analysis of streamlines with PDE arguments for the harmonic potential $\psi$ (since $\Delta\psi=0$ and $v=\nabla\psi$), showing that $\psi(x)=a\,x_1+b$ and hence $v=(a,0,\dots,0)$ with $a\neq0$, and $p$ constant; this holds in any dimension $N\ge2$. A separate 2D PDE-based proof is also provided, reinforcing robustness of the result. The work discusses sharpness via counterexamples when hypotheses are weakened and connects to physical implications for Euler and Navier-Stokes flows, including corollaries about unbounded flows forcing stagnation points or infinity and the role of harmonicity and boundary conditions.
Abstract
Steady incompressible potential flows of an inviscid or viscous fluid are considered in infinite N-dimensional cylinders with tangential boundary conditions. We show that such flows, if away from stagnation, are constant and parallel to the direction of the cylinder. This means equivalently that a harmonic function whose gradient is bounded away from zero in an infinite cylinder with Neumann boundary conditions is an affine function. The proof of this rigidity result uses a combination of ODE and PDE arguments, respectively for the streamlines of the flow and the harmonic potential function.