An extension of Prandtl–Batchelor theory and consequences for chaotic advection

Abstract

We extend the Prandtl–Batchelor theory of steady laminar motion at large Reynolds number to derive conditions that steady three-dimensional Navier–Stokes flows have to satisfy. We combine these results with ergodic theory to show that flows with strong Beltrami property (e.g., ABC flows) cannot be a paradigm for chaotic advection in inertia-dominated boundary-driven three-dimensional flows. Our results indicate that viscous forces are responsible for chaotic advection in steady, three-dimensional boundary-driven Navier–Stokes flows at large Reynolds numbers.