On the stationary Navier-Stokes flow with isotropic streamlines in all latitudes on a sphere or a 2D hyperbolic space

TL;DR

This paper proves the existence of certain stationary Navier-Stokes flows with isotropic streamlines on spheres and hyperbolic spaces, and shows that Poiseuille's flows are incompatible with these geometries.

Contribution

It establishes existence results for stationary Navier-Stokes flows with isotropic streamlines on curved manifolds and rules out Poiseuille flows in these settings.

Findings

Existence of real-analytic stationary flows on spheres and hyperbolic spaces.

Poiseuille's flow profile cannot be stationary in these geometries.

Stationary parallel laminar flows are characterized on hyperbolic spaces.

Abstract

In this paper, we show the existence of real-analytic stationary Navier-Stokes flows with isotropic streamlines in all latitudes in some simply-connected flow region on a rotating round sphere. We also exclude the possibility of having a Poiseuille's flow profile to be one of these stationary Navier-Stokes flows with isotropic streamlines. When the sphere is replaced by a 2-dimensional hyperbolic space, we also give the analog existence result for stationary parallel laminar Navier-Stokes flows along a circular-arc boundary portion of some compact obstacle in the 2-D hyperbolic space. The existence of stationary parallel laminar Navier-Stokes flows along a straight boundary of some obstacle in the 2-D hyperbolic space is also studied. In any one of these cases, we show that a parallel laminar flow with a Poiseuille's flow profile ceases to be a stationary Navier-Stokes flow, due to the curvature of the background manifold.