Dynamics of Partial Differential Equations
Volume 10 (2013)
On the stationary Navier-Stokes flow with isotropic streamlines in all latitudes on a sphere or a 2D hyperbolic space
Pages: 209 – 254
In this paper, we show the existence of real-analytic stationary Navier-Stokes flows with isotropic streamlines in all latitudes in some simplyconnected 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.
Navier-Stokes equation, Riemannian manifold, streamlines
2010 Mathematics Subject Classification
53Z05, 76D03, 76D05