Dynamics of Partial Differential Equations

Volume 10 (2013)

Number 3

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

Pages: 209 – 254

DOI: http://dx.doi.org/10.4310/DPDE.2013.v10.n3.a1

Authors

Chi Hin Chan (Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan)

Tsuyoshi Yoneda (Department of Mathematics, Hokkaido University, Sapporo, Japan)

Abstract

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.

Keywords

Navier-Stokes equation, Riemannian manifold, streamlines

2010 Mathematics Subject Classification

53Z05, 76D03, 76D05

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