Journal of Mathematical Fluid Mechanics

, Volume 11, Issue 1, pp 60–90 | Cite as

Incompressible Viscous Fluid Flows in a Thin Spherical Shell

  • Ranis N. Ibragimov
  • Dmitry E. Pelinovsky
Article

Abstract.

Linearized stability of incompressible viscous fluid flows in a thin spherical shell is studied by using the two-dimensional Navier–Stokes equations on a sphere. The stationary flow on the sphere has two singularities (a sink and a source) at the North and South poles of the sphere. We prove analytically for the linearized Navier–Stokes equations that the stationary flow is asymptotically stable. When the spherical layer is truncated between two symmetrical rings, we study eigenvalues of the linearized equations numerically by using power series solutions and show that the stationary flow remains asymptotically stable for all Reynolds numbers.

Mathematics Subject Classification (2000).

76D05 76E20 34B24 34L16 

Keywords.

Navier–Stokes equations on a sphere associated Legendre equation asymptotic stability of stationary flow numerical approximation of eigenvalues 

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Copyright information

© Birkhaueser 2007

Authors and Affiliations

  • Ranis N. Ibragimov
    • 1
  • Dmitry E. Pelinovsky
    • 1
  1. 1.Department of MathematicsMcMaster UniversityHamiltonCanada

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