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R. Ibragimov and D. Pelinovsky

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Incompressible viscous fluid flows in a thin spherical shell

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Journal of Mathematical Fluid Mechanics 11, 60–90 (2009)

**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.

**Keywords**:

Navier-Stokes equations, spherical shell, stability of stationary flows,
associated Legendre equation, Sturm-Liouville theory