  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.
