  We investigate the equations of anisotropic axisymmetric incompressible
viscous fluids in the exterior of a cylinder of $\R^3$, rotating around an
inhomogeneous vector
  $B(t, r)$. We prove uniform local existence with respect to the Rossby number
in suitable anisotropic Sobolev spaces. We also obtain the propagation of the
isotropic Sobolev regularity.
