Laminar flow in the interior of a rotating hollow sphere as it is set in motion

Theoretical studies of the viscous incompressible flow within a hollow sphere are presented. The Navier-Stokes equations are developed, assuming rotational symmetry, fulfilling the continuity equation with the flow function psi, and eliminating the pressure component; the nonlinear differential equations are analyzed and solved numerically using a Galerkin procedure. The secondary flow lines are shown, and the initial laminar flow is characterized as a three-dimensional secondary flow which passes over into rigid-body rotation as time goes to infinity. The uniqueness and stability of this final state is examined. Experimental determinations of radial velocity are found to agree well with the theoretical predictions. These findings are considered of practical importance in meteorology, geophysics, and the design of fluid-flow engines.