The Dynamics of the Globe Fountain

We present results on the isothermal laminar flow of a thin fluid layer over a sphere as it exits from a small hole at the top of the sphere. Such a flow can be observed in a globe fountain. The fluid is taken to be viscous, incompressible and Newtonian while the flow is assumed to possess azimuthal symmetry. The governing Navier-Stokes equations are solved subject to no-slip and impermeability boundary conditions on the surface and the dynamic and kinematic conditions along the free surface. An approximate analytical solution for the steady-state flow has been derived by expanding the flow variables in powers of a small parameter, d, which represents the shallowness parameter. The leading and first-order terms in the series have been determined and the findings demonstrate that for thin flows the approximate solution is indeed accurate. Various results and comparisons are presented and discussed. Lastly, the analysis was also extended to solve the problem of thin flow over a cylinder and the fundamental differences between the flow over a sphere and that over a cylinder have been identified and explained. The technique and the approach adopted can be used to model and understand similar thin flows that occur in other settings.