Shape and stability of axisymmetric levitated viscous drops

We consider levitation of an axisymmetric drop of molten glass above a spherical porous mould through which air is injected at a constant velocity. Owing to the viscosity contrast, the float height for a given shape is established on a much shorter time scale than the subsequent deformation of the drop under gravity, surface tension and the underlying lubrication pressure. Equilibrium shapes, in which an internal hydrostatic pressure is coupled to the external lubrication pressure through the total curvature and the Young-Laplace equation, are determined using a numerical continuation scheme. The set of solution branches is surprisingly complicated and shows a rich bifurcation structure in the parameter space (Bo rho gV(2/3)/gamma, Ca = mu(a)nu/gamma), where Bo is bond number and Ca is capillary number, rho and V are the drop density and volume, gamma the surface tension, mu(a) the air viscosity and nu the injection velocity. The linear stability of equilibria is determined using a boundary-integral representation for drop deformation that factors out the rapid vertical adjustment of the float height. The results give good agreement with time-dependent simulations. For sufficiently large Ca there are intervals of Bo for which there are no stable solutions and, as Ca increases, these intervals grow and merge. The region of stability decreases as the mould radius a(M) increases with an approximate scaling Ca similar to a(M)(-5), which imposes practical limitations on the use of this geometry for the manufacture of lenses.