Static shapes of levitated viscous drops
Résumé
We consider the levitation of a drop of molten glass above a spherical porous mould, through which air is injected with constant velocity. The glass is assumed to be sufficiently viscous compared to air that motion in the drop is negligible. Thus static equilibrium shapes are determined by the coupling between the lubricating pressure in the supporting air cushion and the Young–Laplace equation. The upper surface of the drop is under constant atmospheric pressure; the static shape of the lower surface of the drop is computed using lubrication theory for the thin air film. Matching of the sessile curvature of the upper surface to the curvature of the mould gives rise to a series of capillary ‘brim' waves near the edge of the drop which scale with powers of a modified capillary number. Several branches of static solutions are found, such that there are multiple solutions for some drop volumes, but no physically reasonable solutions for other drop volumes. Comparison with experiments and full Navier–Stokes calculations suggests that the stability of the process can be predicted from the solution branches for the static shapes, and related to the persistence of brim waves to the centre of the drop. This suggestion remains to be confirmed by a formal stability analysis.