Using stable and unstable profiles to deduce deformation limits of the air-water interface

The shape of the air-water interface deformed by a van der Waals stress induced by a paraboloid shaped solid body is addressed and discussed. Emphasis is placed on the existence limit of solutions to the governing Euler-Lagrange equation for the equilibrium shape. Two legitimate solutions, one stable and one unstable, are found to converge at the existence limit, giving a numerical criterion for establishing critical physical conditions guaranteeing absolute stability. Insight is aided by a study of an analogous mechanical problem that exhibits very similar properties. Among numerical data produced are critical lower height limits of the paraboloid to the air-water surface and associated peak deformation heights and their dependencies on physical parameters. Of further interest to experimentalists in the surface force field are the variations in peak deformation height and total surface force on the solid as a function of position of the paraboloid, paraboloid geometry, and strength of the van der Waals stress.