Applying Methods from Differential Geometry to Devise Stable and Persistent Air Layers Attached to Objects Immersed in Water

Abstract

We describe a few mathematical tools which allow to investigate whether air-water interfaces exist (under prescribed conditions) and are mechanically stable and temporally persistent. In terms of physics, air-water interfaces are governed by the Young-Laplace equation. Mathematically they are surfaces of constant mean curvature which represent solutions of a nonlinear elliptic partial differential equation. Although explicit solutions of this equation can be obtained only in very special cases, it is – under moderately special circumstances – possible to establish the existence of a solution without actually solving the differential equation. We also derive criteria for mechanical stability and temporal persistence of an air layer. Furthermore we calculate the lifetime of a non-persistent air layer. Finally, we apply these tools to two examples which exhibit the symmetries of 2D lattices. These examples can be viewed as abstractions of the biological model represented by the aquatic fern Salvinia.