Exact Solutions of the Laplace–Young Equation

The solution of the Laplace–Young equation determines the equilibrium height of the free surface of a liquid contained in a vessel under the action of gravity and surface tension. There are only two non-trivial exact solutions known; one corresponds to a liquid occupying a semi-infinite domain bounded by a vertical plane wall while the other relates to the case when the liquid is constrained between parallel walls.Atechnique called boundary tracing is introduced; this procedure allows one to modify the geometry of the domain so that both the Laplace–Young equation continues to be satisfied while the necessary contact condition on the boundary remains fulfilled. In this way, new solutions of the equation are derived and such solutions can be found for certain boundaries with one or more sharp corners and for others that possess small-scale irregularities that can be thought of as a model for roughness. The method can be extended to construct new solutions for a variety of other physically significant partial differential equations.