The wetting of a given solid configuration with a given volume of liquid results in a liquid-solid configuration which consists of liquid “puddles” placed onto the solid surface and which minimizes the total free energy. The scope of the paper is limited to the case where the solid is generated by revolving a function \(y = g(x)\) about the \(x\)-axis and the solid-liquid is generated in the same way by a pieces composed function: \(y = g(x)\) in the dry regions; \(y = W(x)\) in the wet regions. The functions \(y = W(x)\), providing the shapes of the puddles, are solutions to an Euler-Lagrange equation, \((y')^ 2 = y^ 2/(H + \lambda y^ 2)^ 2 - 1\).
Analytical properties of these solutions, called Delaunay curves, are discussed and the wetting of several interesting solid configurations is presented: two overlapping spheres; two disconnected spheres (when a puddle bridges between the two spheres); two adjacent spheres containing a spherical cavity; an axisymmetric solid of constant curvature.