Accurate numerical solutions to stationary free surface problems from capillarity

The paper considers two static problems from capillarity. The first one consists in the determination of the surface of a liquid in a capillary tube and the second in the computation of the shape of a sessile or pendent drop of liquid of a given volume, both configurations being considered in the gravitational field. From a mathematical point of view, both problems are nonlinear two-point boundary value problems which include in their formulations additional geometrical unknowns, i.e., the so-called free boundary value problems. Computational, they are laborious problems owing to the fact that they involve nonnormal Jacobian matrices. The resulting numerical difficulties are considerably reduced by treating these problems by a collocation method in conjunction with the continuation with respect to the parameters. The method is implemented by the MATLAB code bvp 4 c. The numerical evaluations of the free surfaces are in reasonable accordance with asymptotic estimations.