Singularity of free-surface curvature in convergent flow: Cusp or corner?

The nature of singularities that arise in the mathematical modeling of free-surface flows and the ways of their analysis and regularization aimed at removing the physically unacceptable features is one of fundamental issues in theoretical fluid dynamics. The present work considers the type of the free-surface curvature singularity emerging in the steady two-dimensional convergent flow of a Newtonian fluid near a free boundary. The unphysical singularities in the flow field, unavoidable in the conventional model, are removed by describing this flow as a particular case of the interface formation/disappearance process in the framework of an earlier developed macroscopic theory of such processes which is applied without any ad hoc alterations. The near-field asymptotic analysis of the problem shows that at finite capillary numbers the singularity of the free-surface curvature is always a sharp corner, not a cusp.