Cusps and cuspidal edges at fluid interfaces: existence and application

One of the intriguing questions in fluid dynamics is on the interrelation between dynamic singularities in the solutions of fluid dynamic equations -- unboundedness of the velocity field in an appropriate norm -- and the geometric ones -- divergence of curvature at fluid interfaces. The present talk focuses on two generic interfacial singularities -- genuine cusps and cuspidal edges -- found here in both two and three dimensions thus establishing a relation between real fluid interfaces and geometric singularity theory. The key new finding is the necessary condition for the existence of geometric singularities, which is a variation of surface tension. It is also established here that the dynamic and geometric singularities entail each other only in the case of three-dimensional cusps. Explicit asymptotic solutions for the flow field and interface shape near steady-state singularities at fluid interfaces are developed as well. The practical motivation for the present study comes from the fundamental role interfacial singularities play in sustaining self-driven conversion of chemical into mechanical energy.