Local cusp solutions of viscous flow

Free-surface cusps are a generic feature of externally driven, viscous flow bounded by a free surface, in that their form is stable under small perturbations. Here we present an alternative to the boundary integral description found recently (J. Eggers, Phys. Rev. Fluids , vol. 8, 2023, 124001), which is based directly on a local analysis of the Stokes equation. The new description has the advantage of greater simplicity and transparency, allowing us to understand the connections with bifurcation theory, as well as with other physical systems displaying similar singularities. To illustrate this, we construct cusp solutions corresponding to higher-order singularities, as well as time-dependent solutions.