Gravitational collapse of liquid layer cavities near boundaries

Collapse of a cavity, or a depression hollow, in a water layer under gravity is modeled with the so-called Shallow Water equations in three dimensional settings, under circular symmetry and its deformation to elliptical cross sections. Self-similar, explicit solutions are found by quadratures in terms of elliptic integrals. We show that the presence of a rigid floor and the proximity of the cavity to this boundary significantly affects the evolution of the free surface, with the collapse evolving to form jet pairs originating at the caustics locations determined by the initial ellipsoidal cavity. The loss of symmetry implied by the deformation to elliptical cross sectional shapes leads to time evolution governed by an integrable two-degree of freedom Hamiltonian system. It is shown that the formation of the singularities is a reflection of the different critical exponents of the fluid velocity components in the solutions, with only the component aligned with the minor axis exhibiting a gradient catastrophe in finite time.