Splash formation at the nose of a smoothly curved body in a stream

In two-dimensional flow past a body close to a free surface, the upwardly d iverted portion may separate to form a splash. We model the nose of such a body by a semi-infinite obstacle of finite draft with a smoothly curved fro nt face. This problem leads to a nonlinear integral equation with a side co ndition, a separation condition and an integral constraint requiring the fa r-upstream free surface to be asymptotically plane. The integral equation, called Villat's equation, connects a natural parametrisation of the curved front face with the parametrisation by the velocity potential near the body . The side condition fixes the position of the separation point, whereas th e separation condition, known as the Brillouin-Villat condition, imposes a continuity relation to be satisfied at separation. For the described flow w e derive the Brillouin-Villat condition in integral form and give a numeric al solution to the problem using a polygonal approximation to the front fac e.