Depression of an Infinite Liquid Surface by an Incompressible Gas Jet

The problem of small angle depressions in a liquid surface due to an impinging two-dimensional potential jet is considered. Using conformal mapping methods and finite Hilbert transforms, the problem is formulated as a non-linear singular integral equation. The integral equation is approximated by a set of non-linear algebraic equations which are solved numerically by a method of repeated linear corrections. In addition, an asymptotic solution (for low jet velocity) is derived. From the numerical solutions of the integral equation, the liquid-surface profiles and the free streamlines of the jet are calculated for four cases. These results verify the appearance of lips on the liquid surface which have been observed experimentally by others.