On some problems of similarity flow of fluid with a free surface

The paper presents the method of solving a class of two-dimensional problems of the similarity flow of an incompressible fluid with a free surface. The fluid is assumed to be non-viscous and weightless. We consider two-dimensional irrotational similarity flows with dimensionless hydrodynamic characteristics depending only on the ratios x/v 0 t, y/v 0 t , where x, y are Cartesian co-ordinates, t is time and v 0 is a constant of the velocity dimension. The proposed method is based upon using the function introduced by Wagner (1932) and can be applied to the problems where the flow region is bounded by free surfaces and uniformly moving (or fixed) rectilinear impermeable boundaries. Introduction of Wagner's function makes it possible to reduce each of the problems under consideration to a non-linear singular integral equation for the real function. The method is illustrated by solving the classical problem of the uniform symmetrical entry of a wedge into a half-plane of a fluid.