Mathematical modeling of the 3D evolution problem for the interface between liquids of different viscosity and density in a uniform and infinite soil

The three-dimensional evolution problem for the interface between liquids of different viscosity and density is formulated for the case of piston displacement. The solution of this problem is reduced to the numerical solution of a system that consists of an integral equation and a differential equation by the method of discrete singularities. The practical convergence of the numerical scheme is studied and a comparison of the numerical results with the well-known two-dimensional solution is given.