The Flow of An Inhomogeneous Fluid Inside a Sphere

Abstract

The research deals with the stationary flow of an inhomogeneous incompressible fluid inside a spherical vessel under the influence of a potential mass force. Using the methods of four-dimensional analysis, the solution to the problem is constructed in an explicit analytical form. Exact solutions of the Euler equations for a homogeneous fluid are obtained only for some of the simplest problems. Researchers usually prove the existence and uniqueness of solutions to various initial - boundary value problems for Euler equations using the methods of a priori estimation. After that, the problem is usually solved by numerical methods. For an inhomogeneous fluid, when the unknown density is a variable, even obtaining a priori estimates becomes much more complicated, not to mention finding exact solutions. Nevertheless, in recent years, new methods of four-dimensional mathematics have been developed, giving previously unknown approaches to the study of nonlinear problems. In this paper, an exact analytical solution of the Euler equations describing the flow of an ideal inhomogeneous fluid inside a sphere is obtained. At the same time, the authors demonstrate new methods of four-dimensional analysis.