Finite difference method implementation for numericalintegration hydrodynamic equations melts

Abstract

The liquid state theory is not a simple section of the modern theory of metallurgical processes. Any substance in liquid state is a difficult object to establish not only quantitative, but also qualitative patterns, being that liquid state is intermediate between solid and gaseous states. Theoretical hydrodynamics has long attracted attention of various specialties’ scientists: comparative simplicity of the basic equations, precise problems formulation and clarity of its experiments inspired hope of getting a dynamic phenomena’s complete description occurring in melts. In describing continuous media’s dynamic properties the following systems of equations were obtained: for a viscous melt - the Navier – Stokes equations, for an ideal melt - the Euler equations, for a weakly compressible melt - the Oberbeck – Boussinesq equations. In fundamental research and in the field of applied research these mathematical models are generally accepted for modeling melt flow. Theoretical processes descriptions occurring in melts are based on the Stokes – Kirchhoff theory, which, with the frame of classical hydrodynamics, revealed phenomenological connections between the molten systems’ kinetic properties. Numerous hydrodynamic paradoxes point to that long and thorny path that has been covered since its inception. First long stage was associated with the study and research of ideal incompressible liquid’s potential flows. Mathematical methods of their research using the theory of complex variable functions seemed almost perfect. Imperfection of the ideal liquid theory was indicated by the famous Euler-Dalamber paradox: the total force acting on a body flowing around a potential flow is equal to zero. Then a mathematical model of a viscous incompressible fluid with its basic Navier-Stokes equations was created. Proposed section outlines various methods for solving and studying the Navier – Stokes equations. At the present stage, a great effort is made to find localized hydrodynamics equations solutions.