The problems of mathematical description of the viscous motion of the metal melt accompanied by overcoming the internal friction caused by the movement of particles and overcoming the forces of their interaction are investigated. Solutions of hydrodynamic equations involving quantum potentials of interparticle interaction of atoms in melts, and quantum effects were taken into account using correlation functions of transport coefficients such as viscosity, since they are quite closely related to the structure of matter and are the most structurally sensitive characteristics of matter. The research consists in the fact that the correlation functionsof viscosity are justified from the point of view of the quantum statistical method. The correlation between correlation functions and radial distribution functions is established. A mathematical model of the flow of high-temperature melts taking into account the nature of the near order in them and the account of the second coefficient of viscosity by methods of statistical physics is described. On the basis of theoretical studies the parameters determining the relationship between viscosity and interatomic potential are calculated. The found dependences allow us to determine the average values of any physical parameters, in particular, the values of shear and volumetric viscosity.