ε-Аpproximation of the temperatures model of inhomogeneous melts with allowance for energy dissipation

Abstract

Accumulated facts and information about the Navier-Stokes equations, together with a large number of experiments and approximate calculations, made it possible to reveal some discrepancies between the mathematical model of a viscous melt and real phenomena in the nature of real molten systems. There are many reasons for this. One of them is the nonlinearity of the Navier-Stokes equations. And for nonlinear equations it is known that in non - stationary problems, a solution satisfying it can exist not on the entire interval t ≥ 0. Over a finite period of time, it can either go to infinity, or crumble.A solution lose regularity and no satisfying the equations and begin branching. It is mathematically proved that if this solution exists for t ≥ 0, then it may not seek to solve the stationary problem when stabilizing the boundary conditions and external influences. The solutions of the nonstationary problem obtained even with a smooth initial regime and smooth external influences can become less regular with time, and then generally go into irregular or turbulent regimes. The actual implementation of this or that branch of the solution depends on extraneous reasons not taken into account in the Navier-Stokes equations. In the proposed paper, we constructed a numerical scheme with good convergence. The regularization of the initial systems of differential equations by ε -approximation is constructed. The Galerkin method is implemented ensuring the correctness of boundary value problems for an incompressible viscous flow both numerically and analytically. A splitting scheme for the Navier-Stokes equations with a weak approximation is constructed. An approximation is constructed for stationary and nonstationary models of an incompressible melt, which leads to nonlinear equations of hydrodynamics to a system of equations of Cauchy-Kovalevskaya type.