Numerical Problems Related to Solving the Navier-Stokes Equations in Connection with the Heat Transfer with the Use of FEM

The paper presents a numerical solution to the unsteady form of the Navier-Stokes (N-S) equations along with the heat transfer equation. The problem of mass transfer with the energy transfer is very common in engineering, for instance, in designing ventilation systems or in founding. Solving the Navier-Stokes equations causes many numerical problems regardless of the chosen discretization method. What is more, numerical methods of solving the N-S equations are characterized by the significant time of calculations.The finite element method is often used when solving heat transfer equations. However, problems arise when it is used for solving the Navier-Stokes equations. They are related, for example, to the LBB condition (Ladyzhenskaya-Babuska-Breezi). In the paper, we have used the stabilized finite element method (FEM) form to solve the N-S equations. It is characterized by the ease of implementation and it does not have a negative influence on the efficiency of calculations. It also makes it possible to bypass the LBB condition. Integration over the time has been done with the help of widely known one-step theta scheme.The paper presents the result, which allows for the quality assessment of the obtained solution with the use of the aforementioned methods. To find the solution, a computer implementation of the presented mathematical problems has been created. On the basis of the obtained results, it is possible to state that the application of stabilization for FEM in the relation to the Navier-Stokes equations is an effective way to find the numerical solution. The authors of this paper are planning to further develop the software to apply it to simulations connected with the problems in the foundry process.