Abstract: In this paper, we formulate a modeling theory and numerical algorithm for a multi-component two-phase fluid system together with the interface between phases and with gravity. We use a diffuse interface model based on Peng-Robinson equation of state (EOS) for the modeling of the fluid. We show that gravity has a significant influence on the phase equilibrium behavior, which is an expected phenomenon but has not been numerically studied in the literature regarding to Peng-Robinson fluid modeled by a diffuse interface model.

Abstract: In this paper, the effects of different numerical integration schemes on the distributed Lagrange multiplier/fictitious domain (DLM/FD) method with body-unfitted meshes are studied for solving different types of interface problems: elliptic-, Stokes- and Stokes/elliptic-interface problems, for which the corresponding mixed finite element approximations are developed. Commonly-used numerical integration schemes, compound type formulas and a specific subgrid integration scheme are presented and the comparison between them is illustrated in numerical experiments, showing that different numerical integration schemes have significant effects on approximation errors of the DLM/FD finite element method for different types of interface problems, especially for Stokes- and Stokes/elliptic-interface problems, and that the subgrid integration scheme always results in numerical solutions with the best accuracy.

Abstract: This paper discusses the application of the double potential method for modeling flow of incompressible fluid. The algorithm allows us to avoid a numeric calculation of pressure. This procedure is not easy for case of an incompressible fluid flow. It may lead to solution instability with approximation by cell center grid methods. Also, the double potential method overcomes a problem of a complex boundary conditions which arises in case of modelling with using the Navier-Stokes equations in the vector potential-vortex formulation. The resulting system of equations is approximated by using the finite volume method and the exponential transformation. As a verification problem, the problem of establishing the Poiseuille flow on three-dimensional cylindrical geometry was applied

Abstract: The formation of the bubbles in the liquid was examined numerically and obtained results were successfully compared with the results provided by experiments. The study covered two different patterns defined by different Morton numbers or gas flow rates. The unsteady three dimensional calculations were carried out in code OpenFoam with the volume of fluid approach. Found numerical results were in a good math to the experiments in respect to bubble shapes, diameters and Reynolds numbers. More accurate comparison was found for lower gas flow rate then for the higher one. The main reason can be that under higher gas flow rate, a complex flow behavior between gas bubbles and surrounding liquid flow is created which after that worsen the accuracy of calculations. The main important output of the study was a comparison of the bubble diameters in time. Especially for higher gas flow rates, bubbles are growing rapidly during its climbing. Nevertheless a satisfactory agreement was found between numerics and experiments.