Application of R-Function Methods and the Nonlinear Galerkin Method in the Mathematical Modeling of Plane Steady Viscous Flows

• Datum: 2026
• Hauptverfasser: Сидоров, Максим, Чумаченко, Світлана, Шпакович, Максим
• Format: Artikel
• Sprache: Ukrainisch
• Veröffentlicht: Кам'янець-Подільський національний університет імені Івана Огієнка 2026
• Online Zugang: https://mcm-math.kpnu.edu.ua/article/view/354829
• Назва журналу: Mathematical and computer modelling. Series: Physical and mathematical sciences

The paper considers a plane–parallel flow of a viscous incompressible fluid in a bounded simply connected domain with a piecewise smooth boundary. Such studies are relevant both from the standpoint of the development of theoretical methods of hydrodynamics and mathematical physics and for solving a wide range of applied problems in modern science and engineering. When analyzing plane-parallel flows, it is convenient to pass from the system of Navier-Stokes equations in natural variables to a problem formulated in terms of the stream function. The stream function is related to the velocity vector and is introduced in such a way that the continuity equation is satisfied identically and the pressure is eliminated from the governing equations by cross differentiation. The mathematical model of the considered process is a nonlinear boundary value problem with a fourth-order elliptic equation for the stream function. For its numerical analysis, it is proposed to use the R-function method with approximation of the unknown component by the nonlinear Galerkin method. The use of the R-function and Galerkin methods made it possible to obtain an approximate solution of the problem in a numerical-analytical form (which simplifies the procedure for determining various flow characteristics, in particular the velocity field, vorticity, and pressure) and to exactly incorporate the geometry of the domain and the boundary conditions into the numerical algorithm. A computational experiment was carried out in a unit square for different values of the Reynolds number. The results are presented in the form of contour lines of the stream function, vorticity, and pressure, as well as the velocity vector field, and in the form of tables comparing numerical characteristics of the flow for different Reynolds numbers. The obtained results are in good agreement with the results of physical experiments and with numerical results reported in the literature.