Lamtyugova S. Mathematical modeling and numerical analysis of problems of flow past bodies by viscous incompressible fluid using the R-functions method

The object of research is the stationary hydrodynamic processes of flow around bodies by viscous incompressible fluid. The purpose of research is to develop methods of mathematical modeling and numerical analysis of steady flow around cylindrical bodies and bodies of revolution by viscous incompressible fluid on the basis of the R-functions method. The methods of research are methods of functional analysis and mathematical physics; mathematical apparatus of the R-functions theory; projection Bubnov-Galerkin method and the method of successive approximations; Gauss quadrature formulas; numerical methods for solving systems of linear and nonlinear equations. Theoretical and practical results consist of the following. The designed methods for solving problems of flow around bodies by viscous incompressible fluid are allow to implement the efficient numerical simulation of viscous incompressible flows with mass transfer. Scientific novelty of the results lies in the fact that for the mathematical modeling of slow flow past cylindrical bodies and the bodies of revolution by viscous incompressible fluid (approximation Oseen and Stokes), a method, based on the joint use of the R-functions and Bubnov-Galerkin method, has been developed. This method accurately takes into account the boundary conditions at the boundary of the streamlined body and the condition at infinity. The method developed can be easily modified in the transition to another area and represents the numerical solution in analytical form, which simplifies its use in further calculations. An iterative numerical method for calculating the problems of flow past bodies by a viscous incompressible fluid was further developed. The original nonlinear problem was replaced by a sequence of linear boundary value problems, for solutions of which at each step of the iterative process the numerical algorithm, based on the R-functions and Bubnov-Galerkin method, has been developed. This method differs from the known ones with the fact that it allows to reduce the consideration of problem of flow past body to a problem in a finite area adjacent to the streamlined body. Besides, the conditions at infinity are taken into account accurately. For developed iterative process the conditions of convergence were obtained. For the calculation of mass transfer of cylindrical bodies and the bodies of revolution with a uniform translational flow a numerical method, based on the joint use of R-functions and Bubnov-Galerkin method, has been developed. The algorithm of developed method does not change with change in the area geometry. In addition, the structure of the solution accurately takes into account the boundary conditions at the boundary of the streamlined body and the condition at infinity. The thesis was carried out in accordance with the plan of research work of the Department of Applied Mathematics of Kharkiv National University of Radio Electronics as part of scientific research on the topic "Development of models, methods and tools of structural and parametric optimization of engineering services with leaks" (№ of state registration 0111U002624, 2011 - 2013). The developed mathematical modeling tools are introduced to the educational process in the Kharkiv National University of Radio Electronics (act of 10.20.2015). Scope - mathematical modeling and computational mathematics.