Artiukh A. Mathematical modeling and numerical analysis of non-steady plane-parallel flows of viscous incompressible fluid by the R-functions method

The object of research is the non-stationary hydrodynamic processes in viscous incompressible fluid. The purpose of research is to develop methods of mathematical modeling and numerical analysis for plane parallel non-steady flows of viscous incompressible heat-conducting fluid. The methods of research are methods of the mathematical physics and functional analysis which are used for proof and investigation of the given methods; the R-function method - for constructing of normalized equations of region boundaries and the solution structures; Galerkin method, the method of successive approximations and methods of the splines theory - for approximation of the unknown components in the solution structures; Gauss quadrature formulas for numerical integration; methods for solving systems of linear and nonlinear equations. Theoretical and practical results are the following. The designed methods allow providing effective numerical modeling of non-steady flows of viscous incompressible heat-conducting fluids. Scientific novelty of the results lies in the fact that the solution method for the linear non-steady Stokes problem in simply connected regions of complex geometry with piecewise smooth boundaries has been developed. The solution method for the non-steady linear problem - computation of flow of viscous heat-conducting fluid in simply connected regions of complex geometry with piecewise smooth boundaries has been developed. The R-function and Galerkin method have been used. The iterative method for solving of nonlinear differential equation for stream function in simply connected regions of complex geometry with piecewise smooth boundaries has been improved for the non-steady case. The iterative method for solving of nonlinear differential equations system for stream function and temperature in simply connected regions of complex geometry with piecewise smooth boundaries has been improved for the non-steady case. The R-function, Galerkin methods and the method of successive approximations have been used. The conditions and estimations of convergence speed to the generalized solution in the appropriate space have been obtained. 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 10.27.2015). Scope - mathematical modeling and computational mathematics.