Метод граничних інтегральних рівнянь в нелінійних крайових задачах динаміки в'язкого газу

The subject of the research is a mathematical model of the processes of flow around carrying systems of arbitrary spatial shape by a viscous gas flow. For more than one and a half centuries, studies have been carried out on the system of partial differential equations of conservation laws in fluid and gas mechanics, known as the Navier-Stokes system of equations, and due to its non-linearity, to date has not seen the development that would guarantee the conditions for the existence and uniqueness of solutions. It is for this reason that many questions and misunderstandings arise regarding the solution of initial-boundary value problems, primarily in aerohydrodynamics, using widespread software packages built on the basis of finite-difference approaches. The purpose of the article is to develop an alternative method of boundary integral equations, which, due to the boundary conditions of flow past/with respect to a continuous medium, leads to a system of linear boundary integral equations with guaranteed uniqueness of solutions. Problem: construction of integral representations of solutions of a system of differential equations of conservation laws by the method of generalized potential theory for differential operators of conservative forms of equations of the corresponding conservation laws of mass, vorticity and momentum. Scientific novelty. Differential operations of vector-tensor analysis are developed and generalized. Generalized integral theorems for second-order differential operators adequate to conservation laws are proved. Results. Based on the created generalized apparatus of vector-tensor analysis, integral representations of the main dynamic and kinematic characteristics of the problem of viscous gas flow past load-bearing systems of arbitrary spatial shape are constructed. The corresponding boundary integral equations obtained by standard limit transform are correctly algorithmized and adapted to convenient numerical implementation. Conclusions. The boundary value problem of viscous gas flow past solid load-bearing systems is reduced to a system of linear equations, due to physical boundary conditions, boundary integral equations that have unique solutions with respect to the kinematic and dynamic characteristics of the problem. In addition, it is proven for the first time that all characteristics depend on the irrotational vector potential of momentum obtained for the first time, which significantly simplifies the integral representations of solutions and their numerical implementation.