Abstract
Рассмотрены плоские нестационарные течения вязкой несжимаемой жидкости в потенциальном поле внешних сил. Получено уравнение в частных производных эллиптического типа, каждое решение которого является функцией тока вихревого течения, описываемого некоторым точным решением уравнений Навье--Стокса. Полученные решения обобщают течения Бельтрами--Тркала и Беллаба. Даны примеры таких новых решений. Они предназначены для верификации численных алгоритмов и компьютерных программ.
Highlights
Starting from the studies by Gromeka and Lamb [1, 2] proposing a new method of writing the Euler equations, a method for integrating the fluid motion equations began to be developed
This paper proposes an elliptic partial differential equation, each solution of which is the stream function of a vortex flow described by an exact solution of the Navier–Stokes equations
We describe the method of obtaining a family of exact solutions to the system (3)–(5)
Summary
Starting from the studies by Gromeka and Lamb [1, 2] proposing a new method of writing the Euler equations, a method for integrating the fluid motion equations began to be developed. The Couette [27], Poiseuille [28, 29], Stokes [30], von Karman [31], Hiemenz [32] flows have proved to be so efficient that they have been studied up to now [11, 33,34,35] These flow motions have in common that they fall within the class of solutions where velocities depend linearly on a part of coordinates [11]. The majority of numerical algorithms work with any initial and boundary conditions; the search for corresponding boundary value problems with a known exact solution can start with a search for the flow parameters satisfying the Navier–Stokes equations, without consideration of any boundary and initial conditions. A method for computing the pressure field for each of such stream functions is proposed
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