Integrability and the Properties of Solutions to Euler and Navier-Stokes Equations | Chapter 07 | Theory and Applications of Mathematical Science Vol. 2

It is known that the Euler and Navier-Stokes equations, which describe flows of ideal and viscid gases, are the set of equations of the conservation laws for energy, linear momentum and mass. As it will be shown, the integrability and properties of the solutions to the Euler and Navier-Stokes equations depend, firstly, on the consistency of equations of the conservation laws and, secondly, on the properties of conservation laws.

It was found that the Euler and Navier-Stokes equations have solutions of  two types, namely, the solutions that are not functions (depend not only on coordinates) and generalized solutions that are functions but realized discretely and hence, functions or their derivatives have discontinuities. A transition from the solutions of first type to generalized solutions describes the process of transition of gas-dynamic medium from non-equilibrium state to the locally-equilibrium one. Such a process is accompanied by the emergence of any observable formations (such as waves, vortices, turbulent pulsations and soon). This discloses the mechanism of such processes as emergence vorticity and turbulence.

Such results were obtained when studying the equations the conservation laws for energy and linear momentum, which turned out to be inconsistent, due to the non-commutativity of the conservation laws.

Author(s) Details

L. I. Petrova
Moscow State University, Russia.

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