Explicit representation of the Euler and Navier --- Stokes equations of incompressible fluid with reduced dimensionality in integral form

M. L. Zaytsev ; V. B. Akkerman

Matematičeskaâ fizika i kompʹûternoe modelirovanie, Tome 25 (2022) no. 1, pp. 5-20.

Résumé

In this paper, we have obtained “stationary” systems of integral-differential equations, which are consequences of the non-stationary Euler and Navier — Stokes equations of an incompressible fluid, and for which there are no time derivatives. If we set a correct problem for them, then we can determine the entire non-stationary flow in the volume without solving the nonstationary problem. It is enough to specify time-varying data only on some surface of this stream. The method of reduction of overdetermined systems of differential equations, proposed earlier by the authors, is used. In this method, in the case of a successful choice of the additional constraint equation, the overdetermined systems of differential equations are reduced to PDE systems of dimension less than that of the original PDE systems. The Euler and Navier — Stokes equations themselves act as the constraint equations, and the dimension is reduced for the integral equations, which are obtained using the Helmholtz theorem on the expansion of an arbitrary vector field into vortex and potential components. The peculiarity of this work lies in the fact that all the equations reduced in dimension are obtained in an explicit form, in contrast to the previous works of the authors, where up to 200–300 equations with reduced dimensions were proposed. In reduced systems of integral-differential equations, there is integration over space, therefore, reduction over spatial variables by given method is impossible. If this method is generalized a little, then we can obtain a reduced system of integral equations, where the unknowns have no derivatives with respect to space, but integration over space remains. Non-stationary new integral equations are also obtained, which determine the evolution of the flow. In Appendices A and B, sufficient conditions for the correctness of the reduced system of integral-differential equations are derived. Conditions are found under which the Euler equations follow directly from these reduced equations. It is shown how the time-varying data on some surface of this stream should be related. The change of variables, which is used in the reduction of the Navier — Stokes equations, is also investigated. The resulting integral equations can be used to study complex vortices in the atmosphere. For example, if devices measuring data on its surfaceare installed on a flying object, then, knowing the vorticity profile $\omega_0(r)$ at a certain moment in time, it is possible, by solving these equations, to track vortexactivity at a distance from this object in real time.

Détail

Keywords: overdetermined systems of differential equations, dimension of differential equations, hydrodynamics, Navier — Stokes equations.
Mots-clés : Euler equations