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Integrals of a One-Dimensional Continuity Equation Along the Trajectories of Liquid Planes

Dowkontt S., Dowkontt G.

Machine Dynamics Research

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2018

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Vol. 42, No. 4

17--31

EN

Authors presents the integration of a one-dimensional continuity equation of a new flow model, along the trajectories of liquid planes. That was made possible by expressing the fluid velocity u with partial derivatives of the function ζ - a function which expresses the position of liquid planes in the flow relative to the positions they occupied when the fluid was at rest. Consequently, the one-dimensional continuity equation has become integrable. This work was signaled in previous authors articles with a description of the process of obtaining formulas, that showing partial derivatives of the function ζ - by integrating a one-dimensional continuum equation along the fluid plane trajectory. In addition, this work is a complement of the previous works, in which the integral of the one-dimensional continuity equation along the straight lines x = const and t = const in the rectangular coordinate system x,t were derived. This work also includes the integral of the differential equation of liquid plane trajectories expressing the function η, which keeps a constant value along these trajectories. In addition, the content of the work consists a necessary explanations and, on the end, additional proof of the derived equations.

2

Integrals of the one-dimensional continuity equation

Dowkontt S., Dowkontt G.

TTS Technika Transportu Szynowego

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2016

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R. 23, nr 12

77--81

EN

The authors analyze the method used by Cauchy and Lagrange to obtain the integral of continuity equation. The authors propose their own method of integration using Schwarz’ theorem. As a result, the authors obtain a greater number of possible solutions with a higher level of generality while also being able to identify the basic disadvantages of the Cauchy-Lagrangian method. Further, the authors conducted a detailed interpretation of the results of their solution.

3

The cauchy and lagrange integral of the euler equation of motion

Dowkontt G., Dowkontt G.

TTS Technika Transportu Szynowego

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2015

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R. 22, nr 12

423--426, CD

PL

Autorzy artykułu dokonują analizy metody zastosowanej przez Cauchy’ego i Lagrange’a dla uzyskania całki równania ruchu Eulera. Na tej podstawie stawiają hipotezę, że całka Cauchy’ego i Lagrange’a nie jest jedyną całką równania ruchu Eulera. Autorzy artykułu przedstawiają krótką procedurę wykorzystującą twierdzenie Schwarza, której zastosowanie doprowadziło do uzyskania rozwiązania równania ruchu Eulera składającego się z dwóch całek. Przedstawione przez autorów rozwiązanie problemu całkowania równania ruchu Eulera stanowi w istocie przypadek jakościowo inny, bo o większym stopniu ogólności.

EN

The authors analyse the method used by Cauchy and Lagrange to obtain the integral of the Euler equation of motion. The authors hypothesize that the Cauchy and Lagrange integral is not the only integral of the Euler equation of motion. The authors present a brief procedure using the Schwarz theorem, which led to asolution of the Euler equation of motion consisting of two integrals. The solution presented by the authors is probably the most general and comprehensive solution to the problem of the integration of the Euler equation of motion.