### N.E. Zhukovsky's strain ellipsoid taking into account terms of the second order of smallness

**Published:**16.04.2018

**Authors:** Ovsyannikov V.M.

**Published in issue: **#5(77)/2018

**DOI: **10.18698/2308-6033-2018-5-1756

**Category:** Mechanics | **Chapter:** Mechanics of Liquid, Gas, and Plasma

The study deals with the physical interpretation of the terms of the second order of smallness found in the continuity equation. The article shows that these terms, which are usually discarded, also contribute to, for instance, periodic wave generation, increasing the vibration intensity computed by L.D. Landau and E.M. Lifshitz more than twofold. N.E. Zhukovsky disregarded the terms of the second order of smallness with respect to strain time or flow time when writing down the continuity equation for plotting the strain ellipsoid. He did calculate a number of additional terms; this is why he could have balanced the amount of substance taking into account terms of the second and third orders of smallness. We also identified that the expression for the curl of the velocity vector in terms of angular velocity is inaccurate: at the level of accuracy defined by terms of the second order of smallness there should be additional terms in the velocity curl as a function of angular velocity. We analysed the continuity equation found in one of the articles by L. Euler that he presented in 1752 at the Royal Prussian Academy of Sciences, and we show that for incompressible fluid those additional terms create local nonconservation. This case should only be considered as a model one that is not possible in reality. For compressible gas this local nonconservation becomes periodic and describes actually existing flows with periodic pressure waves, or sound, generated by the flow. We determined that in equations of compressible gas dynamics these terms of the second order of smallness with respect to motion time found in the non-homogeneous part of the wave equation lead to generation of sound and self-excited vibrations that are not connected to external factors affecting the flow.

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