Using time-linear Cauchy—Helmholtz formulas in the derivations of the continuity equation of Euler, Ostrogradsky, Zhukovsky
Authors: Ovsyannikov V.M.
Published in issue: #5(101)/2020
DOI: 10.18698/2308-6033-2020-5-1978
Category: Mechanics | Chapter: Mechanics of Liquid, Gas, and Plasma
The paper indicates the reason for the appearance of the second and third order terms of smallness of the continuity equation, which in the wave dynamics lead to the appearance of spontaneous self-oscillations. The compatibility of periodic local non-conservation of the amount of substance in the control shape with the integral law of conservation of the total amount of substance in the flow region is demonstrated. It is shown that taking into account terms of the second and third order of smallness is equivalent to the order increase of time differentiality of the continuity equation, which gives an extension of the class of fluid flow movements. Attention is drawn to the similarity of constructions of inhomogeneous terms of the wave equation arising due to the convective terms of the L.D. Landau and E.M. Lifshitz equation of motion and due to the second-order smallness terms of the Euler continuity equation. It is also shown that the loss of members of second and third order of smallness in time in the derivation of the M.V. Ostrogradsky continuity equation occurs due to the neglect of the flow of fluid particles crossing twice the border of the convex control shape along the secant line, coming outside within the finite time interval and not included in the substance balance. The possibility and validity of the appearance of terms of high-order smallness in other physical laws containing the divergence operator is indicated.
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