Using time-linear Cauchy—Helmholtz formulas in the derivations of the continuity equation of Euler, Ostrogradsky, Zhukovsky

Ovsyannikov, V. M.

doi:10.18698/2308-6033-2020-5-1978

Cited by 2 publications

(8 citation statements)

References 3 publications

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“…The use of the exponential Lagrangian law of motion of a liquid particle [23] in the derivation of the continuity equation does not provide terms of a high order of smallness in the continuity equation. The change in the coordinates of the points of the fluid is found from the solution of the Cauchy problem for a system of differential equations…”

Section: Differential (Local) and Integral Conservationmentioning

confidence: 99%

“…There are streamlines that follow the secant line near the boundary of the convex control figure. A liquid particle can cross the border of the control two times [23] in time interval t − t 0 .…”

Section: Differential (Local) and Integral Conservationmentioning

confidence: 99%

“…This is a property of convex figures. These particles reduce the balance of matter, which is controlled by the Gauss-Ostrogradsky theorem The Gauss-Ostrogradsky theorem uses direction cosines [23] and ignores such streamlines. Allowance for double crossing of the boundary in time 𝑡 − 𝑡 gives terms of a high order of smallness.…”

Section: Differential (Local) and Integral Conservationmentioning

confidence: 99%

“…The Gauss-Ostrogradsky theorem for integral sums is discussed in the article [25]. The Gauss-Ostrogradsky theorem uses direction cosines [23] and ignores such streamlines. Allowance for double crossing of the boundary in time t − t 0 gives terms of a high order of smallness.…”

Section: Differential (Local) and Integral Conservationmentioning

confidence: 99%

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Euler’s Equation of Continuity: Additional Terms of High Order of Smallness—An Overview

Ovsyannikov

2021

Fluids

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Professor N.E. Zhukovsky was a famous Russian mechanic and engineer. In 1876 he defended his master’s thesis at Moscow University. At a careful reading of N.E. Zhukovsky’s master’s thesis in 1997, V.A. Bubnov—a professor at the Moscow City Pedagogical University—discovered terms of the second order of smallness in the continuity equation for an incompressible fluid. Zhukovsky calculated them, but did not use the amount of substance in the balance. Ten years later, the author found high-order terms in Euler’s derivation of the 1752 continuity equation for an incompressible fluid. The physical meaning of the additional terms became clear after the derivation in 2006 of the continuity equation with terms of high order of smallness for a compressible gas. The higher order terms of the smallness of the continuity equation penetrate into the inhomogeneous part of the wave equation and lead to the generation of self-oscillations, vibrations, sound, and the initial stage of turbulent pulsations. The stochastic approach ensured success in modeling turbulent flows. The use of high-order terms of smallness of the Euler continuity equation makes it possible to transfer the description of some part of the motions from the stochastic part of the equation to the deterministic part. The article contains a review of works with the derivation of the inhomogeneous wave equation. These works use additional terms of a high order of smallness in the continuity equation.

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Section: Differential (Local) and Integral Conservationmentioning

confidence: 99%

Section: Differential (Local) and Integral Conservationmentioning

confidence: 99%

Section: Differential (Local) and Integral Conservationmentioning

confidence: 99%

Section: Differential (Local) and Integral Conservationmentioning

confidence: 99%

See 2 more Smart Citations

Euler’s Equation of Continuity: Additional Terms of High Order of Smallness—An Overview

Ovsyannikov

2021

Fluids

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“…The inhomogeneous wave equation was derived by Ovsyannikov [10] in 2007. The inhomogeneous part of the wave equation contains quadratic and cubic invariants of the strain rate tensor.…”

Section: Introductionmentioning

confidence: 99%

Accounting for Quadratic and Cubic Invariants in Continuum Mechanics–An Overview

Dmitrenko,

Ovsyannikov

2024

FDMP

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The differential equations of continuum mechanics are the basis of an uncountable variety of phenomena and technological processes in fluid-dynamics and related fields. These equations contain derivatives of the first order with respect to time. The derivation of the equations of continuum mechanics uses the limit transitions of the tendency of the volume increment and the time increment to zero. Derivatives are used to derive the wave equation. The differential wave equation is second order in time. Therefore, increments of volume and increments of time in continuum mechanics should be considered as small but finite quantities for problems of wave formation. This is important for calculating the generation of sound waves and water hammer waves. Therefore, the Euler continuity equation with finite time increments is of interest. The finiteness of the time increment makes it possible to take into account the quadratic and cubic invariants of the strain rate tensor. This is a new branch in hydrodynamics. Quadratic and cubic invariants will be used in differential wave equations of the second and third order in time.

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Using time-linear Cauchy—Helmholtz formulas in the derivations of the continuity equation of Euler, Ostrogradsky, Zhukovsky

Cited by 2 publications

References 3 publications

Euler’s Equation of Continuity: Additional Terms of High Order of Smallness—An Overview

Euler’s Equation of Continuity: Additional Terms of High Order of Smallness—An Overview

Accounting for Quadratic and Cubic Invariants in Continuum Mechanics–An Overview

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