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Open Access

REVIEW

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

Artur V. Dmitrenko1,2,*, Vladislav M. Ovsyannikov2
1 Department of Physics, National Research Nuclear University (MEPhI), Kashirskoye Shosse 31, Moscow, 115409, Russian Federation
2 Department of Thermal Engineering and Department of Waterways, Ports and Hydraulic Structures, Russian University of Transport (MIIT), Obraztsova Street 9, Moscow, 127994, Russian Federation
* Corresponding Author: Artur V. Dmitrenko. Email: email,email

Fluid Dynamics & Materials Processing https://doi.org/10.32604/fdmp.2024.048389

Received 06 December 2023; Accepted 11 March 2024; Published online 22 April 2024

Abstract

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.

Keywords

Quadratic invariant; cubic invariant; continuity equation; generation of periodic waves; N.E. Zhukovsky’s hydraulic shock; turbulence

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