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

Abstract

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.

Direct Comparison of Peak Bulk Flow Rate of Programmable Intermittent Epidural Bolus and Manual Epidural Bolus Using a Closed-End Multiorifice Catheter: An Experimental Study

“…Figure 2 also demonstrates the notable difference between the durations taken to administer the same volume boluses, indicating a significant difference in the speed of the fluid according to the equation of continuity. 9 The highest mean flow rates were seen with the boluses administered by the student and anesthesiologist when no filter was present. The anesthesiologist was able to generate rates that were 49.18, 51.56, 31.84, and 41.82 (mL/min) for the respective 3-, 5-, 10-, and 20-mL syringes.…”

“…Figure 2 also demonstrates the notable difference between the durations taken to administer the same volume boluses, indicating a significant difference in the speed of the fluid according to the equation of continuity. 9…”

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

“…Purpose of the review shows that flow regions with different Lagrangian laws of motion of a liquid particle have different additional terms of a high order of smallness in time. It was realized that quadratic invariant terms generate density and pressure waves that are close to harmonic oscillations [11,12]. Reviews [11,12] describe solutions to problems of the formation of sound waves with potential air flowing around a right angle, cylinder.…”

“…It was realized that quadratic invariant terms generate density and pressure waves that are close to harmonic oscillations [11,12]. Reviews [11,12] describe solutions to problems of the formation of sound waves with potential air flowing around a right angle, cylinder. Monograph [10] contains a description of the formation of waves when air flows into narrow gaps.…”

“…Most of the works take into account only the quadratic invariant, which describes harmonic waves [10,11,12,15]. The quadratic invariant of the strain rate tensor is used in the method of regularization of the system of hydrodynamic equations [15] to increase the stability of iterations of T.G.…”