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

Engineering Journal Science and Innovation ◽

10.18698/2308-6033-2020-5-1978 ◽

2020 ◽

Author(s):

V.M. Ovsyannikov

Keyword(s):

Continuity Equation ◽

Flow Region ◽

Time Interval ◽

Third Order ◽

Amount Of Substance ◽

Wave Dynamics ◽

Lifshitz Equation ◽

Order Increase ◽

Physical Laws ◽

Time Linear

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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KETEROBSERVASIAN SISTEM LINIER DISKRIT

Jurnal Matematika UNAND ◽

10.25077/jmu.4.1.108-114.2015 ◽

2015 ◽

Vol 4 (1) ◽

pp. 108

Author(s):

Midian Manurung

Keyword(s):

Linear System ◽

Discrete Time ◽

State Vector ◽

The State ◽

Input Vector ◽

Linear Control ◽

Time Interval ◽

Initial State ◽

Time Invariant ◽

Time Linear

Given the following discrete time-invariant linear control systems:where x 2 Rnx(t + 1) = Ax(t) + Bu(t);y(t) = Cx(t);is the state vector, u 2 Rmis an input vector, y 2 Rris dened as anoutput, A 2 Rnn, B 2 Rnm, and t 2 Zis dened as time. Linear system is said to beobservable on the nite time interval [t0; t+f] if any initial state xis uniquely determinedby the output y(t) over the same time interval. In order to examine the observabilityof the system, we will use a criteria, that is by determining the observability Gramianmatrix of the system is nonsingular and rank of the observability matrix for the systemis n.

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A New Finite Element Scheme for the Boussinesq Equations

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202597000128 ◽

1997 ◽

Vol 07 (02) ◽

pp. 193-209 ◽

Cited By ~ 6

Author(s):

D. Ambrosi

Keyword(s):

Finite Element ◽

Water Waves ◽

Boussinesq Equations ◽

High Order Accuracy ◽

Third Order ◽

Wave Dynamics ◽

Nonlinear Advection ◽

Spurious Oscillations ◽

Taylor Galerkin Method ◽

The Mathematical Model

In this paper we consider the Boussinesq equations to simulate the motion of water waves with a moderate curvature of the free surface. The mathematical model describing the wave dynamics is introduced together with a short description of its derivation, posing emphasis on the related assumptions. The discrete representation of the Boussinesq equations is faced with numerical difficulties of two kinds: the nonsymmetric character of the (nonlinear) advection–propagation operator and the presence of third-order differential terms accounting for dispersion phenomena. In this paper it is shown how it is possible to use a finite element Taylor–Galerkin method to discretize the equations, ensuring high order accuracy both in time and space and obtaining a numerical solution free of spurious oscillations.

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Time Lag in the System of Financial Transformations

International Journal of Recent Technology and Engineering - 2 ◽

10.35940/ijrte.c3869.098319 ◽

2019 ◽

Vol 8 (3) ◽

pp. 40-45

Keyword(s):

Time Lag ◽

Dynamic Environment ◽

Wave Force ◽

Time Interval ◽

Time Lags ◽

Wave Dynamics ◽

Phase Wave ◽

Study Results ◽

Impact Phase ◽

The Impact

The system of financial transformations has certain inertia. The processes of financial transformations develop in a volatile dynamic environment and require constant search for additional resources. Accordingly, this process occurs in a time interval and is recorded by a given lag gap. The purpose of this article is to study the lag as an indicator of the financial transformations process in the context of the “wave” differentiation, that is wave dynamics during the evaluation of the impact phase (wave push) and decline phase (wave surge strength). Objectives for meeting the purposes: - Build the interdependence of the GDP price deflator and the retail price index in the Russian Federation. - To make a wave projection to the phases of maturity, formation (push) and the effect of recession (surge) using the polynomial dependencies of the monetary, fiscal and business mechanisms. - Consider the time economics, as a category of economic theory for proposals in adjusting the institutions at different levels of housekeeping (economy). Research methods: the technique of wave dynamics, methods of analysis, comparison, induction and deduction were applied in this study. Results: The process of detailing the study of the time lags structure allows concluding that the complexity and ambiguity of the phenomenon of financial transformations imposes the ambiguous effect of time lags on the resultant feature, i.e. the effect of the wave force in the downturn (splash) phase. The ambiguous behavior of indicators, various subsystems instruments of monetary, fiscal and business mechanisms which are transforming and changing in each subsystem with different speed and turnover are of importance. This makes an impact both on the value of time lags and on their influence on the performance indicators of the system

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

Fluids ◽

10.3390/fluids6040162 ◽

2021 ◽

Vol 6 (4) ◽

pp. 162

Author(s):

Vladislav M. Ovsyannikov

Keyword(s):

Wave Equation ◽

Incompressible Fluid ◽

Turbulent Flows ◽

Continuity Equation ◽

Stochastic Approach ◽

High Order ◽

Amount Of Substance ◽

Initial Stage ◽

Master's Thesis ◽

Moscow City

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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DEVELOPMENT OF A CONCEPTUAL DETERMINISTIC RAINFALL-RUNOFF MODEL

Hydrology Research ◽

10.2166/nh.1973.0012 ◽

1973 ◽

Vol 4 (3) ◽

pp. 147-170 ◽

Cited By ~ 195

Author(s):

STEN BERGSTRÖM ◽

ARNE FORSMAN

Keyword(s):

Time Interval ◽

Model Structure ◽

Rainfall Runoff ◽

Soil Moisture Storage ◽

Lumped Parameter ◽

Moisture Storage ◽

Rainfall Runoff Model ◽

Physical Laws ◽

Parameter Values ◽

Runoff Model

This progress report outlines the main principles for the development of a simple conceptual rainfall-runoff model at the Swedish Meteorological and Hydrological Institute. The HBV-2 Model is based on lumped-parameter approximations to the physical laws governing infiltration, percolation and runoff formation. The time interval is one day. The model structure includes a soil moisture storage, an upper zone storage and a lower zone storage. A procedure for evaluating the parameter values is described. Examples of applications to several test catchments in various hydrologic settings are included.

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Interpolation Environment of Tensor Mathematics at the Corpuscular Stage of Computational Experiments in Hydromechanics

EPJ Web of Conferences ◽

10.1051/epjconf/201817302004 ◽

2018 ◽

Vol 173 ◽

pp. 02004

Author(s):

Alexander Bogdanov ◽

Alexander Degtyarev ◽

Vasily Khramushin ◽

Yulia Shichkina

Keyword(s):

Small Time ◽

Computational Experiments ◽

Time Interval ◽

Free Boundaries ◽

Kinematic Parameters ◽

Physical Processes ◽

Discontinuous Solutions ◽

Small Time Interval ◽

Empirical Generalizations ◽

Physical Laws

Stages of direct computational experiments in hydromechanics based on tensor mathematics tools are represented by conditionally independent mathematical models for calculations separation in accordance with physical processes. Continual stage of numerical modeling is constructed on a small time interval in a stationary grid space. Here coordination of continuity conditions and energy conservation is carried out. Then, at the subsequent corpuscular stage of the computational experiment, kinematic parameters of mass centers and surface stresses at the boundaries of the grid cells are used in modeling of free unsteady motions of volume cells that are considered as independent particles. These particles can be subject to vortex and discontinuous interactions, when restructuring of free boundaries and internal rheological states has place. Transition from one stage to another is provided by interpolation operations of tensor mathematics. Such interpolation environment formalizes the use of physical laws for mechanics of continuous media modeling, provides control of rheological state and conditions for existence of discontinuous solutions: rigid and free boundaries, vortex layers, their turbulent or empirical generalizations.

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Probe size and 5th-order aberration

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100125609 ◽

1987 ◽

Vol 45 ◽

pp. 134-135 ◽

Cited By ~ 1

Author(s):

Zhifeng Shao

Keyword(s):

Electron Probe ◽

Spherical Aberration ◽

Wave Length ◽

Cylindrical Symmetry ◽

Electron Wave ◽

Third Order ◽

The Third ◽

Probe Radius ◽

Small Probe ◽

Probe Size

A small electron probe has many applications in many fields and in the case of the STEM, the probe size essentially determines the ultimate resolution. However, there are many difficulties in obtaining a very small probe.Spherical aberration is one of them and all existing probe forming systems have non-zero spherical aberration. The ultimate probe radius is given byδ = 0.43Csl/4ƛ3/4where ƛ is the electron wave length and it is apparent that δ decreases only slowly with decreasing Cs. Scherzer pointed out that the third order aberration coefficient always has the same sign regardless of the field distribution, provided only that the fields have cylindrical symmetry, are independent of time and no space charge is present. To overcome this problem, he proposed a corrector consisting of octupoles and quadrupoles.

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Children’s Recall of Approximations to English

Journal of Speech and Hearing Research ◽

10.1044/jshr.1602.201 ◽

1973 ◽

Vol 16 (2) ◽

pp. 201-212 ◽

Cited By ~ 3

Author(s):

Elizabeth Carrow ◽

Michael Mauldin

Keyword(s):

Language Development ◽

Fourth Order ◽

Third Order ◽

Memory Processes ◽

The Third ◽

General Index ◽

General Linguistic

As a general index of language development, the recall of first through fourth order approximations to English was examined in four, five, six, and seven year olds and adults. Data suggested that recall improved with age, and increases in approximation to English were accompanied by increases in recall for six and seven year olds and adults. Recall improved for four and five year olds through the third order but declined at the fourth. The latter finding was attributed to deficits in semantic structures and memory processes in four and five year olds. The former finding was interpreted as an index of the development of general linguistic processes.

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