scholarly journals Correct Expression of the Material Derivative in Continuum Physics

2020 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Material Derivative ◽  
Correct Expression

The material derivative is important in continuum physics. This Letter shows that the expression $\frac{d }{dt}=\frac{\partial }{\partial t}$+ (v • ∇), used in most literature and textbooks, is incorrect. The correct expression $ \frac{d (:)}{dt} = \frac{\partial }{\partial t}$(:) + v · [∇(:)] is formulated.

scholarly journals Correct Expression of the Material Derivative in Continuum Physics

2020 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Stokes Equation ◽  
Velocity Gradient ◽  
Existence Condition ◽  
Form Solution ◽  
Navier Stokes ◽  
Solution Existence ◽  
Navier Stokes Equation ◽  
Material Derivative ◽  
The Solution Existence ◽  
Correct Expression

The material derivative is important in continuum physics. This Letter shows that the expression $\frac{d }{dt}=\frac{\partial }{\partial t}+(\bm v\cdot \bm \nabla)$, used in most literature and textbooks, is incorrect. The correct expression $ \frac{d (:)}{dt}=\frac{\partial }{\partial t}(:)+\bm v\cdot [\bm \nabla (:)]$ is formulated. The solution existence condition of Navier-Stokes equation has been proposed from its form-solution, the conclusion is that "\emph{The Navier-Stokes equation has a solution if and only if the determinant of flow velocity gradient is not zero, namely $\det (\bm \nabla \bm v)\neq 0$.}"

scholarly journals Correct Expression of Material Derivative in Continuum Physics

2020 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Material Derivative ◽  
Correct Expression

The material derivative is important in continuum physics. It shows that $\frac{d }{dt}=\frac{\partial }{\partial t}+(\bm v\cdot \bm \nabla)$ is a wrong expression. The correct expression $ \frac{d (:)}{dt}=\frac{\partial }{\partial t}(:)+\bm v\cdot [\bm \nabla (:)]$ is to be formulated.

scholarly journals Correct Expression of the Material Derivative in Continuum Physics

2020 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Stokes Equation ◽  
Velocity Gradient ◽  
Existence Condition ◽  
Form Solution ◽  
Navier Stokes ◽  
Solution Existence ◽  
Navier Stokes Equation ◽  
Material Derivative ◽  
The Solution Existence ◽  
Correct Expression

The material derivative is important in continuum physics. This Letter shows that the expression $\frac{d }{dt}=\frac{\partial }{\partial t}+(\bm v\cdot \bm \nabla)$, used in most literature and textbooks, is incorrect. The correct expression $ \frac{d (:)}{dt}=\frac{\partial }{\partial t}(:)+\bm v\cdot [\bm \nabla (:)]$ is formulated. The solution existence condition of Navier-Stokes equation has been proposed from its form-solution, the conclusion is that "\emph{The Navier-Stokes equation has solution if and only if the determinant of flow velocity gradient is not zero, namely $\det (\bm \nabla \bm v)\neq 0$.}"

Energy Approximation

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Main Lemma ◽  
The Other ◽  
Coarse Scale ◽  
Continuous Solutions ◽  
Material Derivative ◽  
Energy Variation ◽  
Energy Approximation ◽  
The Difference ◽  
Derivatives Of

This chapter presents the equations and calculations for energy approximation. It establishes the estimates (261) and (262) of the Main Lemma (10.1) for continuous solutions; these estimates state that we are able to accurately prescribe the energy that the correction adds to the solution, as well as bound the difference between the time derivatives of these two quantities. The chapter also introduces the proposition for prescribing energy, followed by the relevant computations. Each integral contributing to the other term can be estimated. Another proposition for estimating control over the rate of energy variation is given. Finally, the coarse scale material derivative is considered.

scholarly journals On a Generalization of One-Dimensional Kinetics

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1264
Author(s):  
Vladimir V. Uchaikin ◽  
Renat T. Sibatov ◽  
Dmitry N. Bezbatko
Keyword(s):  
Exact Solution ◽  
Asymptotic Behavior ◽  
Constant Velocity ◽  
Telegraph Equation ◽  
Material Derivative ◽  
Symmetric Random Walk ◽  
One Dimensional ◽  
Special Cases ◽  
Fractional Telegraph Equation ◽  
Asymmetric Case

One-dimensional random walks with a constant velocity between scattering are considered. The exact solution is expressed in terms of multiple convolutions of path-distributions assumed to be different for positive and negative directions of the walk axis. Several special cases are considered when the convolutions are expressed in explicit form. As a particular case, the solution of A. S. Monin for a symmetric random walk with exponential path distribution and its generalization to the asymmetric case are obtained. Solution of fractional telegraph equation with the fractional material derivative is presented. Asymptotic behavior of its solution for an asymmetric case is provided.

Shape Sensitivity Formulation for Axisymmetric Thermal Conducting Solids

1993 ◽  
Vol 207 (3) ◽  
pp. 209-216 ◽  
Author(s):  
Boo Youn Lee
Keyword(s):  
Shape Change ◽  
Direct Method ◽  
Design Sensitivity ◽  
Boundary Integral ◽  
Singular Boundary ◽  
Concept Design ◽  
Shape Sensitivity ◽  
Material Derivative ◽  
Thermal Conducting ◽  
Thermal Diffuser

A direct differentiation method is presented for the shape design sensitivity analysis of axisymmetric thermal conducting solids. Based purely on the standard boundary integral equation (BIE) formulation, a new BIE is derived using the material derivative concept. Design derivatives in terms of shape change are directly calculated by solving the derived BIE. The present direct method has a computational advantage over the adjoint variable method, in the sense that it avoids the problem of solving for the adjoint system with the singular boundary condition. Numerical accuracy of the method is studied through three examples. The sensitivities by the present method are compared with analytic sensitivities for two problems of a hollow cylinder and a hollow sphere, and are then compared with those by finite differences for a thermal diffuser problem. As a practical application to numerical optimization, an optimal shape of the thermal diffuser to minimize the weight under a prescribed constraint is found by use of an optimization routine.

Densities of scaling limits of coupled continuous time random walks

2016 ◽  
Vol 19 (6) ◽  
Author(s):  
Marcin Magdziarz ◽  
Tomasz Zorawik
Keyword(s):  
Fractional Differential Equations ◽  
Continuous Time ◽  
Stability Index ◽  
Scaling Limits ◽  
Explicit Formulas ◽  
Material Derivative ◽  
Lévy Walks ◽  
Continuous Time Random Walks ◽  
Fractional Differential ◽  
The Stability

AbstractIn this paper we derive explicit formulas for the densities of Lévy walks. Our results cover both jump-first and wait-first scenarios. The obtained densities solve certain fractional differential equations involving fractional material derivative operators. In the particular case, when the stability index is rational, the densities can be represented as an integral of Meijer

Nonlinear Phenomena in Axially Moving Beams with Speed-Dependent Tension and Tension-Dependent Speed

2021 ◽  
Vol 31 (03) ◽  
pp. 2150037
Author(s):  
Ling Chen ◽  
You-Qi Tang ◽  
Shuang Liu ◽  
Yuan Zhou ◽  
Xing-Guang Liu
Keyword(s):  
Differential Equations ◽  
Fourth Order ◽  
Fourier Transforms ◽  
Time Dependent ◽  
Nonlinear Phenomena ◽  
Maximum Lyapunov Exponent ◽  
Material Derivative ◽  
Domains Of Attraction ◽  
Moving Beam ◽  
Galerkin Truncation

This paper investigates some nonlinear dynamical behaviors about domains of attraction, bifurcations, and chaos in an axially accelerating viscoelastic beam under a time-dependent tension and a time-dependent speed. The axial speed and the axial tension are coupled to each other on the basis of a harmonic variation over constant initial values. The transverse motion of the moving beam is governed by nonlinear integro-partial-differential equations with the rheological model of the Kelvin–Voigt energy dissipation mechanism, in which the material derivative is applied to the viscoelastic constitutive relation. The fourth-order Galerkin truncation is employed to transform the governing equation to a set of nonlinear ordinary differential equations. The nonlinear phenomena of the system are numerically determined by applying the fourth-order Runge–Kutta algorithm. The tristable and bistable domains of attraction on the stable steady state solution with a three-to-one internal resonance are analyzed emphatically by means of the fourth-order Galerkin truncation and the differential quadrature method, respectively. The system parameters on the bifurcation diagrams and the maximum Lyapunov exponent diagram are demonstrated by some numerical results of the displacement and speed of the moving beam. Furthermore, chaotic motion is identified in the forms of time histories, phase-plane portraits, fast Fourier transforms, and Poincaré sections.

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