FORMATION AND USAGE OF DIFFERENTIAL MOMENTUM EQUATION1)

HUANG Shuxin,2)

Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200240, China

Key Laboratory of Hydrodynamics of the Ministry of Education, Shanghai Jiao Tong University, Shanghai 200240, China

State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

 基金资助: 1)上海交通大学基金资助项目(JG010003/006)

Abstract

The differential momentum equation in fluid mechanics is a fundamental equation, which is usually called the motion equation in textbook. The Navier-Stokes equation can be deduced from the momentum equation by adding some assumptions. The present manuscript shows the formation and usage of the equation. The equation was once reported in the work in 1828 of French Augustin L. Cauchy (1789—1857). George G. Stokes (1819—1903) in England could be the first person who used the equation correctly in the flow problem of constant-viscosity fluid according to the literatures. Moreover, English Ronald S. Rivlin (1915—2005) could use the equation in the viscoelastic flow problem firstly.

Keywords： momentum equation; formation and usage; fluid mechanics; teaching

HUANG Shuxin. FORMATION AND USAGE OF DIFFERENTIAL MOMENTUM EQUATION1). Mechanics in Engineering, 2022, 44(2): 390-392 DOI:10.6052/1000-0879-21-277

1 动量方程的历史

$\rho (\frac{\partial v}{\partial t}+v\cdot \nabla v)=\rho f-\nabla p+\nabla \cdot \tau$

$\frac{\partial A}{\partial x}+\frac{\partial F}{\partial y}+\frac{\partial E}{\partial z}+\rho(X-\chi)=0$
$\frac{\partial F}{\partial x}+\frac{\partial B}{\partial y}+\frac{\partial D}{\partial z}+\rho(Y-\delta)=0$
$\frac{\partial E}{\partial x}+\frac{\partial D}{\partial y}+\frac{\partial C}{\partial z}+\rho(Z-\tau)=0$

Stokes[8]在1845年的工作中使用的以应力表示的运动方程是

$\rho(\frac{{\rm D}u}{{\rm D}t}-X)+\frac{\partial P_{1} }{\partial x}+\frac{\partial T_{3} }{\partial y}+\frac{\partial T_{2} }{\partial z}=0$

2 动量方程的使用

Stokes从式(3)出发还给出了流体力学中的基本方程,即常说的NS方程。流体力学主要是研究水和空气的宏观运动的学科,因此NS方程在航空、航海、环境以及工业等领域有着广泛的用途。所以,式(1)在流体力学上的使用和Stokes这个人有关,式(1)或许还可以称为斯托克斯运动方程。

$\tau =2\mu d+2\varPsi_{2} d^{2}$

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