A first integral of Navier–Stokes equations and its applications
A first integral of Navier–Stokes equations and its applications
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纳维-斯托克斯方程的一阶积分及其应用
DOI：
10.1098/rspa.2010.0157
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发表时间：
2011
期刊：
影响因子：
--
通讯作者：
P. Gaskell
中科院分区：
文献类型：
--
作者：
M. Scholle;A. Haas;P. Gaskell
虽然众所周知，伯努利方程是在没有涡量的情况下作为欧拉方程的第一次积分得到的，而在涡量不为零的情况下，它们的第一次积分可以用无粘流的Clebsch变换得到，但粘性流的一般化过程仍然是难以捉摸的。因此，本文构造了定常流Navier-Stokes方程的第一积分。在二维流动的情况下，这是可能的，通过制定的控制方程的复变量，并引入一个新的标量位。相关的边界条件被认为是，并提出了一个扩展的理论到三维。通过计算雷诺数修正的层流剪切流之间的狭窄间隙中产生的平面移动和静止的波浪壁，润滑问题中经常遇到的新方法的能力被证明。它突出了第一个积分作为一个合适的工具，在流体动力学的新的分析和数值方法的发展。
Although it is well known that Bernoulli's equation is obtained as the first integral of Euler's equations in the absence of vorticity and that in the case of non-vanishing vorticity a first integral of them can be found using the Clebsch transformation for inviscid flow, generalization of the procedure for viscous flow has remained elusive. Accordingly, in this paper, a first integral of the Navier–Stokes equations for steady flow is constructed. In the case of a two-dimensional flow, this is made possible by formulating the governing equations in terms of complex variables and introducing a new scalar potential. Associated boundary conditions are considered, and an extension of the theory to three dimensions is proposed. The capabilities of the new approach are demonstrated by calculating a Reynolds number correction to the laminar shear flow generated in the narrow gap between a flat moving and a stationary wavy wall, as is often encountered in lubrication problems. It highlights the first integral as a suitable tool for the development of new analytical and numerical methods in fluid dynamics.