Wave Equation of Micro Element Centroid Velocity in Thermodynamic System

Abstract

As we all know, it is Maxwell’s electromagnetic wave equation and Hertz’s electromagnetic wave experiment that make our mobile communication wonderful today! Since the electromagnetic wave has been widely used after research in the field of electromagnetism, is there a corresponding wave in the field of thermodynamics? Starting from the relevant equations in non-equi- librium thermodynamics works, in some special cases, the wave equation of micro element centroid velocity of thermodynamic system is obtained by pure vector analysis. It is expected that the solution of the nonlinear equation can more accurately describe the causes and images of weather in theory, and then guide the artificial regulation of weather! Hope to find this wave and widely serve mankind! Or discuss it and get a more perfect result.

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Chen, Y.C. (2021) Wave Equation of Micro Element Centroid Velocity in Thermodynamic System. Open Access Library Journal, 8, 1-4. doi: 10.4236/oalib.1108082.

1. 引言

2. 用矢量分析的方法得到非线性波动方程

$\rho \frac{\text{d}\upsilon }{\text{d}t}=-\text{grad}p+\eta \Delta \upsilon +\left(\frac{1}{3}\eta +{\eta }_{\upsilon }\right)\text{graddiv}\upsilon +{\eta }_{r}\text{rot}\left(2\omega -\text{rot}\upsilon \right)$ (1)

$\frac{\text{d}\omega }{\text{d}t}=\frac{-2{\eta }_{r}}{\rho \Theta }\left(2\omega -\text{rot}\upsilon \right)$ (2)

$\rho \frac{\text{d}\upsilon }{\text{d}t}=-\text{grad}p+\eta \Delta \upsilon \text{+}{\eta }_{r}\text{rot}\left(2\omega -\text{rot}\upsilon \right)$ (3)

$\text{rot}\frac{\text{d}\omega }{\text{d}t}=\text{rot}\left(\frac{-2{\eta }_{r}}{\rho \Theta }\left(2\omega -\text{rot}\upsilon \right)\right)$ (4)

$\frac{\text{d}}{\text{d}t}\text{rot}\omega =\frac{2{\eta }_{r}}{\rho \Theta }\left[\text{rot}\left(\text{rot}\upsilon \right)-2\text{rot}\omega \right]$ (5)

$\frac{\text{d}}{\text{d}t}\text{rot}\omega =\frac{2{\eta }_{r}}{\rho \Theta }\left[-\Delta \upsilon -2\text{rot}\omega \right]$ (6)

$\rho \frac{\text{d}\upsilon }{\text{d}t}=-\text{grad}p+\eta \Delta \upsilon \text{+}2{\eta }_{r}\text{rot}\omega +{\eta }_{r}\Delta \upsilon$ (7)

$\text{rot}\omega =\frac{\rho }{2{\eta }_{r}}\frac{\text{d}\upsilon }{\text{d}t}-\frac{\eta }{2{\eta }_{r}}\Delta \upsilon -\frac{1}{2}\Delta \upsilon +\frac{1}{2{\eta }_{r}}\text{grad}p$ (8)

$\Delta \upsilon -\frac{{\rho }^{2}\Theta }{4\eta {\eta }_{r}}\frac{{\text{d}}^{2}\upsilon }{\text{d}{t}^{2}}\text{=}\frac{\rho }{\eta }\frac{\text{d}\upsilon }{\text{d}t}+\frac{1}{\eta }\text{grad}p-\frac{\rho \Theta \left(\eta +{\eta }_{r}\right)}{4\eta {\eta }_{r}}\frac{\text{d}}{\text{d}t}\Delta \upsilon +\frac{\rho \Theta }{4\eta {\eta }_{r}}\frac{\text{d}}{\text{d}t}\text{grad}p$ (9)

${n}^{-2}=\frac{{\rho }^{2}\Theta }{4\eta {\eta }_{r}}$ ，则上方程为：

$\Delta \upsilon -\frac{1}{{n}^{2}}\frac{{\text{d}}^{2}\upsilon }{\text{d}{t}^{2}}\text{=}\frac{\rho }{\eta }\frac{\text{d}\upsilon }{\text{d}t}+\frac{1}{\eta }\text{grad}p-\frac{\rho \Theta \left(\eta +{\eta }_{r}\right)}{4\eta {\eta }_{r}}\frac{\text{d}}{\text{d}t}\Delta \upsilon +\frac{\rho \Theta }{4\eta {\eta }_{r}}\frac{\text{d}}{\text{d}t}\text{grad}p$ (10)

$n=\frac{2}{\rho }\sqrt{\frac{\eta {\eta }_{r}}{\Theta }}$ (11)

3. 结论与展望

Conflicts of Interest

The author declares no conflicts of interest.

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