查看“︁涡量方程”︁的源代码

'''涡量方程'''（{{lang-en|vorticity equation}}）是[[流体力学]]中描述[[流体]]质点[[涡量]]变化的方程。可压缩[[牛顿流体]]的涡量方程表达式为：
:<math>\begin{align}
\frac{D\boldsymbol\omega}{Dt} &= \frac{\partial \boldsymbol \omega}{\partial t} + (\mathbf u \cdot \nabla) \boldsymbol \omega \\
&= (\boldsymbol \omega \cdot \nabla) \mathbf u - \boldsymbol \omega (\nabla \cdot \mathbf u) + \frac{1}{\rho^2}\nabla \rho \times \nabla p + \nabla \times \left( \frac{\nabla \cdot \tau}{\rho} \right) + \nabla \times \left( \frac{B}{\rho} \right) \end{align}</math>
其中{{math|{{sfrac|''D''|''Dt''}}}}表示[[物质导数]]，{{math|'''u'''}}为[[流速]]，{{mvar|ρ}}为流体[[密度]]，{{mvar|p}}为[[压强]]，{{mvar|τ}}为[[粘性应力张量]]，{{math|'''B'''}}为流体所受外力。方程右边第一项表示[[涡旋伸展]]。使用[[爱因斯坦求和约定]]指标记号，上式又可写作

:<math>\begin{align}
\frac{d\omega_i}{dt} &= \frac{\partial \omega_i}{\partial t} + v_j \frac{\partial \omega_i}{\partial x_j} \\
&= \omega_j \frac{\partial v_i}{\partial x_j} - \omega_i \frac{\partial v_j}{\partial x_j} 
+ e_{ijk}\frac{1}{\rho^2}\frac{\partial \rho}{\partial x_j}\frac{\partial p}{\partial x_k}
+ e_{ijk}\frac{\partial}{\partial x_j}\left(\frac{1}{\rho}\frac{\partial \tau_{km}}{\partial x_m}\right)
+ e_{ijk}\frac{\partial B_k }{\partial x_j} \end{align}</math>

对于[[保守力|保守外力]]作用下的[[不可压缩流体]]，涡量方程可以简化为
:<math>\frac{D\boldsymbol\omega}{Dt} = \left(\boldsymbol{\omega} \cdot \nabla\right) \mathbf{u} + \nu \nabla^2 \boldsymbol{\omega}</math>
其中{{mvar|ν}}为[[黏度|运动黏度]]，{{math|∇<sup>2</sup>}}为[[拉普拉斯算符]]。

== 参考文献 ==
*{{cite journal | first1= Utpal |last1=Manna |first2=S.&nbsp;S. |last2=Sritharan | title= Lyapunov Functionals and Local Dissipativity for the Vorticity Equation in {{mvar|L{{isup|p}}}} and Besov spaces | journal = Differential and Integral Equations | volume= 20 |number =5 |year= 2007|pages= 581–598}}
* {{cite book |first1=V. |last1=Barbu |first2=S.&nbsp;S. |last2=Sritharan |chapter={{mvar|M}}-Accretive Quantization of the Vorticity Equation |title=Semi-Groups of Operators: Theory and Applications |editor-first=A.&nbsp;V. |editor-last=Balakrishnan |publisher=Birkhauser |location=Boston |date=2000 |page=296–303 |chapter-url=http://www.nps.edu/Academics/Schools/GSEAS/SRI/BookCH12.pdf |access-date=2016-12-10 |archive-date=2016-03-03 |archive-url=https://web.archive.org/web/20160303201807/http://www.nps.edu/Academics/Schools/GSEAS/SRI/BookCH12.pdf |dead-url=no }}
* {{cite journal|first=A.&nbsp;M. |last=Krigel |title=Vortex evolution |journal=Geophysical, Astrophysical Fluid Dynamics |date=1983 |volume=24 |page=213–223}}

[[Category:流体力学中的方程]]