渦量方程

渦量方程（英語：vorticity equation）是流體力學中描述流體質點渦量變化的方程。可壓縮牛頓流體的渦量方程表達式為：

${\displaystyle {\begin{aligned}{\frac {D{\boldsymbol {\omega }}}{Dt}}&={\frac {\partial {\boldsymbol {\omega }}}{\partial t}}+(\mathbf {u} \cdot \nabla ){\boldsymbol {\omega }}\\&=({\boldsymbol {\omega }}\cdot \nabla )\mathbf {u} -{\boldsymbol {\omega }}(\nabla \cdot \mathbf {u} )+{\frac {1}{\rho ^{2}}}\nabla \rho \times \nabla p+\nabla \times \left({\frac {\nabla \cdot \tau }{\rho }}\right)+\nabla \times \left({\frac {B}{\rho }}\right)\end{aligned}}}$

其中D/Dt表示物質導數，u為流速，ρ為流體密度，p為壓強，τ為粘性應力張量，B為流體所受外力。方程右邊第一項表示渦旋伸展。使用愛因斯坦求和約定指標記號，上式又可寫作

${\displaystyle {\begin{aligned}{\frac {d\omega _{i}}{dt}}&={\frac {\partial \omega _{i}}{\partial t}}+v_{j}{\frac {\partial \omega _{i}}{\partial x_{j}}}\\&=\omega _{j}{\frac {\partial v_{i}}{\partial x_{j}}}-\omega _{i}{\frac {\partial v_{j}}{\partial x_{j}}}+e_{ijk}{\frac {1}{\rho ^{2}}}{\frac {\partial \rho }{\partial x_{j}}}{\frac {\partial p}{\partial x_{k}}}+e_{ijk}{\frac {\partial }{\partial x_{j}}}\left({\frac {1}{\rho }}{\frac {\partial \tau _{km}}{\partial x_{m}}}\right)+e_{ijk}{\frac {\partial B_{k}}{\partial x_{j}}}\end{aligned}}}$

對於保守外力作用下的不可壓縮流體，渦量方程可以簡化為

${\displaystyle {\frac {D{\boldsymbol {\omega }}}{Dt}}=\left({\boldsymbol {\omega }}\cdot \nabla \right)\mathbf {u} +\nu \nabla ^{2}{\boldsymbol {\omega }}}$

其中ν為運動黏度，∇²為拉普拉斯算符。

參考文獻

• Manna, Utpal; Sritharan, S. S. Lyapunov Functionals and Local Dissipativity for the Vorticity Equation in LTemplate:Isup and Besov spaces. Differential and Integral Equations. 2007, 20 (5): 581–598.
• Barbu, V.; Sritharan, S. S. M-Accretive Quantization of the Vorticity Equation (PDF). Balakrishnan, A. V. (編). Semi-Groups of Operators: Theory and Applications. Boston: Birkhauser. 2000: 296–303 [2016-12-10]. （原始內容存檔 (PDF)於2016-03-03）.
• Krigel, A. M. Vortex evolution. Geophysical, Astrophysical Fluid Dynamics. 1983, 24: 213–223.