涡量方程 (英语: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:
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{\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 表示物质导数 ,u 为流速 ,ρ 为流体密度 ,p 为压强 ,τ 为粘性应力张量 ,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
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{\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}}}
对于保守外力 作用下的不可压缩流体 ,涡量方程可以简化为
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{\displaystyle {\frac {D{\boldsymbol {\omega }}}{Dt}}=\left({\boldsymbol {\omega }}\cdot \nabla \right)\mathbf {u} +\nu \nabla ^{2}{\boldsymbol {\omega }}}
其中ν 为运动黏度 ,∇2 为拉普拉斯算符 。
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. (编). Semi-Groups of Operators: Theory and Applications. Boston: Birkhauser. 2000: 296–303 [2016-12-10 ] . (原始内容存档 (PDF) 于2016-03-03).
Krigel, A. M. Vortex evolution. Geophysical, Astrophysical Fluid Dynamics. 1983, 24 : 213–223.