Полная производная по времени от ротора скорости

Last modified by Alexey Popov on 2019/06/26 18:28

$\boldsymbol \varepsilon = \frac{D\boldsymbol \omega(t,\mathbf r(t),\mathbf v(t))}{Dt} = \frac{\partial \boldsymbol \omega}{\partial t} + (\mathbf v \cdot \nabla)\boldsymbol \omega + (\mathbf a \cdot \nabla_{\mathbf v})\boldsymbol \omega$

где
$\boldsymbol \omega = \nabla \times \mathbf v$

$\left(\mathbf {v} \cdot \nabla \right)\boldsymbol \omega =\frac{1}{2} \left ( \nabla \left(\boldsymbol \omega \cdot \mathbf {v} \right)\,+\,\nabla \times \left(\boldsymbol \omega \times \mathbf {v} \right) - \boldsymbol \omega \times \left(\nabla \times \mathbf {v} \right)-\mathbf {v} \times \left(\nabla \times \boldsymbol \omega \right) \right)$

$\mathbf a = \frac{\partial \mathbf v}{\partial t} + \frac{1}{2}\nabla (\mathbf v \cdot \mathbf v) - \mathbf v \times \boldsymbol \omega$

$\boldsymbol \omega \times \mathbf v = -\frac{\partial \mathbf v}{\partial t} - \frac{1}{2}\nabla (\mathbf v \cdot \mathbf v) + \mathbf a$

$\boldsymbol \varepsilon =\frac{1}{2} \left ( \frac{\partial \boldsymbol \omega}{\partial t} + \nabla \left(\boldsymbol \omega \cdot \mathbf {v} \right)\, + \nabla \times \mathbf a -\mathbf {v} \times \left(\nabla \times \boldsymbol \omega \right) \right)$

$2 \boldsymbol \varepsilon = \frac{\partial \boldsymbol \omega}{\partial t} + \nabla \left(\boldsymbol \omega \cdot \mathbf {v} \right)\, + \nabla \times \mathbf a -\mathbf {v} \times \left(\nabla \times \boldsymbol \omega \right)$

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Created by Alexey Popov on 2019/06/17 18:28