Полная производная по времени от ротора ускорения

Last modified by Alexey Popov on 2019/06/07 10:34

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

где 

\nabla_{\mathbf v} = \frac{\partial }{\partial v_x}+\frac{\partial }{\partial v_y}+\frac{\partial }{\partial v_z}

 \left(\mathbf {u} \cdot \nabla \right)(\nabla \times \mathbf a)  =\frac{1}{2} \left ( \nabla \left((\nabla \times \mathbf a)  \cdot \mathbf {u} \right)- (\nabla \times \mathbf a)  \left(\nabla \cdot \mathbf {u} \right)\,+\,\mathbf {u} \left(\nabla \cdot (\nabla \times \mathbf a)  \right) \,+\,\nabla \times \left((\nabla \times \mathbf a)  \times \mathbf {u} \right)  - (\nabla \times \mathbf a) \times \left(\nabla \times \mathbf {u} \right)-\mathbf {u} \times \left(\nabla \times (\nabla \times \mathbf a)  \right) \right)

((\nabla \times \mathbf a) \cdot \nabla_{\mathbf v})(\nabla \times \mathbf a) = \frac{1}{2}\nabla_{\mathbf v} ((\nabla \times \mathbf a) \cdot (\nabla \times \mathbf a) ) - (\nabla \times \mathbf a)  \times (\nabla_{\mathbf v} \times (\nabla \times \mathbf a) )

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Created by Alexey Popov on 2019/06/07 10:31