Полная производная от вектора перемещения

Last modified by Alexey Popov on 2019/06/26 18:19

\mathbf u = \frac{D\mathbf r}{Dt} = \frac{\partial \mathbf r}{\partial t} + \left (\mathbf u \cdot \nabla)\mathbf r

\nabla \left(\mathbf {r} \cdot \mathbf {u} \right)=\left(\mathbf {r} \cdot \nabla \right)\mathbf {u} +\left(\mathbf {u} \cdot \nabla \right)\mathbf {r} +\mathbf {r} \times \left(\nabla \times \mathbf {u} \right)+\mathbf {u} \times \left(\nabla \times \mathbf {r} \right)

\left(\mathbf {u} \cdot \nabla \right)\mathbf {r} = \nabla \left(\mathbf {r} \cdot \mathbf {u} \right)-\left(\mathbf {r} \cdot \nabla \right)\mathbf {u} - \mathbf {r} \times \left(\nabla \times \mathbf {u} \right)-\mathbf {u} \times \left(\nabla \times \mathbf {r} \right)

\nabla \times \left(\mathbf {r} \times \mathbf {u} \right)\ =\ \mathbf {r} \left(\nabla \cdot \mathbf {u} \right)\,-\,\mathbf {u} \left(\nabla \cdot \mathbf {r} \right)\,+\,\left(\mathbf {u} \cdot \nabla \right)\mathbf {r} \,-\,\left(\mathbf {r} \cdot \nabla \right)\mathbf {u}

\left(\mathbf {r} \cdot \nabla \right)\mathbf {u} \ =\ \mathbf {r} \left(\nabla \cdot \mathbf {u} \right)\,-\,\mathbf {u} \left(\nabla \cdot \mathbf {r} \right)\,+\,\left(\mathbf {u} \cdot \nabla \right)\mathbf {r} \,-\,\nabla \times \left(\mathbf {r} \times \mathbf {u} \right)

\left(\mathbf {u} \cdot \nabla \right)\mathbf {r} = \nabla \left(\mathbf {r} \cdot \mathbf {u} \right)- \mathbf {r} \left(\nabla \cdot \mathbf {u} \right)\,+\,\mathbf {u} \left(\nabla \cdot \mathbf {r} \right)\,-\,\left(\mathbf {u} \cdot \nabla \right)\mathbf {r} \,+\,\nabla \times \left(\mathbf {r} \times \mathbf {u} \right) - \mathbf {r} \times \left(\nabla \times \mathbf {u} \right)-\mathbf {u} \times \left(\nabla \times \mathbf {r} \right)

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

\left(\mathbf {u} \cdot \nabla \right)\mathbf {r} =\frac{1}{2} \left ( \nabla \left(\mathbf {r} \cdot \mathbf {u} \right)- \mathbf {r} \left(\nabla \cdot \mathbf {u} \right)\,+\,3\mathbf {u} \,+\,\nabla \times \left(\mathbf {r} \times \mathbf {u} \right) - \mathbf {r} \times \boldsymbol \omega \right)

\mathbf u = \frac{D\mathbf r}{Dt} = \frac{\partial \mathbf r}{\partial t} + \frac{1}{2} \left ( \nabla \left(\mathbf {r} \cdot \mathbf {u} \right)- \mathbf {r} \left(\nabla \cdot \mathbf {u} \right)\,+\,3\mathbf {u} \,+\,\nabla \times \left(\mathbf {r} \times \mathbf {u} \right) - \mathbf {r} \times \boldsymbol \omega \right)

2\mathbf u = 2\frac{\partial \mathbf r}{\partial t} + \nabla \left(\mathbf {r} \cdot \mathbf {u} \right)- \mathbf {r} \left(\nabla \cdot \mathbf {u} \right)\,+\,3\mathbf {u} \,+\,\nabla \times \left(\mathbf {r} \times \mathbf {u} \right) - \mathbf {r} \times \boldsymbol \omega

2\mathbf u \,-\,3\mathbf {u}= 2\frac{\partial \mathbf r}{\partial t} + \nabla \left(\mathbf {r} \cdot \mathbf {u} \right)- \mathbf {r} \left(\nabla \cdot \mathbf {u} \right) \,+\,\nabla \times \left(\mathbf {r} \times \mathbf {u} \right) - \mathbf {r} \times \boldsymbol \omega

\mathbf {u}= -2\frac{\partial \mathbf r}{\partial t} - \nabla \left(\mathbf {r} \cdot \mathbf {u} \right) \,-\,\nabla \times \left(\mathbf {r} \times \mathbf {u} \right) + \mathbf {r} \times \boldsymbol \omega

Дивергенция вектора скорости

2\nabla \cdot \mathbf u =\nabla^2(\mathbf {r} \cdot \mathbf {u}) - \mathbf r \cdot \nabla^2 \mathbf u = 2\nabla \cdot ((\mathbf u \cdot\nabla )\mathbf r)

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Created by Alexey Popov on 2019/02/26 10:33