Ротор вектора скорости

Last modified by Alexey Popov on 2019/02/28 14:52

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

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

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

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

$\nabla \cdot \mathbf u = - \frac{1}{2} \frac{1}{c_s^2} \frac{\partial (\mathbf u \cdot \mathbf u)}{\partial t}$

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

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

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

$\nabla \times \mathbf u = - \frac{1}{c_s^2} \frac{\partial \mathbf a}{\partial t} \times \mathbf {r} + \frac{1}{c_s^2} \frac{\partial^2 \mathbf u}{\partial t^2} \times \mathbf {r} - \frac{1}{c_s^2} \frac{\partial \mathbf u \times \boldsymbol \omega}{\partial t} \times \mathbf {r} -\,\nabla \times (\nabla \times \left(\mathbf {r} \times \mathbf {u} \right)) + \nabla \times (\mathbf {r} \times \boldsymbol \omega)$

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Created by Alexey Popov on 2019/02/27 12:39