Расширенная система уравнений Эйлера для течения жидкости

Last modified by Alexey Popov on 2019/08/06 17:44

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

$\boldsymbol \omega - \frac{1}{c^2}\frac{\nabla (\rho(\mathbf {u}\cdot (\mathbf r \times \mathbf a)))}{\rho}= - \nabla \times \mathbf {u} + \frac{1}{c^2}\nabla(\mathbf {u}\cdot (\mathbf r \times \mathbf a)) + \frac{1}{c^2}\frac{\partial (\mathbf r \times \mathbf a)}{\partial t}$

$\nabla \cdot (\mathbf r \times \mathbf a) + \frac{1}{c^2}\frac{\partial \mathbf {u}\cdot (\mathbf r \times \mathbf a)}{\partial t} = \mathcal{A}$

$\nabla \cdot \mathbf u + \frac{1}{2}\frac{1}{c^2}\frac{\partial (\mathbf u \cdot \mathbf u)}{\partial t} = \mathcal{W}$

$\varphi = -\square (\mathbf r \times \mathbf u)\cdot(\mathbf r \times \mathbf u)$

$\mathbf a = - \frac{D\mathbf u}{Dt} - \nabla \times (\mathbf r \times \mathbf a) - \mathbf r \times \square (\mathbf r \times \mathbf a)$

$\boldsymbol{\omega} = \nabla \times \mathbf u - \frac{1}{c^2}\frac{D (\mathbf r \times \mathbf a)}{Dt} - \mathbf r \times \square \mathbf u$

$\nabla \cdot \mathbf u + \frac{1}{c^2}\frac{\partial (\mathbf u \cdot \mathbf u)}{\partial t} = 0$

$\nabla \cdot (\mathbf r \times \mathbf a) + \frac{1}{c^2}\frac{\partial (\mathbf u \cdot (\mathbf r \times \mathbf a))}{\partial t} = 0$

$\nabla \cdot \boldsymbol{\omega} = \frac{1}{c^4}\frac{\partial^2 \mathbf u \cdot (\mathbf r \times \mathbf a)}{\partial t^2} +\frac{1}{2}\frac{1}{c^2}(\mathbf r \times \mathbf a){\cdot }\nabla ^{2}\mathbf {u} \,-\frac{1}{2}\frac{1}{c^2}\,\mathbf {u} {\cdot }\nabla ^{2}\!(\mathbf r \times \mathbf a) \,\,-\,\frac{1}{c^2}\nabla {\cdot }(\mathbf {u} {\times }\mathbf a)-\,\frac{1}{c^2}\nabla {\cdot }(\mathbf {u} {\times }\frac{D\mathbf u}{Dt}) - \frac{1}{2}\frac{1}{c^2}\nabla ^{2}((\mathbf r \times \mathbf a) \cdot \mathbf {u} )$

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Created by Alexey Popov on 2019/07/26 12:14