Модифицированное уравнение Эйлера

Last modified by Alexey Popov on 2019/02/26 16:25

${\frac {\partial \mathbf {u} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {u} =\mathbf {g} -{\frac {1}{\rho }}\nabla p$

т.к. $\mathbf {u}$ - есть Полная производная от вектора перемещения получим

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

$\boldsymbol \omega = \nabla \times \mathbf u + \frac{1}{c_s^2}\mathbf u \times \mathbf a$

$\nabla \cdot \mathbf {r} \mathbf {u}^2 = - \frac{\partial \mathbf {r} \cdot \mathbf {u}}{\partial t}$

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

$\nabla \times \mathbf {r} \mathbf {u}^2 = 2 \frac{\partial \mathbf {r} \times \mathbf {u}}{\partial t}$

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

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

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

${{\left( {uv} \right)''} = {\left[ {{{\left( {uv} \right)}^\prime }} \right]^\prime } } = {{\left( {u'v + uv'} \right)^\prime } } = {{\left( {u'v} \right)^\prime } + {\left( {uv'} \right)^\prime } } = {u''v + u'v' + u'v' + uv'' = u''v + 2u'v' + uv''.}$

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

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Created by Alexey Popov on 2019/02/26 11:28