« Utilisateur:Ellande/Brouillon » : différence entre les versions

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Ligne 1 :
{{Utilisateur:Alasjourn/Carnet brouillon}}
<math>\mathrm d \left( \rho \vec{v} \right)
=\frac{\partial \left( \rho \vec{v} \right)}{\partial t}{\mathrm d t}
+\frac{\partial \left( \rho \vec{v} \right)}{\partial x}{\mathrm d x}
+\frac{\partial \left( \rho \vec{v} \right)}{\partial y}{\mathrm d y}
+\frac{\partial \left( \rho \vec{v} \right)}{\partial z}{\mathrm d z}
</math>
 
<math>\frac{\mathrm d \left( \rho \vec{v} \right)}{\mathrm d t}
=\frac{\partial \left( \rho \vec{v} \right)}{\partial t}
+\frac{\partial \left( \rho \vec{v} \right)}{\partial x}\frac{\mathrm d x}{\mathrm d t}
+\frac{\partial \left( \rho \vec{v} \right)}{\partial y}\frac{\mathrm d y}{\mathrm d t}
+\frac{\partial \left( \rho \vec{v} \right)}{\partial z}\frac{\mathrm d z}{\mathrm d t}
</math>
 
<math>\frac{\mathrm d \left( \rho \vec{v} \right)}{\mathrm d t}
=\frac{\partial \left( \rho \vec{v} \right)}{\partial t}
+v_x\frac{\partial \left( \rho \vec{v} \right)}{\partial x}
+v_y\frac{\partial \left( \rho \vec{v} \right)}{\partial y}
+v_z\frac{\partial \left( \rho \vec{v} \right)}{\partial z}
</math>
 
<math>\frac{\mathrm d \left( \rho \vec{v} \right)}{\mathrm d t}
=\frac{\partial \left( \rho \vec{v} \right)}{\partial t}
+(\vec{v}\cdot \overrightarrow \mathrm {grad})\left( \rho \vec{v} \right)
=\frac{\partial \left( \rho \vec{v} \right)}{\partial t}
+(\vec{v}\cdot \vec \nabla)\left( \rho \vec{v} \right)
 
</math>
-----<math>\frac{\mathrm d \left( \rho \vec{v} \right)}{\mathrm d t}
=\rho\frac{\partial \vec{v}} {\partial t}
+\frac{\partial \rho}{\partial t}\vec{v}
+\rho \, v_x\frac{\partial \vec{v} }{\partial x}
+\frac{\partial \rho}{\partial x} v_x \vec{v}
+\rho \, v_y\frac{\partial \vec{v} }{\partial y}
+\frac{\partial \rho}{\partial y} v_y \vec{v}
+\rho \, v_z\frac{\partial \vec{v} }{\partial z}
+\frac{\partial \rho}{\partial z} v_z \vec{v}
 
</math>
 
<math>\frac{\mathrm d \left( \rho \vec{v} \right)}{\mathrm d t}
=\rho \left (
\frac{\partial \vec{v}} {\partial t}
+ v_x\frac{\partial \vec{v} }{\partial x}
+ v_y\frac{\partial \vec{v} }{\partial y}
+v_z\frac{\partial \vec{v} }{\partial z}
\right )
+\left (
\frac{\partial \rho}{\partial t}
+\frac{\partial \rho}{\partial x} v_x
+\frac{\partial \rho}{\partial y} v_y
+\frac{\partial \rho}{\partial z} v_z
\right )
\vec{v}
 
</math>
 
<math>\frac{\mathrm d \left( \rho \vec{v} \right)}{\mathrm d t}
=\rho \left (
\frac{\partial \vec{v}} {\partial t}
+(\vec{v} \cdot \vec{\nabla} )\cdot \vec{v}
\right )
+ \underbrace{\left (
\frac{\partial \rho}{\partial t}
+ \vec{\nabla}\cdot (\rho\vec{v}) \right )
}_{=0}
\vec{v}
 
</math>
 
<math>\frac{\mathrm d \left( \rho \vec{v} \right)}{\mathrm d t}
= \rho \frac{\partial \vec{v}} {\partial t}
+\rho(\vec{v} \cdot \vec{\nabla} )\cdot \vec{v}
 
 
 
</math>
-----<math>\frac{\mathrm d \left( \rho \vec{v} \right)}{\mathrm d t}
=\frac{\partial \left( \rho \vec{v} \right)}{\partial t} + \vec{\nabla} \cdot \left(\rho \vec{v} \otimes \vec{v} \right)
 
</math>
 
-----
 
<math>\frac{\partial \left( \rho \vec{v} \right)}{\partial t} + \vec{\nabla} \cdot \left(\rho \vec{v} \otimes \vec{v} \right) = - \vec{\nabla} p + \vec{\nabla} \cdot \overline{\overline {\tau}} + \rho \vec{f}</math>
 
<math>\vec{\nabla} \cdot \left(\rho \vec{v} \otimes \vec{v} \right)
= \vec{\nabla} \cdot \left ( \begin{pmatrix} \rho v_x \\ \rho v_y \\ \rho v_z \end{pmatrix}
\times
\otimes
\begin{pmatrix} v_x & v_y & v_z \end{pmatrix}
\right )
Ligne 16 ⟶ 100 :
<math>\vec{\nabla} \cdot \left(\rho \vec{v} \otimes \vec{v} \right)
 
=
= \vec{\nabla} \cdot
\begin{pmatrix} \frac{\partial (\rho v_x v_x)}{\partial x}
+\frac{\partial (\rho v_x v_y)}{\partial y}
+\frac{\partial (\rho v_x v_z)}{\partial z}
\\ \rho v_y v_x &frac{\partial (\rho v_y v_y & v_x)}{\rho v_ypartial v_zx}
+\frac{\partial (\rho v_z v_x & \rho v_zv_y v_y & )}{\rho v_z v_zpartial y}
+\frac{\partial (\rho v_y v_z)}{\partial z}
\\ \frac{\partial (\rho v_z v_x)}{\partial x}
+\frac{\partial (\rho v_z v_y)}{\partial y}
+\frac{\partial (\rho v_z v_z)}{\partial z}
 
\end{pmatrix}</math>
 
<math>\mathrmvec{div\nabla} \,cdot \overline{left(\overlinerho \vec{v} \mathbfotimes \vec{H}}v} \right)
= \mathrm{div}\begin{pmatrix} A_{11} & A_{12} & A_{13}\\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{pmatrix}
= \begin{pmatrix}
\frac{\partial A_{11}}{\partial x} + \frac{\partial A_{12}}{\partial x} + \frac{\partial A_{13}}{\partial x}
\\ \frac{\partial A_{21}}{\partial x} + \frac{\partial A_{22}}{\partial x} + \frac{\partial A_{23}}{\partial x}
\\ \frac{\partial A_{31}}{\partial x} + \frac{\partial A_{32}}{\partial x} + \frac{\partial A_{33}}{\partial x}
\end{pmatrix}
 
=
</math>
\begin{pmatrix} \frac{\partial (\rho v_x v_x)}{\partial x}
+\frac{\partial (\rho v_x v_y)}{\partial y}
+\frac{\partial (\rho v_x v_z)}{\partial z}
\\ \frac{\partial (\rho v_y v_x)}{\partial x}
+\frac{\partial (\rho v_y v_y)}{\partial y}
+\frac{\partial (\rho v_y v_z)}{\partial z}
\\ \frac{\partial (\rho v_z v_x)}{\partial x}
+\frac{\partial (\rho v_z v_y)}{\partial y}
+\frac{\partial (\rho v_z v_z)}{\partial z}
 
\end{pmatrix}</math>
 
http://www-ljk.imag.fr/Publications/Basilic/com.lmc.publi.PUBLI_Phdthesis@1172c0fd434_17c9986/Chapitres/Annexes.pdf
Ligne 44 ⟶ 137 :
= \frac{\partial(\rho\overrightarrow v)}{\mathrm \partial t}
+ (\overrightarrow{v} \cdot \overrightarrow{\nabla}) \cdot (\rho\overrightarrow v) </math>
 
<math>\left(\overrightarrow{v} \cdot \overrightarrow\nabla\right)\rho\overrightarrow{v} =
\begin{pmatrix}
{\left( \overrightarrow{v} \cdot \overrightarrow\nabla \right) \rho\, v_x}
\\ {\left( \overrightarrow{v} \cdot \overrightarrow\nabla \right) \rho\, v_y}
\\ {\left( \overrightarrow{v} \cdot \overrightarrow\nabla \right) \rho\, v_z} \end{pmatrix}
= \begin{pmatrix}
{v_x \frac{\partial \rho\, v_x}{\partial x} + v_y \frac{\partial \rho\, v_x}{\partial y}+ v_z \frac{\partial \rho\, v_x}{\partial z}}
\\ {v_x \frac{\partial \rho\, v_y}{\partial x} + v_y \frac{\partial \rho\, v_y}{\partial y}+ v_z \frac{\partial \rho\, v_y}{\partial z}}
\\ {v_x \frac{\partial \rho\, v_z}{\partial x} + v_y \frac{\partial \rho\, v_z}{\partial y}+ v_z \frac{\partial \rho\, v_z}{\partial z}} \end{pmatrix}
</math>
 
<math> \frac{\mathrm d(\rho\overrightarrow v)}{\mathrm dt}
Ligne 62 ⟶ 166 :
+ \frac{\partial(\rho v_z v_z)}{\mathrm \partial z}
\end{pmatrix}
</math> ???????