« Utilisateur:Ellande/Brouillon » : différence entre les versions
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Ligne 114 :
<math>\vec{\nabla} \cdot \left(\rho \vec{v} \otimes \vec{v} \right)
\begin{pmatrix}
▲=
+\
+\rho\frac{\partial
▲+\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}▼
+\rho\frac{\partial v_x }{\partial z}v_z
+\rho\frac{\partial v_z}{\partial z}v_x
\\ \frac{\partial \rho }{\partial x}v_y v_x
+\rho\frac{\partial v_y }{\partial x}v_x
+\rho\frac{\partial v_x}{\partial x}v_y
+\rho\frac{\partial v_y }{\partial y}v_y
+\rho\frac{\partial v_y}{\partial y}v_y
+\rho\frac{\partial v_y }{\partial z}v_z
+\rho\frac{\partial v_z}{\partial v_z}v_y
\\ \frac{\partial \rho }{\partial x}v_z v_x
+\rho\frac{\partial v_z }{\partial x}v_x
+\rho\frac{\partial v_x}{\partial x}v_z
+\frac{\partial \rho }{\partial y}v_z v_y
+\rho\frac{\partial v_z }{\partial y}v_y
+\rho\frac{\partial v_y}{\partial y}v_z
+\frac{\partial \rho }{\partial z}v_z v_z
+\rho\frac{\partial v_z }{\partial z}v_z
+\rho\frac{\partial v_z}{\partial z}v_z
\end{pmatrix}</math>
<math>\vec{\nabla} \cdot \left(\rho \vec{v} \otimes \vec{v} \right)
=
\begin{pmatrix}
(\rho\,\mathrm {div}\,\vec v + \overrightarrow \mathrm{grad}\,\rho \cdot \vec v) v_x
+ \rho \,\overrightarrow \mathrm{grad}\, v_x \cdot \vec v
\\ (\rho\,\mathrm {div}\,\vec v + \overrightarrow \mathrm{grad}\,\rho \cdot \vec v) v_y
+ \rho \,\overrightarrow \mathrm{grad}\, v_y \cdot \vec v
\\ (\rho\,\mathrm {div}\,\vec v + \overrightarrow \mathrm{grad}\,\rho \cdot \vec v) v_z
+ \rho \,\overrightarrow \mathrm{grad}\, v_z \cdot \vec v
\end{pmatrix}</math>
<math>\vec{\nabla} \cdot \left(\rho \vec{v} \otimes \vec{v} \right)
=
\begin{pmatrix}
\mathrm {div}\,(\rho\vec v) \, v_x
+ \rho \,\overrightarrow \mathrm{grad}\, v_x \cdot \vec v
\\ \mathrm {div}\,(\rho\vec v) \, v_y
+ \rho \,\overrightarrow \mathrm{grad}\, v_y \cdot \vec v
\\ \mathrm {div}\,(\rho\vec v) \, v_z
+ \rho \,\overrightarrow \mathrm{grad}\, v_z \cdot \vec v
\end{pmatrix}</math>
<math>\vec{\nabla} \cdot \left(\rho \vec{v} \otimes \vec{v} \right)
=\mathrm {div}\,(\rho\vec v)\,\vec v +
\begin{pmatrix}
\rho \,\overrightarrow \mathrm{grad}\, v_x \cdot \vec v
\\ \rho \,\overrightarrow \mathrm{grad}\, v_y \cdot \vec v
\\ \rho \,\overrightarrow \mathrm{grad}\, v_z \cdot \vec v
\end{pmatrix}</math>
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