|
== Notes ==
<references />
== Recherches ==
:
:
:<math>\frac{\mathrm d ( \rho\overrightarrow{v} )}{\mathrm d t}
=\frac{\partial ( \rho\overrightarrow{v} )}{\partial t}
+(\vec{v}\cdot \overrightarrow \mathrm {grad})( \rho\overrightarrow{v} )
=\frac{\partial ( \rho\overrightarrow{v} )}{\partial t}
+(\vec{v}\cdot \overrightarrow \nabla)\left( \rho \overrightarrow{v} \right)
</math>
:<math>\frac{\mathrm d ( \rho\overrightarrow{v} )}{\mathrm d t}
=\rho\frac{\partial \overrightarrow{v}} {\partial t}
+\frac{\partial \rho}{\partial t}\overrightarrow{v}
+\rho \, v_x\frac{\partial \overrightarrow{v} }{\partial x}
+\frac{\partial \rho}{\partial x} v_x \overrightarrow{v}
+\rho \, v_y\frac{\partial \overrightarrow{v} }{\partial y}
+\frac{\partial \rho}{\partial y} v_y \overrightarrow{v}
+\rho \, v_z\frac{\partial \overrightarrow{v} }{\partial z}
+\frac{\partial \rho}{\partial z} v_z \overrightarrow{v}
</math>
:<math>\frac{\mathrm d ( \rho\overrightarrow{v} )}{\mathrm d t}
=\rho\frac{\partial \overrightarrow{v}} {\partial t}
+\frac{\partial \rho}{\partial t}\overrightarrow{v}
+\rho \left(
v_x\frac{\partial \overrightarrow{v} }{\partial x}
+ v_y\frac{\partial \overrightarrow{v} }{\partial y}
+ v_z\frac{\partial \overrightarrow{v} }{\partial z}
\right)
+
\left(
\frac{\partial \rho}{\partial x} v_x
+\frac{\partial \rho}{\partial y} v_y
+\frac{\partial \rho}{\partial z} v_z
\right)\overrightarrow{v}
</math>
:<math>\frac{\mathrm d ( \rho\overrightarrow{v} )}{\mathrm d t}
=\rho\frac{\partial \overrightarrow{v}} {\partial t}
+\frac{\partial \rho}{\partial t}\overrightarrow{v}
+\rho \left( \overrightarrow{v}\cdot \overrightarrow\mathrm{grad}
\right)\overrightarrow{v}
+
\left(\overrightarrow{v}\cdot \overrightarrow\mathrm{grad} \, \rho
\right)\overrightarrow{v}
</math>
:<math>\frac{\mathrm d ( \rho\overrightarrow{v} )}{\mathrm d t}
+\rho \, \overrightarrow v\, \hbox{div}\ \overrightarrow{v}
=\rho
\left(\frac{\partial \overrightarrow{v}} {\partial t}
+ \left( \overrightarrow{v}\cdot \overrightarrow\mathrm{grad}
\right)\overrightarrow{v}
\right)
+\left(
\frac{\partial \rho}{\partial t}
+\overrightarrow{v}\cdot \overrightarrow\mathrm{grad} \, \rho
+\rho \, \hbox{div}\, \overrightarrow v
\right)\overrightarrow{v}
</math>
:<math>\frac{\mathrm d ( \rho\overrightarrow{v} )}{\mathrm d t}
+\rho \, \overrightarrow v\, \hbox{div}\ \overrightarrow{v}
=\rho
\left(\frac{\partial \overrightarrow{v}} {\partial t}
+ \left( \overrightarrow{v}\cdot \overrightarrow\mathrm{grad}
\right)\overrightarrow{v}
\right)
+\underbrace{ \left(
\frac{\partial \rho}{\partial t}
+\hbox{div} \left(\rho \,\overrightarrow v\right)
\right)}_{=0}\overrightarrow{v}
</math>
:
:<math>\frac{\mathrm d \left( \rho \overrightarrow{v} \right)}{\mathrm d t}
=\rho \frac{\mathrm d \overrightarrow{v}}{\mathrm d t}
+\frac{\mathrm d \rho}{\mathrm d t}
\overrightarrow{v}
</math>
:<math>\frac{\mathrm d \left( \rho \overrightarrow{v} \right)}{\mathrm d t}
+\rho \, \overrightarrow v\, \hbox{div}\ \overrightarrow{v}
=\rho \frac{\mathrm d \overrightarrow{v}}{\mathrm d t}
+
\underbrace{ \left(\frac{\mathrm d \rho}{\mathrm d t}
+\rho \ \hbox{div}\, \overrightarrow v\right) }_{=0}\overrightarrow{v}
</math>
:
:<math>\frac{\mathrm d \left( \rho \overrightarrow{v} \right)}{\mathrm d t}
=\rho \left (
\frac{\partial \overrightarrow{v}} {\partial t}
+ v_x\frac{\partial \overrightarrow{v} }{\partial x}
+ v_y\frac{\partial \overrightarrow{v} }{\partial y}
+v_z\frac{\partial \overrightarrow{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 )
\overrightarrow{v}
</math>
:<math>\frac{\mathrm d \left( \rho \overrightarrow{v} \right)}{\mathrm d t}
=\rho \left (
\frac{\partial \overrightarrow{v}} {\partial t}
+(\overrightarrow{v} \cdot \overrightarrow{\nabla} )\cdot \overrightarrow{v}
\right )
+\frac{\mathrm d \rho}{\mathrm d t}
\overrightarrow{v}
</math>
:<math>\frac{\mathrm d \left( \rho \overrightarrow{v} \right)}{\mathrm d t}
+\rho \, \overrightarrow v\, \hbox{div}\ \overrightarrow{v}
= \rho \left (
\frac{\partial \overrightarrow{v}} {\partial t}
+(\overrightarrow{v} \cdot \overrightarrow{\nabla} )\cdot \overrightarrow{v}
\right )
+\rho \, \overrightarrow v\, \hbox{div}\ \overrightarrow{v}
</math>
:<math>\frac{\mathrm d \left( \rho \overrightarrow{v} \right)}{\mathrm d t}
+\rho \, \overrightarrow v\, \hbox{div}\ \overrightarrow{v}
= \rho \frac{\partial \overrightarrow{v}} {\partial t}
+\rho\,(\overrightarrow{v} \cdot \overrightarrow{\nabla} )\cdot \overrightarrow{v}
</math>
<math>\rho \frac{\partial \overrightarrow{v}} {\partial t}
+\rho(\overrightarrow{v} \cdot \overrightarrow{\hbox{grad}} )\cdot \overrightarrow{v}
+\rho \, \overrightarrow v\, \hbox{div}\ \overrightarrow{v}
= f_V+ \overrightarrow \hbox{div}\ \overline\overline\mathrm{\tau}
</math>
<math>\overrightarrow{\nabla} \cdot \left(\rho \overrightarrow{v} \otimes \overrightarrow{v} \right)
=\mathrm {div}\,(\rho\overrightarrow v)\,\overrightarrow v +
\rho (\overrightarrow v\cdot \overrightarrow \mathrm{grad})\, \overrightarrow v
</math>
[https://books.google.fr/books?id=PdxOsIkhZCcC&pg=PA24&dq=equation+de+continuit%C3%A9+bilan+masse&hl=fr&sa=X&ved=0ahUKEwi_2Z_soqPOAhUJ2RoKHZ2dABQQ6AEIMDAD#v=onepage&q=equation%20de%20continuit%C3%A9%20bilan%20masse&f=false][http://perso.mines-albi.fr/~louisnar/MECADEF/PolyMecaDef.pdf]
[[Fichier:Composantes tenseur des contraintes.png|vignette|400x400px|Illustration]]
Les forces surfaciques qui s'appliquent sur les faces d'un élément de volume sont modélisées par
:
| 