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« Utilisateur:Ellande/Brouillon2 » : différence entre les versions

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Ligne 57 :
\end{pmatrix}
</math>
 
On peut exprimer les forces de surface :
 
<math>\overrightarrow{F_S}
=
\int\!\!\!\!\!\!\!\subset\!\!\!\supset\!\!\!\!\!\!\!\int_S
\overline{\overline{\tau}} \times \overrightarrow{\mathrm d S}
= \int\!\!\!\!\!\int\!\!\!\!\!\int_V
\overrightarrow \hbox{div}\ \overline\overline\mathrm{\tau} {\mathrm d V}
 
 
</math>,
 
avec
 
<math>\overrightarrow \hbox{div}\ \overline\overline\mathrm{\tau} = \begin{pmatrix}
Ligne 63 ⟶ 77 :
\frac{\partial\tau_{zx}}{\partial x} + \frac{\partial\tau_{zy}}{\partial y} + \frac{\partial\sigma_{zz}}{\partial z}\\
\end{pmatrix}
</math>.
 
<math>\int\!\!\!\!\!\int\!\!\!\!\!\int_V \left(\frac{\partial (\rho {\overrightarrow {v})}}{\partial t}
+ \overrightarrow{\mathrm{div}} \left(\rho {\overrightarrow {v}} \otimes \vec{v} \right)
\right) \mathrm dV
= \int\!\!\!\!\!\int\!\!\!\!\!\int_V f_V\, \mathrm d V
+\int\!\!\!\!\!\!\!\subset\!\!\!\supset\!\!\!\!\!\!\!\int_S
\overline{\overline{\tau}} \times \overrightarrow{\mathrm d S}
 
</math>
 
== Forme locale ==
En exprimant la forme globale
 
<math>\int\!\!\!\!\!\int\!\!\!\!\!\int_V \left(\frac{\partial (\rho {\overrightarrow {v})}}{\partial t}
+ \overrightarrow{\mathrm{div}} \left(\rho {\overrightarrow {v}} \otimes \vec{v} \right)
\right) \mathrm dV
= \int\!\!\!\!\!\int\!\!\!\!\!\int_V f_V\, \mathrm d V
+\int\!\!\!\!\!\int\!\!\!\!\!\int_V
\overrightarrow \hbox{div}\ \overline\overline\mathrm{\tau} {\mathrm d V}
 
</math>,
 
il vient la forme conservative de l'équation de bilan de la quantité de mouvement :
 
<math>\frac{\partial (\rho \overrightarrow v)}{\partial t}
+ \overrightarrow{\mathrm{div}} \left(\rho {\overrightarrow {v}} \otimes \vec{v} \right)
= f_V+ \overrightarrow \hbox{div}\ \overline\overline\mathrm{\tau}
 
</math>,
 
qui peut aussi s'exprimer sous deux formes non-conservatives :
 
<math>\frac{\mathrm {d} (\rho \,\overrightarrow v)}{\mathrm {d} t}
+\rho \, \overrightarrow v\, \hbox{div}\ \overrightarrow{v}
= f_V+ \overrightarrow \hbox{div}\ \overline\overline\mathrm{\tau}
</math> et <math>\rho \,\frac{\partial \overrightarrow{v}} {\partial t}
+ \rho \left( \overrightarrow{v}\cdot \overrightarrow\mathrm{grad}
\right)\overrightarrow{v}
= f_V+ \overrightarrow \hbox{div}\ \overline\overline\mathrm{\tau}
 
</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}
+ \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 \, \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 \left( \rho \vec{v} \right)}{\mathrm d t}
=\rho\frac{\partial \left( \rho \vec{v}} \right)}{\partial t}
+\frac{\partial \left( \rho \vec{v} \right)}{\partial xt}\fracvec{\mathrm d x}{\mathrm d tv}
+\frac{\partialrho \left( \rho \vecoverrightarrow{v} \right)}{\partialcdot y}\frac{overrightarrow\mathrm d y}{\mathrm d tgrad}
\right)\vec{v}
+\frac{\partial \left( \rho \vec{v} \right)}{\partial z}\frac{\mathrm d z}{\mathrm d t}
+
\left(\overrightarrow{v}\cdot \overrightarrow\mathrm{grad} \, \rho
\right)\vec{v}
 
</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 \left(
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 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\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>\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>\vec{\nabla} \cdot \left(\rho \vec{v} \otimes \vec{v} \right)
=\mathrm {div}\,(\rho\vec v)\,\vec v +
\rho (\vec v\cdot \overrightarrow \mathrm{grad})\, \vec v
</math>
Récupérée de « https://fr.wikiversity.org/wiki/Utilisateur:Ellande/Brouillon2 »