.. _demoDivergenceTenseurContCyl: .. role:: red .. role:: blue Démonstration ============= On calcule terme à terme les coordonnées du tenseur des contraintes en coordonnées cylindriques soit :math:`\vec{\nabla}.\bar{\bar{\tau}}` :On doit donc calculer mes 3*3*3 termes qui se projettent sur :math:`\left({\vec{i}}_r,{\vec{i}}_\theta,{\vec{i}}_z\right)`. Ces termes sont : :math:`\left( {\vec{i}}_r\frac{\partial}{\partial r}+{\vec{i}}_\theta\frac{1}{r}\frac{\partial}{\partial\theta}+{\vec{i}}_z\frac{\partial}{\partial z}\right).\left(\tau_{rr}\ {\vec{i}}_r{\vec{i}}_r+\tau_{r\theta}{\vec{i}}_r{\vec{i}}_\theta+\tau_{rz}{\vec{i}}_r{\vec{i}}_z\right)` :math:`+\left({\vec{i}}_r\frac{\partial}{\partial r}+{\vec{i}}_\theta\frac{1}{r}\frac{\partial}{\partial\theta}+{\vec{i}}_z\frac{\partial}{\partial z}\right).\left(\tau_{\theta r}\ {\vec{i}}_\theta{\vec{i}}_r+\tau_{\theta\theta}{\vec{i}}_\theta{\vec{i}}_\theta+\tau_{\theta z}{\vec{i}}_\theta{\vec{i}}_z\right)` :math:`+\left({\vec{i}}_r\frac{\partial}{\partial r}+{\vec{i}}_\theta\frac{1}{r}\frac{\partial}{\partial\theta}+{\vec{i}}_z\frac{\partial}{\partial z}\right).\left(\tau_{zr}\ {\vec{i}}_z{\vec{i}}_r+\tau_{z\theta}{\vec{i}}_z{\vec{i}}_\theta+\tau_{zz}{\vec{i}}_z{\vec{i}}_z\right)` Calcul du 1er terme ------------------- :math:`\left({\vec{i}}_r\frac{\partial}{\partial r}+{\vec{i}}_\theta\frac{1}{r}\frac{\partial}{\partial\theta}+{\vec{i}}_z\frac{\partial}{\partial z}\right).\left(\tau_{rr}\ {\vec{i}}_r{\vec{i}}_r+\tau_{r\theta}{\vec{i}}_r{\vec{i}}_\theta+\tau_{rz}{\vec{i}}_r{\vec{i}}_z\right)` :math:`={\vec{i}}_r.\frac{\partial\tau_{rr}\ {\vec{i}}_r{\vec{i}}_r+\tau_{r\theta}{\vec{i}}_r{\vec{i}}_\theta+\tau_{rz}{\vec{i}}_r{\vec{i}}_z}{\partial r}` :math:`+{\vec{i}}_\theta\frac{1}{r}\frac{\partial\tau_{rr}\ {\vec{i}}_r{\vec{i}}_r+\tau_{r\theta}{\vec{i}}_r{\vec{i}}_\theta+\tau_{rz}{\vec{i}}_r{\vec{i}}_z}{\partial\theta}` :math:`+{\vec{i}}_z.\frac{\partial\tau_{rr}\ {\vec{i}}_r{\vec{i}}_r+\tau_{r\theta}{\vec{i}}_r{\vec{i}}_\theta+\tau_{rz}{\vec{i}}_r{\vec{i}}_z}{\partial z}` Calcul du terme en :math:`{\vec{i}}_r` """""""""""""""""""""""""""""""""""""" :math:`={\vec{i}}_r.\frac{\partial\tau_{rr}\ {\vec{i}}_r{\vec{i}}_r+\tau_{r\theta}{\vec{i}}_r{\vec{i}}_\theta+\tau_{rz}{\vec{i}}_r{\vec{i}}_z}{\partial r}` :math:`={\vec{i}}_r.\frac{\partial\tau_{rr}\ {\vec{i}}_r{\vec{i}}_r}{\partial r}+{\vec{i}}_r.\frac{\partial\tau_{r\theta}{\vec{i}}_r{\vec{i}}_\theta}{\partial r}+{\vec{i}}_r.\frac{\partial\tau_{rz}{\vec{i}}_r{\vec{i}}_z}{\partial r}` :math:`={\vec{i}}_r\frac{\partial\tau_{rr}\ }{\partial r}+{\vec{i}}_\theta\frac{\partial\tau_{r\theta}}{\partial r}+{\vec{i}}_z\frac{\partial\tau_{rz}}{\partial r}` :red:`soit :` :math:`={\vec{i}}_r\frac{\partial\tau_{rr}\ }{\partial r}+{\vec{i}}_\theta\frac{\partial\tau_{r\theta}}{\partial r}+{\vec{i}}_z\frac{\partial\tau_{rz}}{\partial r}` Calcul du terme en :math:`{\vec{i}}_{\theta}` """"""""""""""""""""""""""""""""""""""""""""" :math:`{\vec{i}}_\theta.\ \frac{1}{r}\frac{\partial\tau_{rr}\ {\vec{i}}_r{\vec{i}}_r+\tau_{r\theta}{\vec{i}}_r{\vec{i}}_\theta+\tau_{rz}{\vec{i}}_r{\vec{i}}_z}{\partial\theta}` :math:`={\vec{i}}_\theta.\ \left(\frac{1}{r}\frac{\partial\tau_{rr}\ {\vec{i}}_r{\vec{i}}_r}{\partial\theta}+\frac{1}{r}\frac{\partial\tau_{r\theta}{\vec{i}}_r{\vec{i}}_\theta}{\partial\theta}+\frac{1}{r}\frac{\partial\tau_{rz}{\vec{i}}_r{\vec{i}}_z}{\partial\theta}\right)` :math:`={\vec{i}}_\theta.\ \left(\frac{\tau_{rr}}{r}\frac{\partial\ {\vec{i}}_r{\vec{i}}_r}{\partial\theta}+\frac{{\vec{i}}_r{\vec{i}}_r}{r}\frac{\partial\tau_{rr}\ }{\partial\theta}\right) + {\vec{i}}_\theta.\ \left(\frac{\tau_{r\theta}}{r}\frac{\partial{\vec{i}}_r{\vec{i}}_\theta}{\partial\theta}+\frac{{\vec{i}}_r{\vec{i}}_\theta}{r}\frac{\partial\tau_{r\theta}}{\partial\theta}\right)` :math:`+{\vec{i}}_\theta.\ \left(\frac{\tau_{rz}}{r}\frac{\partial{\vec{i}}_r{\vec{i}}_z}{\partial\theta}+\frac{{\vec{i}}_r{\vec{i}}_z}{r}\frac{\partial\tau_{rz}}{\partial\theta}\right)` :math:`= {\vec{i}}_\theta.\ \left(\frac{\tau_{rr}}{r}{\vec{i}}_r\frac{\partial\ {\vec{i}}_r}{\partial\theta}+\frac{\tau_{rr}}{r}\frac{\partial\ {\vec{i}}_r}{\partial\theta}{\vec{i}}_r+\frac{{\vec{i}}_r{\vec{i}}_r}{r}\frac{\partial\tau_{rr}\ }{\partial\theta}\right)+{\vec{i}}_\theta.\ \left(\frac{\tau_{r\theta}{\vec{i}}_r}{r}\frac{\partial{\vec{i}}_\theta}{\partial\theta} + \frac{\tau_{r\theta}}{r}\frac{\partial{\vec{i}}_r}{\partial\theta}{\vec{i}}_\theta+\frac{{\vec{i}}_r{\vec{i}}_\theta}{r}\frac{\partial\tau_{r\theta}}{\partial\theta}\right)` :math:`+{\vec{i}}_\theta.\ \left(\frac{\tau_{rz}{\vec{i}}_r}{r}\frac{\partial{\vec{i}}_z}{\partial\theta}+\frac{\tau_{rz}}{r}\frac{\partial{\vec{i}}_r}{\partial\theta}{\vec{i}}_z+\frac{{\vec{i}}_r{\vec{i}}_z}{r}\frac{\partial\tau_{rz}}{\partial\theta}\right)` :math:`={\vec{i}}_\theta.\ \left(\frac{\tau_{rr}}{r}\frac{\partial\ {\vec{i}}_r}{\partial\theta}{\vec{i}}_r\right)+{\vec{i}}_\theta.\ \left(\frac{\tau_{r\theta}}{r}\frac{\partial{\vec{i}}_r}{\partial\theta}{\vec{i}}_\theta+\right)+{\vec{i}}_\theta.\ \left(\frac{\tau_{rz}}{r}\frac{\partial{\vec{i}}_r}{\partial\theta}{\vec{i}}_z\right)` :red:`soit :` :math:`= \frac{\tau_{rr}}{r}{\vec{i}}_r+\frac{\tau_{r\theta}}{r}{\vec{i}}_\theta+\frac{\tau_{rz}}{r}{\vec{i}}_z` Calcul du terme en :math:`{\vec{i}}_z` """""""""""""""""""""""""""""""""""""" Le terme en :math:`{\vec{i}}_z` est : :math:`{\vec{i}}_z.\frac{\partial\tau_{rr}\ {\vec{i}}_r{\vec{i}}_r+\tau_{r\theta}{\vec{i}}_r{\vec{i}}_\theta+\tau_{rz}{\vec{i}}_r{\vec{i}}_z}{\partial z}` Pour chaque terme, le produit scalaire :math:`{\vec{i}}_r . {\vec{i}}_z =0` qui apparait: :red:`soit :` :math:`= 0` :blue:`Donc le 1er terme s'écrit` :math:`={\vec{i}}_r\left(\frac{\partial\tau_{rr}\ }{\partial r}+\frac{\tau_{rr}}{r}\right)+{\vec{i}}_\theta\left(\frac{\partial\tau_{r\theta}}{\partial r}+\frac{\tau_{r\theta}}{r}\right)+{\vec{i}}_z\left(\frac{\partial\tau_{rz}}{\partial r}+\frac{\tau_{rz}}{r}\right)` Calcul du 2eme terme -------------------- C'est le terme :math:`\left({\vec{i}}_r\frac{\partial}{\partial r}+{\vec{i}}_\theta\frac{1}{r}\frac{\partial}{\partial\theta}+{\vec{i}}_z\frac{\partial}{\partial z}\right).\left(\tau_{\theta r}\ {\vec{i}}_\theta{\vec{i}}_r+\tau_{\theta\theta}{\vec{i}}_\theta{\vec{i}}_\theta+\tau_{\theta z}{\vec{i}}_\theta{\vec{i}}_z\right)` que l'on développe: :math:`{\vec{i}}_r.\frac{\partial\left(\tau_{\theta r}\ {\vec{i}}_\theta{\vec{i}}_r+\tau_{\theta\theta}{\vec{i}}_\theta{\vec{i}}_\theta+\tau_{\theta z}{\vec{i}}_\theta{\vec{i}}_z\right)}{\partial r} + {\vec{i}}_\theta\frac{1}{r}.\frac{\partial\left(\tau_{\theta r}\ {\vec{i}}_\theta{\vec{i}}_r+\tau_{\theta\theta}{\vec{i}}_\theta{\vec{i}}_\theta+\tau_{\theta z}{\vec{i}}_\theta{\vec{i}}_z\right)}{\partial\theta}` :math:`+{\vec{i}}_z.\frac{\partial\left(\tau_{\theta r}\ {\vec{i}}_\theta{\vec{i}}_r+\tau_{\theta\theta}{\vec{i}}_\theta{\vec{i}}_\theta+\tau_{\theta z}{\vec{i}}_\theta{\vec{i}}_z\right)}{\partial z}` Calcul du terme en :math:`{\vec{i}}_r` """""""""""""""""""""""""""""""""""""" Développons le terme :math:`{\vec{i}}_r.\frac{\partial\left(\tau_{\theta r}\ {\vec{i}}_\theta{\vec{i}}_r+\tau_{\theta\theta}{\vec{i}}_\theta{\vec{i}}_\theta+\tau_{\theta z}{\vec{i}}_\theta{\vec{i}}_z\right)}{\partial r}` :math:`={\vec{i}}_r.\frac{\partial\tau_{\theta r}\ {\vec{i}}_\theta{\vec{i}}_r}{\partial r}+{\vec{i}}_r.\frac{\partial\tau_{\theta\theta}{\vec{i}}_\theta{\vec{i}}_\theta}{\partial r}+{\vec{i}}_r.\frac{\partial\tau_{\theta z}{\vec{i}}_\theta{\vec{i}}_z}{\partial r}` Pour chaque terme, le produit scalaire :math:`{\vec{i}}_r . {\vec{i}}_{\theta} =0` qui apparait: :red:`soit :` :math:`= 0` Calcul du terme en :math:`{\vec{i}}_{\theta}` """"""""""""""""""""""""""""""""""""""""""""" :math:`={\vec{i}}_\theta\frac{1}{r}.\frac{\partial\left(\tau_{\theta r}\ {\vec{i}}_\theta{\vec{i}}_r+\tau_{\theta\theta}{\vec{i}}_\theta{\vec{i}}_\theta+\tau_{\theta z}{\vec{i}}_\theta{\vec{i}}_z\right)}{\partial\theta} = {\vec{i}}_\theta\frac{1}{r}.\frac{\partial\tau_{\theta r}\ {\vec{i}}_\theta{\vec{i}}_r}{\partial\theta}+{\vec{i}}_\theta\frac{1}{r}.\frac{\partial\tau_{\theta\theta}{\vec{i}}_\theta{\vec{i}}_\theta}{\partial\theta}+{\vec{i}}_\theta\frac{1}{r}.\frac{\partial\tau_{\theta z}{\vec{i}}_\theta{\vec{i}}_z}{\partial\theta}` :math:`= {\vec{i}}_\theta\frac{1}{r}.\left(\ {\vec{i}}_\theta{\vec{i}}_r\frac{\partial\tau_{\theta r}}{\partial\theta}+\tau_{\theta r}\frac{\partial\ {\vec{i}}_\theta{\vec{i}}_r}{\partial\theta}\right) + {\vec{i}}_\theta\frac{1}{r}.\left(\tau_{\theta\theta}\frac{\partial{\vec{i}}_\theta{\vec{i}}_\theta}{\partial\theta}+{\vec{i}}_\theta{\vec{i}}_\theta\frac{\partial\tau_{\theta\theta}}{\partial\theta}\right)` :math:`{\vec{i}}_\theta\frac{1}{r}.\left({\vec{i}}_\theta{\vec{i}}_z\frac{\partial\tau_{\theta z}}{\partial\theta}+\tau_{\theta z}\frac{\partial{\vec{i}}_\theta{\vec{i}}_z}{\partial\theta}\right)` :math:`={\vec{i}}_\theta\frac{1}{r}.\left(\ {\vec{i}}_\theta{\vec{i}}_r\frac{\partial\tau_{\theta r}}{\partial\theta}+\tau_{\theta r}{\vec{i}}_\theta\frac{\partial\ {\vec{i}}_r}{\partial\theta}+\tau_{\theta r}\frac{\partial\ {\vec{i}}_\theta}{\partial\theta}{\vec{i}}_r\right)` :math:`+ {\vec{i}}_\theta\frac{1}{r}.\left(\tau_{\theta\theta}\frac{\partial{\vec{i}}_\theta}{\partial\theta}{\vec{i}}_\theta+\tau_{\theta\theta}{\vec{i}}_\theta\frac{\partial{\vec{i}}_\theta}{\partial\theta}+{\vec{i}}_\theta{\vec{i}}_\theta\frac{\partial\tau_{\theta\theta}}{\partial\theta}\right)` :math:`+{\vec{i}}_\theta\frac{1}{r}.\left({\vec{i}}_\theta{\vec{i}}_z\frac{\partial\tau_{\theta z}}{\partial\theta}+\tau_{\theta z}{\vec{i}}_\theta\frac{\partial{\vec{i}}_z}{\partial\theta}+\tau_{\theta z}\frac{\partial{\vec{i}}_\theta}{\partial\theta}{\vec{i}}_z\right)` :math:`= {\vec{i}}_\theta\frac{1}{r}.\left(\ {\vec{i}}_\theta{\vec{i}}_r\frac{\partial\tau_{\theta r}}{\partial\theta}+\tau_{\theta r}{\vec{i}}_\theta{\vec{i}}_\theta-\tau_{\theta r}{\vec{i}}_r{\vec{i}}_r\right)` :math:`+{\vec{i}}_\theta\frac{1}{r}.\left(-\tau_{\theta\theta}{\vec{i}}_\theta{\vec{i}}_r+{\vec{i}}_\theta{\vec{i}}_\theta\frac{\partial\tau_{\theta\theta}}{\partial\theta}\right)+{\vec{i}}_\theta\frac{1}{r}.\left({\vec{i}}_\theta{\vec{i}}_z\frac{\partial\tau_{\theta z}}{\partial\theta}\right)` :math:`\frac{1}{r}\left(\ {\vec{i}}_r\frac{\partial\tau_{\theta r}}{\partial\theta}+\tau_{\theta r}{\vec{i}}_\theta\right)+\frac{1}{r}\left(-\tau_{\theta\theta}{\vec{i}}_r+{\vec{i}}_\theta\frac{\partial\tau_{\theta\theta}}{\partial\theta}\right)+\frac{1}{r}\left({\vec{i}}_z\frac{\partial\tau_{\theta z}}{\partial\theta}\right)` :red:`Soit :` :math:`=\frac{{\vec{i}}_r}{r}\left(\frac{\partial\tau_{\theta r}}{\partial\theta}\ -\tau_{\theta\theta}\right)+\frac{{\vec{i}}_\theta}{r}\left(\frac{\partial\tau_{\theta\theta}}{\partial\theta}+\tau_{\theta r}\right)+\frac{{\vec{i}}_z}{r}\left(\frac{\partial\tau_{\theta z}}{\partial\theta}\right)` Calcul du terme en :math:`{\vec{i}}_z` :math:`{\vec{i}}_z.\frac{\partial\left(\tau_{\theta r}\ {\vec{i}}_\theta{\vec{i}}_r+\tau_{\theta\theta}{\vec{i}}_\theta{\vec{i}}_\theta+\tau_{\theta z}{\vec{i}}_\theta{\vec{i}}_z\right)}{\partial z}` Ce terme ne fait apparaitre que des termes :math:`\vec{i}_z. \vec{i}_\theta = 0` :red:`Soit : = 0` :blue:`Donc le 2ème terme s'écrit` :math:`=\frac{{\vec{i}}_r}{r}\left(\frac{\partial\tau_{\theta r}}{\partial\theta}\ -\tau_{\theta\theta}\right)+\frac{{\vec{i}}_\theta}{r}\left(\frac{\partial\tau_{\theta\theta}}{\partial\theta}+\tau_{\theta r}\right)+\frac{{\vec{i}}_z}{r}\left(\frac{\partial\tau_{\theta z}}{\partial\theta}\right)` Calcul du 3eme terme -------------------- C'est le terme :math:`\left({\vec{i}}_r\frac{\partial}{\partial r}+{\vec{i}}_\theta\frac{1}{r}\frac{\partial}{\partial\theta}+{\vec{i}}_z\frac{\partial}{\partial z}\right).\left(\tau_{zr}\ {\vec{i}}_z{\vec{i}}_r+\tau_{z\theta}{\vec{i}}_z{\vec{i}}_\theta+\tau_{zz}{\vec{i}}_z{\vec{i}}_z\right)` qui donne les 3 termes suivants : :math:`={\vec{i}}_r.\frac{\partial\left(\tau_{zr}\ {\vec{i}}_z{\vec{i}}_r+\tau_{z\theta}{\vec{i}}_z{\vec{i}}_\theta+\tau_{zz}{\vec{i}}_z{\vec{i}}_z\right)}{\partial r}+{\vec{i}}_\theta.\frac{1}{r}\frac{\partial\left(\tau_{zr}\ {\vec{i}}_z{\vec{i}}_r+\tau_{z\theta}{\vec{i}}_z{\vec{i}}_\theta+\tau_{zz}{\vec{i}}_z{\vec{i}}_z\right)}{\partial\theta}` :math:`+{\vec{i}}_z.\frac{\partial\left(\tau_{zr}\ {\vec{i}}_z{\vec{i}}_r+\tau_{z\theta}{\vec{i}}_z{\vec{i}}_\theta+\tau_{zz}{\vec{i}}_z{\vec{i}}_z\right)}{\partial z}` Calcul du terme en :math:`\vec{i}_r` """""""""""""""""""""""""""""""""""" Il s'agit du terme :math:`{\vec{i}}_r.\frac{\partial\left(\tau_{zr}\ {\vec{i}}_z{\vec{i}}_r+\tau_{z\theta}{\vec{i}}_z{\vec{i}}_\theta+\tau_{zz}{\vec{i}}_z{\vec{i}}_z\right)}{\partial r}` qui se décompose en 3 termes qui sont : :math:`={\vec{i}}_r.\frac{\partial\tau_{zr}\ {\vec{i}}_z{\vec{i}}_r}{\partial r}+{\vec{i}}_r.\frac{\partial\tau_{z\theta}{\vec{i}}_z{\vec{i}}_\theta}{\partial r}+{\vec{i}}_r.\frac{\partial\tau_{zz}{\vec{i}}_z{\vec{i}}_z}{\partial r}` Les vecteurs de base étant constants, on a pour chaque terme le produit scalaire : :math:`\vec{i}_r.`\vec{i}_z = 0` :red:`Soit : = 0` Calcul du terme en :math:`\vec{i}_\theta` """"""""""""""""""""""""""""""""""""""""" C'est le terme :math:`{\vec{i}}_\theta.\frac{1}{r}\frac{\partial\left(\tau_{zr}\ {\vec{i}}_z{\vec{i}}_r+\tau_{z\theta}{\vec{i}}_z{\vec{i}}_\theta+\tau_{zz}{\vec{i}}_z{\vec{i}}_z\right)}{\partial\theta}` qui se décompose :math:`={\vec{i}}_\theta.\frac{1}{r}\frac{\partial\tau_{zr}\ {\vec{i}}_z{\vec{i}}_r}{\partial\theta}+{\vec{i}}_\theta.\frac{1}{r}\frac{\partial\tau_{z\theta}{\vec{i}}_z{\vec{i}}_\theta}{\partial\theta}+{\vec{i}}_\theta.\frac{1}{r}\frac{\partial\tau_{zz}{\vec{i}}_z{\vec{i}}_z}{\partial\theta}` :math:`={\vec{i}}_\theta.\frac{1}{r}\left(\ {\vec{i}}_z{\vec{i}}_r\frac{\partial\tau_{zr}}{\partial\theta}+\tau_{zr}\ {\vec{i}}_z\frac{\partial{\vec{i}}_r}{\partial\theta}\right)+{\vec{i}}_\theta.\frac{1}{r}\left({\vec{i}}_z{\vec{i}}_\theta\frac{\partial\tau_{z\theta}}{\partial\theta}+\tau_{z\theta}{\vec{i}}_z\frac{\partial{\vec{i}}_\theta}{\partial\theta}\right)` :math:`+{\vec{i}}_\theta.\frac{1}{r}\left({\vec{i}}_z{\vec{i}}_z\frac{\partial\tau_{zz}}{\partial\theta}\right)` Chaque terme et facteur du produit scalaire :math:`\vec{i}_\theta . \vec{i}_z = 0` :red:`Soit : = 0` Calcul du terme en :math:`\vec{i}_z` """""""""""""""""""""""""""""""""""" C'est le terme :math:`{\vec{i}}_z.\frac{\partial\left(\tau_{zr}\ {\vec{i}}_z{\vec{i}}_r+\tau_{z\theta}{\vec{i}}_z{\vec{i}}_\theta+\tau_{zz}{\vec{i}}_z{\vec{i}}_z\right)}{\partial z}` :math:`={\vec{i}}_z.\frac{\partial\left(\tau_{zr}\ {\vec{i}}_z{\vec{i}}_r\right)}{\partial z}+{\vec{i}}_z.\frac{\partial\tau_{z\theta}{\vec{i}}_z{\vec{i}}_\theta}{\partial z}+{\vec{i}}_z.\frac{\partial\tau_{zz}{\vec{i}}_z{\vec{i}}_z}{\partial z}` :math:`{\vec{i}}_z.{\vec{i}}_z{\vec{i}}_r\frac{\partial\tau_{zr}\ }{\partial z}+{\vec{i}}_z.{\vec{i}}_z{\vec{i}}_\theta\frac{\partial\tau_{z\theta}}{\partial z}+{\vec{i}}_z.{\vec{i}}_z{\vec{i}}_z\frac{\partial\tau_{zz}}{\partial z}` :red:`Soit :` :math:`={\vec{i}}_r\frac{\partial\tau_{zr}\ }{\partial z}+{\vec{i}}_\theta\frac{\partial\tau_{z\theta}}{\partial z}+{\vec{i}}_z\frac{\partial\tau_{zz}}{\partial z}` :blue:`Donc le 3eme terme s'écrit:` :math:`{\vec{i}}_r\frac{\partial\tau_{zr}\ }{\partial z}+{\vec{i}}_\theta\frac{\partial\tau_{z\theta}}{\partial z}+{\vec{i}}_z\frac{\partial\tau_{zz}}{\partial z}` Récapitulatif ------------- Regroupons les 3 termes: :math:`\vec{\nabla}.{\bar{\bar{\tau}}}_{vis}=` :math:`={\vec{i}}_r\left(\frac{\partial\tau_{rr}\ }{\partial r}+\frac{\tau_{rr}}{r}\right)+{\vec{i}}_\theta\left(\frac{\partial\tau_{r\theta}}{\partial r}+\frac{\tau_{r\theta}}{r}\right)+{\vec{i}}_z\left(\frac{\partial\tau_{rz}}{\partial r}+\frac{\tau_{rz}}{r}\right)` :math:`+ \frac{{\vec{i}}_r}{r}\left(\frac{\partial\tau_{\theta r}}{\partial\theta}\ -\tau_{\theta\theta}\right)+\frac{{\vec{i}}_\theta}{r}\left(\frac{\partial\tau_{\theta\theta}}{\partial\theta}+\tau_{\theta r}\right)+\frac{{\vec{i}}_z}{r}\left(\frac{\partial\tau_{\theta z}}{\partial\theta}\right)` :math:`+{\vec{i}}_r\frac{\partial\tau_{zr}\ }{\partial z}+{\vec{i}}_\theta\frac{\partial\tau_{z\theta}}{\partial z}+{\vec{i}}_z\frac{\partial\tau_{zz}}{\partial z}` :blue:`On arrive alors :` :math:`\vec{\nabla}.{\bar{\bar{\tau}}}_{vis}=\left\{\begin{matrix}\frac{\partial\tau_{rr}\ }{\partial r}+\frac{1}{r}\frac{\partial\tau_{\theta r}}{\partial\theta}+\frac{\partial\tau_{zr}\ }{\partial z}+\frac{\tau_{rr\ -\tau_{\theta\theta}}}{r}\\\frac{\partial\tau_{r\theta}}{\partial r}+\frac{1}{r}\frac{\partial\tau_{\theta\theta}}{\partial\theta}+\frac{\partial\tau_{z\theta}}{\partial z}+\frac{\tau_{r\theta+}\tau_{\theta r}}{r}\\\frac{\partial\tau_{rz}}{\partial r}+\frac{1}{r}\frac{\partial\tau_{\theta z}}{\partial\theta}+\frac{\partial\tau_{zz}}{\partial z}+\frac{\tau_{rz}}{r}\\\end{matrix}\right.`