Démonstration

On calcule terme à terme les coordonnées du tenseur des contraintes en coordonnées cylindriques soit \(\vec{\nabla}.\bar{\bar{\tau}}\)

:On doit donc calculer mes 3*3*3 termes qui se projettent sur \(\left({\vec{i}}_r,{\vec{i}}_\theta,{\vec{i}}_z\right)\).

Ces termes sont :

\(\left( {\vec{i}}_r\frac{\partial}{\partial r}+{\vec{i}}_\theta\frac{1}{r}\frac{\partial}{\partial\theta}+{\vec{i}}_z\frac{\partial}{\partial z}\right).\left(\tau_{rr}\ {\vec{i}}_r{\vec{i}}_r+\tau_{r\theta}{\vec{i}}_r{\vec{i}}_\theta+\tau_{rz}{\vec{i}}_r{\vec{i}}_z\right)\) \(+\left({\vec{i}}_r\frac{\partial}{\partial r}+{\vec{i}}_\theta\frac{1}{r}\frac{\partial}{\partial\theta}+{\vec{i}}_z\frac{\partial}{\partial z}\right).\left(\tau_{\theta r}\ {\vec{i}}_\theta{\vec{i}}_r+\tau_{\theta\theta}{\vec{i}}_\theta{\vec{i}}_\theta+\tau_{\theta z}{\vec{i}}_\theta{\vec{i}}_z\right)\) \(+\left({\vec{i}}_r\frac{\partial}{\partial r}+{\vec{i}}_\theta\frac{1}{r}\frac{\partial}{\partial\theta}+{\vec{i}}_z\frac{\partial}{\partial z}\right).\left(\tau_{zr}\ {\vec{i}}_z{\vec{i}}_r+\tau_{z\theta}{\vec{i}}_z{\vec{i}}_\theta+\tau_{zz}{\vec{i}}_z{\vec{i}}_z\right)\)

Calcul du 1er terme

\(\left({\vec{i}}_r\frac{\partial}{\partial r}+{\vec{i}}_\theta\frac{1}{r}\frac{\partial}{\partial\theta}+{\vec{i}}_z\frac{\partial}{\partial z}\right).\left(\tau_{rr}\ {\vec{i}}_r{\vec{i}}_r+\tau_{r\theta}{\vec{i}}_r{\vec{i}}_\theta+\tau_{rz}{\vec{i}}_r{\vec{i}}_z\right)\)

\(={\vec{i}}_r.\frac{\partial\tau_{rr}\ {\vec{i}}_r{\vec{i}}_r+\tau_{r\theta}{\vec{i}}_r{\vec{i}}_\theta+\tau_{rz}{\vec{i}}_r{\vec{i}}_z}{\partial r}\) \(+{\vec{i}}_\theta\frac{1}{r}\frac{\partial\tau_{rr}\ {\vec{i}}_r{\vec{i}}_r+\tau_{r\theta}{\vec{i}}_r{\vec{i}}_\theta+\tau_{rz}{\vec{i}}_r{\vec{i}}_z}{\partial\theta}\) \(+{\vec{i}}_z.\frac{\partial\tau_{rr}\ {\vec{i}}_r{\vec{i}}_r+\tau_{r\theta}{\vec{i}}_r{\vec{i}}_\theta+\tau_{rz}{\vec{i}}_r{\vec{i}}_z}{\partial z}\)

Calcul du terme en \({\vec{i}}_r\)

\(={\vec{i}}_r.\frac{\partial\tau_{rr}\ {\vec{i}}_r{\vec{i}}_r+\tau_{r\theta}{\vec{i}}_r{\vec{i}}_\theta+\tau_{rz}{\vec{i}}_r{\vec{i}}_z}{\partial r}\)

\(={\vec{i}}_r.\frac{\partial\tau_{rr}\ {\vec{i}}_r{\vec{i}}_r}{\partial r}+{\vec{i}}_r.\frac{\partial\tau_{r\theta}{\vec{i}}_r{\vec{i}}_\theta}{\partial r}+{\vec{i}}_r.\frac{\partial\tau_{rz}{\vec{i}}_r{\vec{i}}_z}{\partial r}\)

\(={\vec{i}}_r\frac{\partial\tau_{rr}\ }{\partial r}+{\vec{i}}_\theta\frac{\partial\tau_{r\theta}}{\partial r}+{\vec{i}}_z\frac{\partial\tau_{rz}}{\partial r}\)

soit : \(={\vec{i}}_r\frac{\partial\tau_{rr}\ }{\partial r}+{\vec{i}}_\theta\frac{\partial\tau_{r\theta}}{\partial r}+{\vec{i}}_z\frac{\partial\tau_{rz}}{\partial r}\)

Calcul du terme en \({\vec{i}}_{\theta}\)

\({\vec{i}}_\theta.\ \frac{1}{r}\frac{\partial\tau_{rr}\ {\vec{i}}_r{\vec{i}}_r+\tau_{r\theta}{\vec{i}}_r{\vec{i}}_\theta+\tau_{rz}{\vec{i}}_r{\vec{i}}_z}{\partial\theta}\)

\(={\vec{i}}_\theta.\ \left(\frac{1}{r}\frac{\partial\tau_{rr}\ {\vec{i}}_r{\vec{i}}_r}{\partial\theta}+\frac{1}{r}\frac{\partial\tau_{r\theta}{\vec{i}}_r{\vec{i}}_\theta}{\partial\theta}+\frac{1}{r}\frac{\partial\tau_{rz}{\vec{i}}_r{\vec{i}}_z}{\partial\theta}\right)\)

\(={\vec{i}}_\theta.\ \left(\frac{\tau_{rr}}{r}\frac{\partial\ {\vec{i}}_r{\vec{i}}_r}{\partial\theta}+\frac{{\vec{i}}_r{\vec{i}}_r}{r}\frac{\partial\tau_{rr}\ }{\partial\theta}\right) + {\vec{i}}_\theta.\ \left(\frac{\tau_{r\theta}}{r}\frac{\partial{\vec{i}}_r{\vec{i}}_\theta}{\partial\theta}+\frac{{\vec{i}}_r{\vec{i}}_\theta}{r}\frac{\partial\tau_{r\theta}}{\partial\theta}\right)\)

\(+{\vec{i}}_\theta.\ \left(\frac{\tau_{rz}}{r}\frac{\partial{\vec{i}}_r{\vec{i}}_z}{\partial\theta}+\frac{{\vec{i}}_r{\vec{i}}_z}{r}\frac{\partial\tau_{rz}}{\partial\theta}\right)\)

\(= {\vec{i}}_\theta.\ \left(\frac{\tau_{rr}}{r}{\vec{i}}_r\frac{\partial\ {\vec{i}}_r}{\partial\theta}+\frac{\tau_{rr}}{r}\frac{\partial\ {\vec{i}}_r}{\partial\theta}{\vec{i}}_r+\frac{{\vec{i}}_r{\vec{i}}_r}{r}\frac{\partial\tau_{rr}\ }{\partial\theta}\right)+{\vec{i}}_\theta.\ \left(\frac{\tau_{r\theta}{\vec{i}}_r}{r}\frac{\partial{\vec{i}}_\theta}{\partial\theta} + \frac{\tau_{r\theta}}{r}\frac{\partial{\vec{i}}_r}{\partial\theta}{\vec{i}}_\theta+\frac{{\vec{i}}_r{\vec{i}}_\theta}{r}\frac{\partial\tau_{r\theta}}{\partial\theta}\right)\)

\(+{\vec{i}}_\theta.\ \left(\frac{\tau_{rz}{\vec{i}}_r}{r}\frac{\partial{\vec{i}}_z}{\partial\theta}+\frac{\tau_{rz}}{r}\frac{\partial{\vec{i}}_r}{\partial\theta}{\vec{i}}_z+\frac{{\vec{i}}_r{\vec{i}}_z}{r}\frac{\partial\tau_{rz}}{\partial\theta}\right)\)

\(={\vec{i}}_\theta.\ \left(\frac{\tau_{rr}}{r}\frac{\partial\ {\vec{i}}_r}{\partial\theta}{\vec{i}}_r\right)+{\vec{i}}_\theta.\ \left(\frac{\tau_{r\theta}}{r}\frac{\partial{\vec{i}}_r}{\partial\theta}{\vec{i}}_\theta+\right)+{\vec{i}}_\theta.\ \left(\frac{\tau_{rz}}{r}\frac{\partial{\vec{i}}_r}{\partial\theta}{\vec{i}}_z\right)\)

soit : \(= \frac{\tau_{rr}}{r}{\vec{i}}_r+\frac{\tau_{r\theta}}{r}{\vec{i}}_\theta+\frac{\tau_{rz}}{r}{\vec{i}}_z\)

Calcul du terme en \({\vec{i}}_z\)

Le terme en \({\vec{i}}_z\) est : \({\vec{i}}_z.\frac{\partial\tau_{rr}\ {\vec{i}}_r{\vec{i}}_r+\tau_{r\theta}{\vec{i}}_r{\vec{i}}_\theta+\tau_{rz}{\vec{i}}_r{\vec{i}}_z}{\partial z}\)

Pour chaque terme, le produit scalaire \({\vec{i}}_r . {\vec{i}}_z =0\) qui apparait:

soit : \(= 0\)

Donc le 1er terme s’écrit \(={\vec{i}}_r\left(\frac{\partial\tau_{rr}\ }{\partial r}+\frac{\tau_{rr}}{r}\right)+{\vec{i}}_\theta\left(\frac{\partial\tau_{r\theta}}{\partial r}+\frac{\tau_{r\theta}}{r}\right)+{\vec{i}}_z\left(\frac{\partial\tau_{rz}}{\partial r}+\frac{\tau_{rz}}{r}\right)\)

Calcul du 2eme terme

C’est le terme \(\left({\vec{i}}_r\frac{\partial}{\partial r}+{\vec{i}}_\theta\frac{1}{r}\frac{\partial}{\partial\theta}+{\vec{i}}_z\frac{\partial}{\partial z}\right).\left(\tau_{\theta r}\ {\vec{i}}_\theta{\vec{i}}_r+\tau_{\theta\theta}{\vec{i}}_\theta{\vec{i}}_\theta+\tau_{\theta z}{\vec{i}}_\theta{\vec{i}}_z\right)\)

que l’on développe:

\({\vec{i}}_r.\frac{\partial\left(\tau_{\theta r}\ {\vec{i}}_\theta{\vec{i}}_r+\tau_{\theta\theta}{\vec{i}}_\theta{\vec{i}}_\theta+\tau_{\theta z}{\vec{i}}_\theta{\vec{i}}_z\right)}{\partial r} + {\vec{i}}_\theta\frac{1}{r}.\frac{\partial\left(\tau_{\theta r}\ {\vec{i}}_\theta{\vec{i}}_r+\tau_{\theta\theta}{\vec{i}}_\theta{\vec{i}}_\theta+\tau_{\theta z}{\vec{i}}_\theta{\vec{i}}_z\right)}{\partial\theta}\)

\(+{\vec{i}}_z.\frac{\partial\left(\tau_{\theta r}\ {\vec{i}}_\theta{\vec{i}}_r+\tau_{\theta\theta}{\vec{i}}_\theta{\vec{i}}_\theta+\tau_{\theta z}{\vec{i}}_\theta{\vec{i}}_z\right)}{\partial z}\)

Calcul du terme en \({\vec{i}}_r\)

Développons le terme \({\vec{i}}_r.\frac{\partial\left(\tau_{\theta r}\ {\vec{i}}_\theta{\vec{i}}_r+\tau_{\theta\theta}{\vec{i}}_\theta{\vec{i}}_\theta+\tau_{\theta z}{\vec{i}}_\theta{\vec{i}}_z\right)}{\partial r}\)

\(={\vec{i}}_r.\frac{\partial\tau_{\theta r}\ {\vec{i}}_\theta{\vec{i}}_r}{\partial r}+{\vec{i}}_r.\frac{\partial\tau_{\theta\theta}{\vec{i}}_\theta{\vec{i}}_\theta}{\partial r}+{\vec{i}}_r.\frac{\partial\tau_{\theta z}{\vec{i}}_\theta{\vec{i}}_z}{\partial r}\)

Pour chaque terme, le produit scalaire \({\vec{i}}_r . {\vec{i}}_{\theta} =0\) qui apparait:

soit : \(= 0\)

Calcul du terme en \({\vec{i}}_{\theta}\)

\(={\vec{i}}_\theta\frac{1}{r}.\frac{\partial\left(\tau_{\theta r}\ {\vec{i}}_\theta{\vec{i}}_r+\tau_{\theta\theta}{\vec{i}}_\theta{\vec{i}}_\theta+\tau_{\theta z}{\vec{i}}_\theta{\vec{i}}_z\right)}{\partial\theta} = {\vec{i}}_\theta\frac{1}{r}.\frac{\partial\tau_{\theta r}\ {\vec{i}}_\theta{\vec{i}}_r}{\partial\theta}+{\vec{i}}_\theta\frac{1}{r}.\frac{\partial\tau_{\theta\theta}{\vec{i}}_\theta{\vec{i}}_\theta}{\partial\theta}+{\vec{i}}_\theta\frac{1}{r}.\frac{\partial\tau_{\theta z}{\vec{i}}_\theta{\vec{i}}_z}{\partial\theta}\)

\(= {\vec{i}}_\theta\frac{1}{r}.\left(\ {\vec{i}}_\theta{\vec{i}}_r\frac{\partial\tau_{\theta r}}{\partial\theta}+\tau_{\theta r}\frac{\partial\ {\vec{i}}_\theta{\vec{i}}_r}{\partial\theta}\right) + {\vec{i}}_\theta\frac{1}{r}.\left(\tau_{\theta\theta}\frac{\partial{\vec{i}}_\theta{\vec{i}}_\theta}{\partial\theta}+{\vec{i}}_\theta{\vec{i}}_\theta\frac{\partial\tau_{\theta\theta}}{\partial\theta}\right)\)

\({\vec{i}}_\theta\frac{1}{r}.\left({\vec{i}}_\theta{\vec{i}}_z\frac{\partial\tau_{\theta z}}{\partial\theta}+\tau_{\theta z}\frac{\partial{\vec{i}}_\theta{\vec{i}}_z}{\partial\theta}\right)\)

\(={\vec{i}}_\theta\frac{1}{r}.\left(\ {\vec{i}}_\theta{\vec{i}}_r\frac{\partial\tau_{\theta r}}{\partial\theta}+\tau_{\theta r}{\vec{i}}_\theta\frac{\partial\ {\vec{i}}_r}{\partial\theta}+\tau_{\theta r}\frac{\partial\ {\vec{i}}_\theta}{\partial\theta}{\vec{i}}_r\right)\)

\(+ {\vec{i}}_\theta\frac{1}{r}.\left(\tau_{\theta\theta}\frac{\partial{\vec{i}}_\theta}{\partial\theta}{\vec{i}}_\theta+\tau_{\theta\theta}{\vec{i}}_\theta\frac{\partial{\vec{i}}_\theta}{\partial\theta}+{\vec{i}}_\theta{\vec{i}}_\theta\frac{\partial\tau_{\theta\theta}}{\partial\theta}\right)\)

\(+{\vec{i}}_\theta\frac{1}{r}.\left({\vec{i}}_\theta{\vec{i}}_z\frac{\partial\tau_{\theta z}}{\partial\theta}+\tau_{\theta z}{\vec{i}}_\theta\frac{\partial{\vec{i}}_z}{\partial\theta}+\tau_{\theta z}\frac{\partial{\vec{i}}_\theta}{\partial\theta}{\vec{i}}_z\right)\)

\(= {\vec{i}}_\theta\frac{1}{r}.\left(\ {\vec{i}}_\theta{\vec{i}}_r\frac{\partial\tau_{\theta r}}{\partial\theta}+\tau_{\theta r}{\vec{i}}_\theta{\vec{i}}_\theta-\tau_{\theta r}{\vec{i}}_r{\vec{i}}_r\right)\)

\(+{\vec{i}}_\theta\frac{1}{r}.\left(-\tau_{\theta\theta}{\vec{i}}_\theta{\vec{i}}_r+{\vec{i}}_\theta{\vec{i}}_\theta\frac{\partial\tau_{\theta\theta}}{\partial\theta}\right)+{\vec{i}}_\theta\frac{1}{r}.\left({\vec{i}}_\theta{\vec{i}}_z\frac{\partial\tau_{\theta z}}{\partial\theta}\right)\)

\(\frac{1}{r}\left(\ {\vec{i}}_r\frac{\partial\tau_{\theta r}}{\partial\theta}+\tau_{\theta r}{\vec{i}}_\theta\right)+\frac{1}{r}\left(-\tau_{\theta\theta}{\vec{i}}_r+{\vec{i}}_\theta\frac{\partial\tau_{\theta\theta}}{\partial\theta}\right)+\frac{1}{r}\left({\vec{i}}_z\frac{\partial\tau_{\theta z}}{\partial\theta}\right)\)

Soit : \(=\frac{{\vec{i}}_r}{r}\left(\frac{\partial\tau_{\theta r}}{\partial\theta}\ -\tau_{\theta\theta}\right)+\frac{{\vec{i}}_\theta}{r}\left(\frac{\partial\tau_{\theta\theta}}{\partial\theta}+\tau_{\theta r}\right)+\frac{{\vec{i}}_z}{r}\left(\frac{\partial\tau_{\theta z}}{\partial\theta}\right)\)

Calcul du terme en \({\vec{i}}_z\)

\({\vec{i}}_z.\frac{\partial\left(\tau_{\theta r}\ {\vec{i}}_\theta{\vec{i}}_r+\tau_{\theta\theta}{\vec{i}}_\theta{\vec{i}}_\theta+\tau_{\theta z}{\vec{i}}_\theta{\vec{i}}_z\right)}{\partial z}\)

Ce terme ne fait apparaitre que des termes \(\vec{i}_z. \vec{i}_\theta = 0\)

Soit : = 0

Donc le 2ème terme s’écrit \(=\frac{{\vec{i}}_r}{r}\left(\frac{\partial\tau_{\theta r}}{\partial\theta}\ -\tau_{\theta\theta}\right)+\frac{{\vec{i}}_\theta}{r}\left(\frac{\partial\tau_{\theta\theta}}{\partial\theta}+\tau_{\theta r}\right)+\frac{{\vec{i}}_z}{r}\left(\frac{\partial\tau_{\theta z}}{\partial\theta}\right)\)

Calcul du 3eme terme

C’est le terme \(\left({\vec{i}}_r\frac{\partial}{\partial r}+{\vec{i}}_\theta\frac{1}{r}\frac{\partial}{\partial\theta}+{\vec{i}}_z\frac{\partial}{\partial z}\right).\left(\tau_{zr}\ {\vec{i}}_z{\vec{i}}_r+\tau_{z\theta}{\vec{i}}_z{\vec{i}}_\theta+\tau_{zz}{\vec{i}}_z{\vec{i}}_z\right)\)

qui donne les 3 termes suivants :

\(={\vec{i}}_r.\frac{\partial\left(\tau_{zr}\ {\vec{i}}_z{\vec{i}}_r+\tau_{z\theta}{\vec{i}}_z{\vec{i}}_\theta+\tau_{zz}{\vec{i}}_z{\vec{i}}_z\right)}{\partial r}+{\vec{i}}_\theta.\frac{1}{r}\frac{\partial\left(\tau_{zr}\ {\vec{i}}_z{\vec{i}}_r+\tau_{z\theta}{\vec{i}}_z{\vec{i}}_\theta+\tau_{zz}{\vec{i}}_z{\vec{i}}_z\right)}{\partial\theta}\)

\(+{\vec{i}}_z.\frac{\partial\left(\tau_{zr}\ {\vec{i}}_z{\vec{i}}_r+\tau_{z\theta}{\vec{i}}_z{\vec{i}}_\theta+\tau_{zz}{\vec{i}}_z{\vec{i}}_z\right)}{\partial z}\)

Calcul du terme en \(\vec{i}_r\)

Il s’agit du terme \({\vec{i}}_r.\frac{\partial\left(\tau_{zr}\ {\vec{i}}_z{\vec{i}}_r+\tau_{z\theta}{\vec{i}}_z{\vec{i}}_\theta+\tau_{zz}{\vec{i}}_z{\vec{i}}_z\right)}{\partial r}\)

qui se décompose en 3 termes qui sont :

\(={\vec{i}}_r.\frac{\partial\tau_{zr}\ {\vec{i}}_z{\vec{i}}_r}{\partial r}+{\vec{i}}_r.\frac{\partial\tau_{z\theta}{\vec{i}}_z{\vec{i}}_\theta}{\partial r}+{\vec{i}}_r.\frac{\partial\tau_{zz}{\vec{i}}_z{\vec{i}}_z}{\partial r}\)

Les vecteurs de base étant constants, on a pour chaque terme le produit scalaire : \(\vec{i}_r.\)vec{i}_z = 0` Soit : = 0

Calcul du terme en \(\vec{i}_\theta\)

C’est le terme \({\vec{i}}_\theta.\frac{1}{r}\frac{\partial\left(\tau_{zr}\ {\vec{i}}_z{\vec{i}}_r+\tau_{z\theta}{\vec{i}}_z{\vec{i}}_\theta+\tau_{zz}{\vec{i}}_z{\vec{i}}_z\right)}{\partial\theta}\)

qui se décompose \(={\vec{i}}_\theta.\frac{1}{r}\frac{\partial\tau_{zr}\ {\vec{i}}_z{\vec{i}}_r}{\partial\theta}+{\vec{i}}_\theta.\frac{1}{r}\frac{\partial\tau_{z\theta}{\vec{i}}_z{\vec{i}}_\theta}{\partial\theta}+{\vec{i}}_\theta.\frac{1}{r}\frac{\partial\tau_{zz}{\vec{i}}_z{\vec{i}}_z}{\partial\theta}\)

\(={\vec{i}}_\theta.\frac{1}{r}\left(\ {\vec{i}}_z{\vec{i}}_r\frac{\partial\tau_{zr}}{\partial\theta}+\tau_{zr}\ {\vec{i}}_z\frac{\partial{\vec{i}}_r}{\partial\theta}\right)+{\vec{i}}_\theta.\frac{1}{r}\left({\vec{i}}_z{\vec{i}}_\theta\frac{\partial\tau_{z\theta}}{\partial\theta}+\tau_{z\theta}{\vec{i}}_z\frac{\partial{\vec{i}}_\theta}{\partial\theta}\right)\) \(+{\vec{i}}_\theta.\frac{1}{r}\left({\vec{i}}_z{\vec{i}}_z\frac{\partial\tau_{zz}}{\partial\theta}\right)\)

Chaque terme et facteur du produit scalaire \(\vec{i}_\theta . \vec{i}_z = 0\) Soit : = 0

Calcul du terme en \(\vec{i}_z\)

C’est le terme \({\vec{i}}_z.\frac{\partial\left(\tau_{zr}\ {\vec{i}}_z{\vec{i}}_r+\tau_{z\theta}{\vec{i}}_z{\vec{i}}_\theta+\tau_{zz}{\vec{i}}_z{\vec{i}}_z\right)}{\partial z}\)

\(={\vec{i}}_z.\frac{\partial\left(\tau_{zr}\ {\vec{i}}_z{\vec{i}}_r\right)}{\partial z}+{\vec{i}}_z.\frac{\partial\tau_{z\theta}{\vec{i}}_z{\vec{i}}_\theta}{\partial z}+{\vec{i}}_z.\frac{\partial\tau_{zz}{\vec{i}}_z{\vec{i}}_z}{\partial z}\)

\({\vec{i}}_z.{\vec{i}}_z{\vec{i}}_r\frac{\partial\tau_{zr}\ }{\partial z}+{\vec{i}}_z.{\vec{i}}_z{\vec{i}}_\theta\frac{\partial\tau_{z\theta}}{\partial z}+{\vec{i}}_z.{\vec{i}}_z{\vec{i}}_z\frac{\partial\tau_{zz}}{\partial z}\)

Soit : \(={\vec{i}}_r\frac{\partial\tau_{zr}\ }{\partial z}+{\vec{i}}_\theta\frac{\partial\tau_{z\theta}}{\partial z}+{\vec{i}}_z\frac{\partial\tau_{zz}}{\partial z}\)

Donc le 3eme terme s’écrit: \({\vec{i}}_r\frac{\partial\tau_{zr}\ }{\partial z}+{\vec{i}}_\theta\frac{\partial\tau_{z\theta}}{\partial z}+{\vec{i}}_z\frac{\partial\tau_{zz}}{\partial z}\)

Récapitulatif

Regroupons les 3 termes: \(\vec{\nabla}.{\bar{\bar{\tau}}}_{vis}=\)

\(={\vec{i}}_r\left(\frac{\partial\tau_{rr}\ }{\partial r}+\frac{\tau_{rr}}{r}\right)+{\vec{i}}_\theta\left(\frac{\partial\tau_{r\theta}}{\partial r}+\frac{\tau_{r\theta}}{r}\right)+{\vec{i}}_z\left(\frac{\partial\tau_{rz}}{\partial r}+\frac{\tau_{rz}}{r}\right)\)

\(+ \frac{{\vec{i}}_r}{r}\left(\frac{\partial\tau_{\theta r}}{\partial\theta}\ -\tau_{\theta\theta}\right)+\frac{{\vec{i}}_\theta}{r}\left(\frac{\partial\tau_{\theta\theta}}{\partial\theta}+\tau_{\theta r}\right)+\frac{{\vec{i}}_z}{r}\left(\frac{\partial\tau_{\theta z}}{\partial\theta}\right)\)

\(+{\vec{i}}_r\frac{\partial\tau_{zr}\ }{\partial z}+{\vec{i}}_\theta\frac{\partial\tau_{z\theta}}{\partial z}+{\vec{i}}_z\frac{\partial\tau_{zz}}{\partial z}\)

On arrive alors :

\(\vec{\nabla}.{\bar{\bar{\tau}}}_{vis}=\left\{\begin{matrix}\frac{\partial\tau_{rr}\ }{\partial r}+\frac{1}{r}\frac{\partial\tau_{\theta r}}{\partial\theta}+\frac{\partial\tau_{zr}\ }{\partial z}+\frac{\tau_{rr\ -\tau_{\theta\theta}}}{r}\\\frac{\partial\tau_{r\theta}}{\partial r}+\frac{1}{r}\frac{\partial\tau_{\theta\theta}}{\partial\theta}+\frac{\partial\tau_{z\theta}}{\partial z}+\frac{\tau_{r\theta+}\tau_{\theta r}}{r}\\\frac{\partial\tau_{rz}}{\partial r}+\frac{1}{r}\frac{\partial\tau_{\theta z}}{\partial\theta}+\frac{\partial\tau_{zz}}{\partial z}+\frac{\tau_{rz}}{r}\\\end{matrix}\right.\)