Annexe B
Fondamental : Expression des diverses équations en coordonnées cylindriques pour un fluide classique incompressible.
Soit \(\vec{V}\left(\begin{array}{c} V_r\\V_\theta\\V_3\end{array}\right)\)le vecteur vitesse du fluide.
Expression de la conservation du volume (incompressibilité du fluide) :
\(\nabla \vec V=0\Leftrightarrow \frac 1r\frac{\partial }{\partial r}(rV_r)+\frac 1r\frac{\partial V_\theta}{\partial \theta}+\frac{\partial V_3}{\partial x_3}=0\)
Expression de la conservation de la quantité de mouvement (équation de Navier-Stokes) :
\(\rho\frac{d \vec V}{dt}=-\nabla P^*+\mu\Delta \vec V \Leftrightarrow\)
\(\left\{ \begin{array}{c} \rho\left(\frac{\partial V_r}{\partial t}+V_r\frac{\partial V_r}{\partial r}+\frac{V_\theta}{r}\frac{\partial V_r}{\partial \theta}-\frac{V_\theta^2}{r}+V_3\frac{\partial V_r}{\partial x_3}\right)=-\frac{\partial P^*}{\partial r}+\mu\left(\frac 1r \frac{\partial}{\partial r}\left( r \frac{\partial V_r}{\partial r}\right)+\frac{1}{r^2}\frac{\partial^2 V_r}{\partial \theta^2}+\frac{\partial^2 V_r}{\partial x_3^2}-\frac{V_r}{r^2}-\frac{2}{r^2}\frac{\partial V_\theta}{\partial \theta}\right) \\ \rho\left(\frac{\partial V_\theta}{\partial t}+V_r\frac{\partial V_\theta}{\partial r}+\frac{V_\theta}{r}\frac{\partial V_\theta}{\partial \theta}-\frac{V_r V_\theta}{r}+V_3\frac{\partial V_\theta}{\partial x_3}\right)=-\frac{\partial P^*}{\partial \theta}+\mu\left(\frac 1r \frac{\partial}{\partial r}\left( r \frac{\partial V_\theta}{\partial r}\right)+\frac{1}{r^2}\frac{\partial^2 V_\theta}{\partial \theta^2}+\frac{\partial^2 V_\theta}{\partial x_3^2}-\frac{V_\theta}{r^2}-\frac{2}{r^2}\frac{\partial V_r}{\partial \theta}\right)\\ \rho\left(\frac{\partial V_3}{\partial t}+V_r\frac{\partial V_3}{\partial r}+\frac{V_\theta}{r}\frac{\partial V_3}{\partial \theta}+V_3\frac{\partial V_3}{\partial x_3}\right)=-\frac{\partial P^*}{\partial x_3}+\mu\left(\frac 1r \frac{\partial}{\partial r}\left( r \frac{\partial V_3}{\partial r}\right)+\frac{1}{r^2}\frac{\partial^2 V_3}{\partial \theta^2}+\frac{\partial^2 V_3}{\partial x_3^2}\right)\end{array}\right.\)
Expression des composantes du tenseur des contraintes visqueuses :
\(\tau_{rr}=2\mu\left( \frac{\partial V_r}{\partial r}\right), \tau_{\theta\theta}=2\mu\left(\frac 1r\frac{\partial V_\theta}{\partial x_\theta}\right), \tau_{33}=2\mu\left( \frac{\partial V_3}{\partial x_3}\right)\)
\(\tau_{r\theta}=\tau_{\theta r}=\mu\left( \frac{\partial V_\theta}{\partial r}-\frac{V_\theta}{r}+\frac1r \frac{\partial V_r}{\partial x_\theta}\right), \tau_{3\theta}=\tau_{\theta 3}=\mu\left( \frac 1r \frac{\partial V_3}{\partial \theta}+\frac{\partial V_\theta}{\partial x_3}\right), \tau_{r3}=\tau_{3r}=\mu\left( \frac{\partial V_r}{\partial x_3}+\frac{\partial V_3}{\partial r}\right)\)