Annexe A
Fondamental : Expression des diverses équations en coordonnées cartésiennes pour un fluide classique incompressible.
Soit \(\vec{V}\left(\begin{array}{c} V_1\\V_2\\V_3\end{array}\right)\)le vecteur vitesse du fluide.
Expression de la conservation du volume (incompressibilité du fluide) :
\(\nabla \vec V=0\Leftrightarrow \frac{\partial V_1}{\partial x_1}+\frac{\partial V_2}{\partial x_2}+\frac{\partial V_3}{\partial x_3}=0\)
Expression de la conservation de la quantité de mouvement (équation de Navier-Stokes) :
\(\rho\frac{d \vec V}{dt}=-\nabla P^*+\mu\Delta \vec V \Leftrightarrow\)
\(\begin{Bmatrix}\rho\left(\frac{\partial V_1}{\partial t}+V_1\frac{\partial V_1}{\partial x_1}+V_2\frac{\partial V_1}{\partial x_2}+V_3\frac{\partial V_1}{\partial x_3}\right)=-\frac{\partial P^*}{\partial x_1}+\mu\left(\frac{\partial^2 V_1}{\partial x_1^2}+\frac{\partial^2 V_1}{\partial x_2^2}+\frac{\partial^2 V_1}{\partial x_3^2}\right)\\\rho\left( \frac{\partial V_2}{\partial t}+V_1\frac{\partial V_2}{\partial x_1}+V_2\frac{\partial V_2}{\partial x_2}+V_3\frac{\partial V_2}{\partial x_3}\right)=-\frac{\partial P^*}{\partial x_2}+\mu\left( \frac{\partial^2 V_2}{\partial x_1^2}+\frac{\partial^2 V_2}{\partial x_2^2}+\frac{\partial^2 V_2}{\partial x_3^2}\right)\\\rho\left( \frac{\partial V_3}{\partial t}+V_1\frac{\partial V_3}{\partial x_1}+V_2\frac{\partial V_3}{\partial x_2}+V_3\frac{\partial V_3}{\partial x_3}\right)=-\frac{\partial P^*}{\partial x_3}+\mu\left( \frac{\partial^2 V_3}{\partial x_1^2}+\frac{\partial^2 V_3}{\partial x_2^2}+\frac{\partial^2 V_3}{\partial x_3^2}\right)\end{Bmatrix}\)
Expression des composantes du tenseur des contraintes visqueuses :
\(\tau_{11}=2\mu\left( \frac{\partial V_1}{\partial x_1}\right), \tau_{22}=2\mu\left( \frac{\partial V_2}{\partial x_2}\right), \tau_{33}=2\mu\left( \frac{\partial V_3}{\partial x_3}\right)\)
\(\tau_{12}=\tau_{21}=\mu\left( \frac{\partial V_1}{\partial x_2}+\frac{\partial V_2}{\partial x_1}\right), \tau_{23}=\tau_{32}=\mu\left( \frac{\partial V_2}{\partial x_3}+\frac{\partial V_3}{\partial x_2}\right), \tau_{13}=\tau_{31}=\mu\left( \frac{\partial V_1}{\partial x_3}+\frac{\partial V_3}{\partial x_1}\right)\)