Mecânica dos fluidos/Equações básicas em coordenadas cilíndricas para o líquido Newtoniano

Equações básicas em coordenadas cilíndricas

Como vimos anteriormente, a equação da continuidade em um sistema de coordenadas cilíndricas em a forma seguinte:

\frac{\partial ρ}{\partial t} + (\nabla \cdot (ρ \vec{v})) = 0

Considerando ρ constante, ela se simplifica para

ρ (\frac{1}{r} \frac{\partial}{\partial r} (r v_{r}) + \frac{1}{r} \frac{\partial}{\partial θ} (v_{θ}) + \frac{\partial}{\partial z} (v_{z})) = 0

E assim

\frac{1}{r} \frac{\partial}{\partial r} (r v_{r}) + \frac{1}{r} \frac{\partial}{\partial θ} (v_{θ}) + \frac{\partial}{\partial z} (v_{z}) = 0

Componente r:

ρ (\frac{\partial v_{r}}{\partial t} + v_{r} \frac{\partial v_{r}}{\partial r} + \frac{v_{θ}}{r} \frac{\partial v_{r}}{\partial θ} - \frac{v_{θ}^{2}}{r} + v_{z} \frac{\partial v_{r}}{\partial z}) =

= ρ g_{r} - \frac{\partial p}{\partial r} + μ_{0} [\frac{\partial}{\partial r} (\frac{1}{r} \frac{\partial}{\partial r} (r v_{r})) + \frac{1}{r^{2}} \frac{\partial^{2} v_{r}}{\partial θ^{2}} - \frac{2}{r^{2}} \frac{\partial v_{θ}}{\partial θ} + \frac{\partial^{2} v_{r}}{\partial z^{2}}]

Componente Θ:

ρ (\frac{\partial v_{θ}}{\partial t} + v_{r} \frac{\partial v_{θ}}{\partial r} + \frac{v_{θ}}{r} \frac{\partial v_{θ}}{\partial θ} + \frac{v_{r} v_{θ}}{r} + v_{z} \frac{\partial v_{θ}}{\partial z}) =

= ρ g_{θ} - \frac{1}{r} \frac{\partial p}{\partial θ} + μ_{0} [\frac{\partial}{\partial r} (\frac{1}{r} \frac{\partial}{\partial r} (r v_{θ})) + \frac{1}{r^{2}} \frac{\partial^{2} v_{θ}}{\partial θ^{2}} + \frac{2}{r^{2}} \frac{\partial v_{θ}}{\partial θ} + \frac{\partial^{2} v_{θ}}{\partial z^{2}}]

Componente z:

ρ (\frac{\partial v_{z}}{\partial t} + v_{r} \frac{\partial v_{z}}{\partial r} + \frac{v_{θ}}{r} \frac{\partial v_{z}}{\partial θ} + v_{z} \frac{\partial v_{z}}{\partial z}) =

= ρ g_{z} - \frac{\partial p}{\partial z} + μ_{0} [\frac{1}{r} \frac{\partial}{\partial r} (r \frac{\partial v_{z}}{\partial r}) + \frac{1}{r^{2}} \frac{\partial^{2} v_{z}}{\partial θ^{2}} + \frac{\partial^{2} v_{z}}{\partial z^{2}}]