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

Equações básicas em coordenadas cilíndricas

Equação da continuidade

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

${\frac {\partial \rho }{\partial t}}\;+\;(\nabla \cdot (\rho {\vec {v}}))\;=\;0$

Considerando ρ constante, ela se simplifica para

$\rho \left({\frac {1}{r}}\;{\frac {\partial }{\partial r}}\;(rv_{r})\;+\;{\frac {1}{r}}\;{\frac {\partial }{\partial \theta }}\;(v_{\theta })\;+\;{\frac {\partial }{\partial z}}\;(v_{z})\right)\;=\;0$

E assim

${\frac {1}{r}}\;{\frac {\partial }{\partial r}}\;(rv_{r})\;+\;{\frac {1}{r}}\;{\frac {\partial }{\partial \theta }}\;(v_{\theta })\;+\;{\frac {\partial }{\partial z}}\;(v_{z})\;=\;0$

Equações de Navier-Stokes

Componente r:

$\rho \left({\frac {\partial v_{r}}{\partial t}}\;+\;v_{r}\;{\frac {\partial v_{r}}{\partial r}}\;+\;{\frac {v_{\theta }}{r}}\;{\frac {\partial v_{r}}{\partial \theta }}\;-\;{\frac {v_{\theta }^{2}}{r}}\;+\;v_{z}\;{\frac {\partial v_{r}}{\partial z}}\right)\;=\;$

$\;=\;\rho g_{r}\;-\;{\frac {\partial p}{\partial r}}\;+\;\mu _{0}\;\left[{\frac {\partial }{\partial r}}\left({\frac {1}{r}}\;{\frac {\partial }{\partial r}}(rv_{r})\right)\;+\;{\frac {1}{r^{2}}}\;{\frac {\partial ^{2}v_{r}}{\partial \theta ^{2}}}\;-\;{\frac {2}{r^{2}}}\;{\frac {\partial v_{\theta }}{\partial \theta }}\;+\;{\frac {\partial ^{2}v_{r}}{\partial z^{2}}}\right]$

Componente Θ:

$\rho \left({\frac {\partial v_{\theta }}{\partial t}}\;+\;v_{r}\;{\frac {\partial v_{\theta }}{\partial r}}\;+\;{\frac {v_{\theta }}{r}}\;{\frac {\partial v_{\theta }}{\partial \theta }}\;+\;{\frac {v_{r}v_{\theta }}{r}}\;+\;v_{z}\;{\frac {\partial v_{\theta }}{\partial z}}\right)\;=\;$

$\;=\;\rho g_{\theta }\;-\;{\frac {1}{r}}\;{\frac {\partial p}{\partial \theta }}\;+\;\mu _{0}\;\left[{\frac {\partial }{\partial r}}\left({\frac {1}{r}}\;{\frac {\partial }{\partial r}}(rv_{\theta })\right)\;+\;{\frac {1}{r^{2}}}\;{\frac {\partial ^{2}v_{\theta }}{\partial \theta ^{2}}}\;+\;{\frac {2}{r^{2}}}\;{\frac {\partial v_{\theta }}{\partial \theta }}\;+\;{\frac {\partial ^{2}v_{\theta }}{\partial z^{2}}}\right]$

Componente z:

$\rho \left({\frac {\partial v_{z}}{\partial t}}\;+\;v_{r}\;{\frac {\partial v_{z}}{\partial r}}\;+\;{\frac {v_{\theta }}{r}}\;{\frac {\partial v_{z}}{\partial \theta }}\;+\;v_{z}\;{\frac {\partial v_{z}}{\partial z}}\right)\;=\;$

$\;=\;\rho g_{z}\;-\;{\frac {\partial p}{\partial z}}\;+\;\mu _{0}\;\left[{\frac {1}{r}}\;{\frac {\partial }{\partial r}}\left(r\;{\frac {\partial v_{z}}{\partial r}}\right)\;+\;{\frac {1}{r^{2}}}\;{\frac {\partial ^{2}v_{z}}{\partial \theta ^{2}}}\;+\;{\frac {\partial ^{2}v_{z}}{\partial z^{2}}}\right]$