... | ... | @@ -4,9 +4,13 @@ We write in this page the continuity and Navier-Stokes equations in the form tha |
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$`{\partial \rho \over \partial t}+ \nabla \cdot (\rho \vec v)= 0 `$
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where $`\rho`$ is the volume density and $`\vec v=(v_R,v_{\varphi},v_{\theta},)`$ is the fluid velocity vector.
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where $`\rho`$ is the volume density and $`\vec v=(v_R,v_{\varphi},v_{\theta},)`$ is the fluid velocity vector with $v_\varphi=r\sin (\theta)(\omega+\Omega)$
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where $\omega$ is the azimuthal angular velocity in the rotating frame.
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## Navier-Stokes equations
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## Navier-Stokes equations for the momenta
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The Navier-Stokes equations for the radial momentum $`J_r= \rho v_r`$, the polar momentum $`J_\theta = \rho r v_\theta`$ and the angular momentum $`J_\varphi = \rho r\sin(\theta) v_\varphi = \rho r^2 \sin^2\theta (\omega+\Omega)`$ read:
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$`\left\lbrace \begin{array}{lll}
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{\partial J_r \over \partial t}+ \nabla \cdot (J_r\vec v) & = & \rho [{v_\theta
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