... | ... | @@ -4,4 +4,22 @@ We write in this page the continuity and Navier-Stokes equations in the form tha |
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$`{\partial \rho \over \partial t}+ \nabla (\rho \vec v)= 0 `$
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where $`\rho`$ is the volume density and $`\vec v=(v_R,v_{\theta},v_{\varphi})`$ |
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\ No newline at end of file |
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where $`\rho`$ is the volume density and $`\vec v=(v_R,v_{\theta},v_{\varphi})`$ is the fluid velocity vector.
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## Navier-Stokes equations
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$`\left\lbrace \begin{array}{lll}
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{\partial J_r \over \partial t}+ \nabla \cdot (J_r\vec v) & = & \rho [{v_\theta
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^2 \over r }+ {v_\varphi^2\over r} -{\partial \Phi \over \partial r}
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\\
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& & +{1\over \rho} (f_r-{\partial P \over \partial r})] \\
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{\partial J_\theta \over \partial t}+ \nabla \cdot (J_\theta \vec v) & = &
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\rho r[{v_\varphi^2\cot(\theta)\over r} -{1\over r}{\partial \Phi \over {\parti
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al \theta}} \\ & & +{1\over \rho }
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( f_\theta -{1\over r }{\partial P \over {\partial \theta}} ) ] \\
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{\partial J_\varphi \over \partial t}+ \nabla \cdot (J_\varphi \vec v) & = &
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\rho r \sin (\theta) [-{1\over {r \sin \theta}}{\partial \Phi \over \partial \va
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rphi} \\ & & +{1\over \rho}(f_\varphi
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-{1\over {r \sin \theta}}{\partial P \over \partial \varphi} )]
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\end{array} \right.`$
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