... | ... | @@ -32,16 +32,16 @@ $`\left\lbrace \begin{array}{lllll} |
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D _{RR} & & & = & {\partial v_R \over \partial R}\\
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D _{\varphi \varphi} & & & = & {1\over R}{\partial v_\varphi \over \partial \varphi}+{v_R\over R}\\
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D_{\theta \theta} & & & = & {1\over R\sin \varphi}{\partial v_\theta \over \partial \theta}+{v_R\over R}+{v_\varphi \cot(\varphi)\over R}\\
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D _{R\varphi} & = & D _{\varphi R} & = & ({1\over R}{\partial v_R \over \partial \varphi}+R{\partial \over \partial R}{v_{\varphi}\over R}) \\
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D _{\theta\varphi}& = & D _{\varphi\theta} & = & ({1\over R\sin (\varphi)}{\partial v_\varphi \over \partial \theta}+{\sin (\varphi) \over R} {\partial \over \partial \varphi }{v_\theta \over \sin (\varphi)}) \\
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D _{R\theta} & = & D _{\theta R} & = & (R{\partial \over \partial R}{v_\theta \over R}+{1\over R\sin (\varphi)} {\partial v_R
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D _{R\varphi} & = & D _{\varphi R} & = & {1\over 2} ({1\over R}{\partial v_R \over \partial \varphi}+R{\partial \over \partial R}{v_{\varphi}\over R}) \\
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D _{\theta\varphi}& = & D _{\varphi\theta} & = &{1\over 2} ({1\over R\sin (\varphi)}{\partial v_\varphi \over \partial \theta}+{\sin (\varphi) \over R} {\partial \over \partial \varphi }{v_\theta \over \sin (\varphi)}) \\
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D _{R\theta} & = & D _{\theta R} & = &{1\over 2} (R{\partial \over \partial R}{v_\theta \over R}+{1\over R\sin (\varphi)} {\partial v_R
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\over \partial \theta})
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\end{array} \right. `$
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We can now compute the viscous heating (see Eq. 27.30 pag. 92 of Mihalas and Mihalas 1984):
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$`Q^+ = 2\rho \nu (D_{RR}^2+D_{\varphi \varphi}^2+D_{\theta \theta}^2
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+{1\over 2}(D_{R\varphi}^2+D_{\theta\varphi}^2+D_{R\theta}^2)-
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+(D_{R\varphi}^2+D_{\theta\varphi}^2+D_{R\theta}^2)-
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{1\over 3}(\nabla \cdot \vec v)^2)`$
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We will consider in the following the heating from the central star.
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... | ... | @@ -69,7 +69,7 @@ the stress tensor in the 2 dimensional case is: |
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in the hypothesis the the gas has only Keplerian speed (no radial velocity and
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$`v_{\theta} = \Omega_K R`$ then:
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$`Q^+ = \Sigma \nu D_{R\theta}^2 = {9\over 4}\Sigma \nu \Omega_K^2`$
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$`Q^+ = \Sigma \nu D_{R\theta}^2 = {9\over 8}\Sigma \nu \Omega_K^2`$
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this simple prescription can also be implemented in a 2 dimensional disk.
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