... | ... | @@ -30,19 +30,19 @@ In spherical coordinates the components of the sress tensor |
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from (see Tassoul 1978) are:
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$`\left\lbrace \begin{array}{lllll}
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\tau _{RR} & & & = & 2\rho \nu ({\partial v_R \over \partial R}-{1\over 3}\nabla \cdot \vec v )\\
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\tau _{\varphi \varphi} & & & = & 2\rho\nu ({1\over R}{\partial v_\varphi \over \partial \varphi}+{v_R\over R}-{1\over 3}\nabla \cdot \vec v )\\
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\tau_{\theta \theta} & & & = & 2\rho\nu ({1\over R\sin \varphi}{\partial v_\theta \over \partial \theta}+{v_R\over R}+{v_\varphi \cot(\varphi)\over R} -{1\over 3}\nabla \cdot \vec v )\\
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\tau _{R\varphi} & = & \tau _{\varphi R} & = & 2\rho\nu[{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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\tau _{\theta\varphi}& = &\tau _{\varphi\theta} & = & 2\rho\nu [{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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\tau _{R\theta} & = & \tau _{\theta R} & = & 2\rho\nu [{1\over 2}(R{\partial \over \partial R}{v_\theta \over R}+{1\over R\sin (\varphi)} {\partial v_R
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D _{RR} & & & = & 2\rho \nu ({\partial v_R \over \partial R})\\
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D _{\varphi \varphi} & & & = & 2\rho\nu ({1\over R}{\partial v_\varphi \over \partial \varphi}+{v_R\over R})\\
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D_{\theta \theta} & & & = & 2\rho\nu ({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} & = & 2\rho\nu[{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} & = & 2\rho\nu [{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} & = & 2\rho\nu [{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 from Mihalas and Mihalas 1984):
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$`Q^+ = 2\rho \nu (\tau _{RR}^2+\tau _{\varphi \varphi}^2+\tau_{\theta \theta}^2
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+{1\over 2}(\tau _{R\varphi}^2+\tau _{\theta\varphi}^2+\tau _{R\theta}^2)-
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$`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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{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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... | ... | @@ -68,6 +68,6 @@ $`\begin{array}{lll} |
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where $`Q^+`$ is the viscous heating and $Q^-$ is the radiative cooling.
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the stress tensor in the 2 dimensional case is:
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$`Q^+ = 2\rho \nu (\tau _{RR}^2+\tau_{\theta \theta}^2
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+{1\over 2}\tau _{R\theta}^2-
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$`Q^+ = 2\rho \nu (D_{RR}^2+D_{\theta \theta}^2
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+{1\over 2}D{R\theta}^2-
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{1\over 3}(\nabla \cdot \vec v)^2)`$ |