... | ... | @@ -26,17 +26,16 @@ In order to compute the viscous heating we need to know the viscous stress tenso |
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$`\bar{ \bar \tau} = 2\rho \nu(\bar{ \bar D} -{1\over 3}(\nabla \cdot \vec v)\bar {\bar I})`$
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where $`\bar {\bar D}`$ is the strain tensor an $`\nu`$ the shear viscosity.
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In spherical coordinates the components of the sress tensor
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from (see Tassoul 1978) are:
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In spherical coordinates the components of the strain tensor are:
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$`\left\lbrace \begin{array}{lllll}
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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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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 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 from Mihalas and Mihalas 1984):
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