... | ... | @@ -20,7 +20,28 @@ We indicate with $`\kappa_p`$ and $`\kappa_r`$ respectively the Planck and t |
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We consider an ideal gas of pressure $`P`$ with equation of state:
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$`P = (\gamma-1)e`$ for a gas with adiabatic index $`\gamma`$.
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The terms $`P \nabla \cdot \vec v `$ and $`Q^+`$ are respectively the compressional heating and the viscous heating ( see Mihalas and Mihalas 1984). We do not include here the heating from the central star.
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The terms $`P \nabla \cdot \vec v `$ and $`Q^+= \bar{ \bar \tau}\nabla v`$ are respectively the compressional heating and the viscous heating ( see Mihalas and Mihalas 1984).
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The stress tensor is given by:
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$`\bar{ \bar \tau} = 2\rho \nu(\bar{ \bar D} -{1\over 3}(\nabla \vec v)\bar {\bar I})`$
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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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$`\left\lbrace \begin{array}{lllll}
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\tau _{RR} & & & = & 2\rho \nu ({\partial v_R \over \partial R}-{1\over 3}\nabla \cdot \vec v )\\
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\tau _{\varphi \varphi} & & & = & 2\rho\nu ({1\over R}{\partial v_\varphi \over \partial \varphi}+{v_R\over R}-{1\over 3}\nabla \cdot \vec v )\\
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\tau_{\theta \theta} & & & = & 2\rho\nu ({1\over R\sin \varphi}{\partial v_\theta \over \partial \theta}+{v_R\over R}+{v_\varphi \cot(\varphi)\over R} -{1\over 3}\nabla \cdot \vec v )\\
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\tau _{R\varphi} & = & \tau _{\varphi R} & = & 2\rho\nu[{1\over 2}({1\over R}{\partial v_R \over \partial \varphi}+R{\partial \over \partial R}{v_{\varphi}\over R})] \\
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\tau _{\theta\varphi}& = &\tau _{\varphi\theta} & = & 2\rho\nu [{1\over 2}({1\over R\sin (\varphi)}{\partial v_\varphi \over \partial \theta}+{\sin (\varphi) \over R} {\partial \over \partial \varphi }{v_\theta \over \sin (\varphi)})] \\
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\tau _{R\theta} & = & \tau _{\theta R} & = & 2\rho\nu [{1\over 2}(R{\partial \over \partial R}{v_\theta \over R}+{1\over R\sin (\varphi)} {\partial v_R
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\over \partial \theta})]
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\end{array} \right. `$
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We do not include here the heating from the central star.
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## How we derive the equation for the internal energy
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