*ZZHmoty: :$\frac{D}{Dt}\int_V \rho dV = 0$

:$\frac{\partial \rho}{\partial t}+\Delta\cdot(\rho v) = 0$ a

:

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*ZZHybnosti :$\frac{D}{Dt}\int_V\rho v dV = \int_V \rho g dV - 2\int_V \rho \Omega \times v dV - \int_V \rho \Omega \times(\Omega \times r)dV+ \int_S \tau \cdot n dS$

:$\nabla \cdot \tau^T+\rho g-2\rho\Omega\times v-\rho\Omega\times(\Omega\times r)=\rho\frac{Dv}{Dt}=\rho(\frac{\partial v}{\partial t}+v\cdot \nabla v)$ a

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*ZZMomentu hybnosti :$\frac{D}{Dt}\int_V r\times\rho v dV = \int_V r\times \rho g dV - 2\int_V r\times(\rho \Omega \times v) dV - \int_V r\times( \rho \Omega \times(\Omega \times r))dV+ \int_S r\times \tau \cdot n dS$

:$I\times \tau = 0$ <=> $\tau = \tau^T$

*ZZEnergie :

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Gibbsova relace :$\rho T\frac{Ds}{Dt}=\rho\frac{D\epsilon}{Dt}+p\nabla\cdot v$

a z toho :$\rho T\frac{Ds}{Dt}=\nabla\cdot(k\cdot \nabla T)+\sigma : \nabla v+H$

a :

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Dále tepelná rovnice ve stavových proměnných

:$\rho T \frac{Ds}{Dt} = \rho T\frac{\partial s}{\partial T}_V \frac{DT}{Dt}+\rho T\frac{\partial s}{\partial V}_T \frac{DV}{Dt}=\rho c_V \frac{DT}{Dt}+\mu\nabla\cdot v$ kde

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kde termální expanze $\alpha = -\frac{1}{p}(\frac{\partial p}{\partial T})_p$ a izotermální objemový modul $K_T = -V(\frac{\partial p}{\partial V})_T$

Pak získáme $\mu=T\alpha K_T = \rho c_V T \gamma$ pro Grüneisenův parametr $\gamma = \alpha K_T / \rho c_V$

a z toho plyne termální rovnice :$\rho c_V \frac{\partial T}{\partial t} = \nabla \cdot (k\cdot \nabla T)-\rho c_V v\cdot \nabla T - \rho c_V T\gamma \nabla\cdot v+\sigma: \nabla v+H$

*Termální rovnice v proměnných p,T

:$\rho c_p \frac{\partial T}{\partial t} = \nabla \cdot (k\cdot \nabla T)-\rho c_p v\cdot \nabla T - \alpha T(\frac{\partial p}{\partial t}+v \cdot \nabla p)+\sigma: \nabla v+H$ kde

:$c_p = c_V(1+\gamma \alpha T)$ a rovnice kontinuity

:$\nabla \cdot v = -\frac{1}{\rho}\frac{D\rho}{Dt}=-K_T^{-1}\frac{Dp}{Dt}+\alpha \frac{DT}{Dt}$

*Termální rovnice v kontinuu s dominantním hydrostatickým tlakem

platí :$\nabla p_0 = \rho_0 g_0 - \rho_0\Omega \times(\Omega\times r)$

nechť tedy :$\frac{\partial p}{\partial t}+v cdot \nabla p = -v_r\rho g$

v radiální složce, kde g je :$g=|\vec g| = \vec g_0-\Omega\times(\Omega\times r)$

pak dostaneme obvyklou termální rovnici :$\rho c_p \frac{\partial T}{\partial t} = \nabla \cdot (k\cdot \nabla T)-\rho c_p v\cdot \nabla T - \rho v_r\alpha Tg+\sigma: \nabla v+H$

kde figuruje adiabatický gradient / ohřívání :$(\frac{\partial T}{\partial r})_s = -\frac{\alpha Tg}{c_p}$