*ZZHmoty:
:\frac{D}{Dt}\int_V \rho dV = 0
:\frac{\partial \rho}{\partial t}+\Delta\cdot(\rho v) = 0
a
:\[\rho(v-\nu)]_-^+\cdot n =0
*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
\[\rho v(v-\nu)-\tau]_-^+\cdot n = 0
*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
:\frac{D}{Dt}\int_V (\rho\epsilon+1/2 \rho v \cdot v)dV = \int_S v\cdot \tau \cdot n dS+\int_V \rho g \cdot v dV-2\int_V \rho(\Omega\times v)\cdot v dV - \int_V \rho(\Omega\times(\Omega\times r))\cdot v dV - \int_S \q\cdot n dS + \int_V HdV
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
:\[k\cdot\nabla T]_-^+\cdot n = -\[v\cdot \tau]_-^+\cdot n+\[(\rho\epsilon+1/2\rho v\cdot v)(v-\nu)]_-^+\cdot n
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 \mu = T(\frac{\partial s}{\partial V})_T = T (\dfrac{\partial p}{\partial T})_V = T \[\frac{1}{V}(\frac{\partial V}{\partial T})_p]\[-V(\frac{\partial p}{\partial V})_T]=\alpha K_T
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}