*ZZHmoty: :DDt∫VρdV=0\frac{D}{Dt}\int_V \rho dV = 0

:∂ρ∂t+Δ⋅(ρv)=0\frac{\partial \rho}{\partial t}+\Delta\cdot(\rho v) = 0 a

:

ParseError: KaTeX parse error: Undefined control sequence: \[ at position 1: \̲[̲\rho(v-\nu)]_-^…

*ZZHybnosti :DDt∫VρvdV=∫VρgdV−2∫VρΩ×vdV−∫VρΩ×(Ω×r)dV+∫Sτ⋅ndS\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

:∇⋅τT+ρg−2ρΩ×v−ρΩ×(Ω×r)=ρDvDt=ρ(∂v∂t+v⋅∇v)\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

ParseError: KaTeX parse error: Undefined control sequence: \[ at position 1: \̲[̲\rho v(v-\nu)-\…

*ZZMomentu hybnosti :DDt∫Vr×ρvdV=∫Vr×ρgdV−2∫Vr×(ρΩ×v)dV−∫Vr×(ρΩ×(Ω×r))dV+∫Sr×τ⋅ndS\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×τ=0I\times \tau = 0 <=> τ=τT\tau = \tau^T

*ZZEnergie :

ParseError: KaTeX parse error: Undefined control sequence: \q at position 215: … v dV - \int_S \̲q̲\cdot n dS + \i…

Gibbsova relace :ρTDsDt=ρDϵDt+p∇⋅v\rho T\frac{Ds}{Dt}=\rho\frac{D\epsilon}{Dt}+p\nabla\cdot v

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

a :

ParseError: KaTeX parse error: Undefined control sequence: \[ at position 1: \̲[̲k\cdot\nabla T]…

Dále tepelná rovnice ve stavových proměnných

:ρTDsDt=ρT∂s∂TVDTDt+ρT∂s∂VTDVDt=ρcVDTDt+μ∇⋅v\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

ParseError: KaTeX parse error: Undefined control sequence: \[ at position 85: …tial T})_V = T \̲[̲\frac{1}{V}(\fr…

kde termální expanze α=−1p(∂p∂T)p\alpha = -\frac{1}{p}(\frac{\partial p}{\partial T})_p a izotermální objemový modul KT=−V(∂p∂V)TK_T = -V(\frac{\partial p}{\partial V})_T

Pak získáme μ=TαKT=ρcVTγ\mu=T\alpha K_T = \rho c_V T \gamma pro Grüneisenův parametr γ=αKT/ρcV\gamma = \alpha K_T / \rho c_V

a z toho plyne termální rovnice :ρcV∂T∂t=∇⋅(k⋅∇T)−ρcVv⋅∇T−ρcVTγ∇⋅v+σ:∇v+H\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

:ρcp∂T∂t=∇⋅(k⋅∇T)−ρcpv⋅∇T−αT(∂p∂t+v⋅∇p)+σ:∇v+H\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

:cp=cV(1+γαT)c_p = c_V(1+\gamma \alpha T) a rovnice kontinuity

:∇⋅v=−1ρDρDt=−KT−1DpDt+αDTDt\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 :∇p0=ρ0g0−ρ0Ω×(Ω×r)\nabla p_0 = \rho_0 g_0 - \rho_0\Omega \times(\Omega\times r)

nechť tedy :∂p∂t+vcdot∇p=−vrρg\frac{\partial p}{\partial t}+v cdot \nabla p = -v_r\rho g

v radiální složce, kde g je :g=∣g⃗∣=g⃗0−Ω×(Ω×r)g=|\vec g| = \vec g_0-\Omega\times(\Omega\times r)

pak dostaneme obvyklou termální rovnici :ρcp∂T∂t=∇⋅(k⋅∇T)−ρcpv⋅∇T−ρvrαTg+σ:∇v+H\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 :(∂T∂r)s=−αTgcp(\frac{\partial T}{\partial r})_s = -\frac{\alpha Tg}{c_p}