Benutzer:Tiddens

$-\Pi _{yx}=\eta ({\frac {\partial v_{x}}{\partial y}}+{\frac {\partial v_{y}}{\partial x}})$.

Gravitationskraft $F_{g}=\int _{V}{\vec {g}}\rho dV$.

Hydrostatische Druckkraft $F_{p}=-\int _{\partial V}pdA=-\int _{V}\Delta pdV$.

Reibungskraft $F_{v}=-\int _{\partial V}\Pi _{ij}dA=-\int _{V}\Delta \Pi _{ij}dV=\eta \int _{V}\nabla ^{2}{\vec {v}}dV$.

Massenerhaltung ${\frac {\partial }{\partial t}}m={\frac {\partial }{\partial t}}\int _{V}\rho dV=-\int _{\partial V}\rho {\vec {v}}d{\vec {A}}=-\int _{V}\Delta (\rho {\vec {v}})dV\Rightarrow {\frac {\partial \rho }{\partial t}}+\nabla \cdot (\rho {\vec {v}})=0$ mit rho const$\nabla \cdot {\vec {v}}=0$

Impulserhaltung ${\dot {p}}={\dot {m{\vec {v}}}}=m{\dot {\vec {v}}}+{\vec {v}}{\dot {m}}=m{\dot {\vec {v}}}=\int _{V}\rho dV{\frac {d{\vec {v}}}{dt}}=F={\vec {F}}_{g}+{\vec {F}}_{p}+{\vec {F}}_{v}=\int _{V}(\rho {\vec {g}}-\nabla p+\eta \nabla ^{2}{\vec {v}})dV\Rightarrow$

Navier-Stokes-Gleichung $\rho {\frac {d{\vec {v}}}{dt}}=\rho {\vec {g}}-\nabla p+\eta \nabla ^{2}{\vec {v}}$

Aus $df={\frac {\partial f}{\partial t}}dt+{\frac {\partial f}{\partial r}}dr\Rightarrow {\frac {df}{dt}}={\frac {\partial f}{\partial t}}+{\frac {\partial f}{\partial r}}{\frac {dr}{dt}}\Rightarrow {\frac {df}{dt}}={\frac {\partial f}{\partial t}}+v{\frac {\partial f}{\partial r}}$ folgt:

Navier-Stokes-Gleichung $\rho {\frac {\partial {\vec {v}}}{\partial t}}+\rho ({\vec {v}}\cdot \nabla ){\vec {v}}=\rho {\vec {g}}-\nabla p+\eta \nabla ^{2}{\vec {v}}$

Falls (kompressibles Fluid) $\nabla \cdot {\vec {v}}\neq 0\Rightarrow \eta \nabla ^{2}{\vec {v}}\rightarrow \eta \nabla ^{2}{\vec {v}}+{\frac {1}{3}}\eta \nabla (\nabla \cdot {\vec {v}})$

laminarer Fluss $\rho {\frac {\partial {\vec {v}}}{\partial t}}=\rho {\vec {g}}-\nabla p+\eta \nabla ^{2}{\vec {v}}$

stationärer Fluss $0=\rho {\vec {g}}-\nabla p+\eta \nabla ^{2}{\vec {v}}$

Porosität $\Phi ={\frac {V_{p}}{V}}$

Trockendichte $\rho _{b}=\rho _{s}(1-{\frac {V_{V}}{V}})=\rho _{s}(1-\Phi )$

Volumetrischer Fluidgehalt $\theta ={\frac {V_{Fluid}}{V}}$

gravimetrischer Fluidgehalt $\theta ={\frac {M_{Fluid}}{M_{Matrix}}}$

Sättigung: $S={\frac {V_{w}}{V_{p}}}={\frac {\theta }{\Phi }}$

Potentielle Energie eines Fluidelements: $\Psi =p-p_{0}+\rho g(z-z_{0})=\Psi _{p}+\Psi _{g}$

Potentielle Energie pro Einheitsgewicht $h:=\Psi {\frac {V}{gm}}={\frac {\Psi }{\rho g}}={\frac {p-p_{0}}{\rho g}}+(z-z_{0})$

Darcy Gesetz: ${\frac {Q}{A}}=K{\frac {h_{1}-h_{2}}{L}}=q=-K\nabla h$

laminarer, langsamer, stationärer Fluss mit g=0 $\nabla p=\eta \nabla ^{2}{\vec {v}}\Rightarrow \nabla ^{2}{\vec {v}}^{\mu }={\frac {1}{\eta }}\nabla p^{\mu }$

$\rightarrow$(Haagen-Poisuille)$\Rightarrow {\vec {v}}^{\mu }=-{\frac {k^{\mu }}{\eta }}\nabla p^{\nu }$

makroskopische Betrachtung (Darcy-Gesetz für Newtonsche Fluide) ${\vec {q}}A=<{\vec {v}}^{\mu }>A_{W}$ mit ${\frac {A_{W}}{A}}={\frac {V_{W}}{V}}=\theta$$\Rightarrow {\vec {q}}=\theta <{\vec {v}}^{\mu }>=-{\frac {\theta }{\eta }}<k^{\mu }\nabla p^{\mu }>$and$\theta <k^{\mu }\nabla p^{\mu }>\rightarrow k\nabla p$ and $<{\vec {v}}^{\mu }>\rightarrow {\vec {v}}$$\Rightarrow q=-{\frac {k}{\eta }}\nabla p$

${\vec {q}}=\theta {\vec {v}}\rightarrow _{(gesaettigt)}{\vec {q}}=\Phi {\vec {v}}$

allgemeine Gleichung für g≠0${\vec {q}}=-{\frac {k}{\eta }}(\nabla p-\rho {\vec {g}})=_{g=const}-{\frac {k}{\eta }}\nabla (p+\rho gz)=-{\frac {k}{\eta }}\nabla \Psi =-{\frac {k}{\eta }}\nabla (h\rho g)=-{\frac {k\rho g}{\eta }}\nabla (h)$

Reynoldszahl: ${\mathit {Re}}={\frac {\varrho \cdot v\cdot d}{\eta }}$

Massenkontinuitätsgleichung ${\frac {\partial }{\partial t}}(V_{W}\rho )={\frac {\partial }{\partial t}}\int _{V}\rho dV=-\int \nabla (\rho {\vec {v}}^{\mu })dV=-\nabla \cdot \int \rho {\vec {v}}^{\mu }dV=-\nabla V_{W}\rho <{\vec {v}}^{\mu }>\Rightarrow {\frac {\partial }{\partial t}}(\theta \rho )+\nabla \cdot (\rho q)=0$

für gesättigte Verhältnisse ${\frac {\partial }{\partial t}}(\theta \rho )=_{\rho =const}=\rho {\frac {\partial \theta (p)}{\partial t}}=\rho {\frac {d\theta }{dp}}{\frac {\partial p}{\partial t}}$ wobei ${\frac {d\theta }{dp}}=c_{W}(p)$: Wasserkapazitätsfunktion; integriert man diese erhält man die Wasserretentionskurve $\sigma (p)$

für ungesättigte Verhältnisse ${\frac {\partial }{\partial t}}(\theta \rho )=\rho (\theta \kappa _{p,W}+(1-\theta )\kappa _{p,s}){\frac {\partial p}{\partial t}}$ wobei $\theta \kappa _{p,W}+(1-\theta )\kappa _{p,s}=s$; $\kappa _{p,W}$:Wasser-Kompressibilitätskoeffizient; $\kappa _{p,s}$:Matrix-Kompressibilitätskoeffizient; $\Rightarrow {\frac {\partial }{\partial t}}(\theta \rho )=\rho s{\frac {\partial p}{\partial t}}$

Richardsgleichung ${\frac {\partial }{\partial t}}(\theta \rho )+\nabla \cdot (\rho q)=0$ und ${\vec {q}}=-K\nabla h$ und ${\vec {q}}=-{\frac {k}{\eta }}(\nabla p-\rho q)$ $\Rightarrow \nabla \cdot (\rho {\frac {k}{\eta }}(\nabla p-\rho {\vec {g}}))={\frac {\partial }{\partial t}}(\theta \rho )$ und $\nabla \cdot (K\nabla h)={\frac {\partial \theta }{\partial t}}$

Spezialfälle der Richardsgleichung

für gespannten Aquifer: Kontinuitätsgleichgung mit s $\nabla \cdot (K\nabla h)=s_{s}{\frac {\partial h}{\partial t}}$ mit $s_{s}:=s\rho g$

mit einem zusätzlichen Quell- bzw. Senkenterm $\nabla \cdot (K\nabla h)={\frac {\partial \theta }{\partial t}}+\gamma$

für Anisotropie $K\rightarrow {\vec {\vec {K}}}$ d.h. $\nabla \cdot ({\vec {\vec {K}}}\nabla h)=s_{s}{\frac {\partial h}{\partial t}}$

stationäre Bedingungen, isotroper aquifer $\nabla \cdot (K\nabla )=0$

homogener, aber anisotroper Aquifer ${\vec {\vec {K}}}\rightarrow K_{x},K_{y},K_{z}$ d.h. $K_{x}{\frac {\partial ^{2}h}{\partial x^{2}}}+K_{y}{\frac {\partial ^{2}h}{\partial y^{2}}}+K_{z}{\frac {\partial ^{2}h}{\partial z^{2}}}=s_{s}{\frac {\partial h}{\partial t}}$

homogener, isotroper Aquifer ${\vec {\vec {K}}}\rightarrow K$ d.h. ${\frac {K}{s_{s}}}({\frac {\partial ^{2}h}{\partial x^{2}}}+{\frac {\partial ^{2}h}{\partial y^{2}}}+{\frac {\partial ^{2}h}{\partial z^{2}}})={\frac {K}{s_{s}}}\nabla ^{2}h={\frac {\partial h}{\partial t}}$

homogener, isotroper Aquifer mit stationären Bedingungen $\nabla ^{2}h=0$

homogener, isotroper Aquifer mit stationären Bedingungen mit Quell-Term (Laplace Gleichung) $\nabla ^{2}h=\gamma$

Laplace Glichung (Oberflächenspannung): Fehler beim Parsen (Unbekannte Funktion „\laplace“): {\displaystyle \laplace p=\frac{2\sigma}{r}}

Kappilares Steigen $H={\frac {2\sigma \cos \gamma }{\rho gR}}$