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Steady-State flowQuasi-isothermal flow

LaTeX Math Inline
body--uriencoded--\displaystyle \frac%7B\partial p%7D%7B\partial t%7D = 0

LaTeX Math Inline
body--uriencoded--\displaystyle \frac%7B\partial T%7D%7B\partial t%7D =0 \rightarrow T(t,l) = T(l)

Homogenous flow

Constant cross-section pipe area

LaTeX Math Inline
bodyA
along hole

LaTeX Math Inline
body--uriencoded--\displaystyle \frac%7B\partial p%7D%7B\partial \tau_x%7D =\frac%7B\partial p%7D%7B\partial \tau_y%7D =0 \rightarrow p(t, \tau_x,\tau_y,l) = p(l)

LaTeX Math Inline
bodyA(l) = A = \rm const

Constant inclinationConstant friction along hole

LaTeX Math Inline
body--uriencoded--\displaystyle \theta(l) = \theta = %7B\rm const%7D \rightarrow \cos \theta = \frac%7Bdz%7D%7Bdl%7D = %7B\rm const%7D

LaTeX Math Inline
bodyf(l) = f = \rm const

Linear density

LaTeX Math Inline
body--uriencoded--\rho = \rho%5e* \cdot ( 1 + c%5e* \cdot p) \Rightarrow c%5e* \, \rho%5e* = c_0 \, \rho_0
 which leads to 
LaTeX Math Inline
body--uriencoded--\displaystyle c(p) = \frac%7Bc%5e*%7D%7B1 + c%5e* \cdot p %7D
 and 
LaTeX Math Inline
body--uriencoded--c%5e* \, \rho%5e* = c_0 \, \rho_0


Equations

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Pressure profile along the pipe


LaTeX Math Block
anchorPressureProfile
alignmentleft
L = \frac{1}{2 \, G \, c^*  \rho^*}  \cdot \ln \frac{G \, \rho^2-F}{G \, \rho_0^2-F}
-\frac{d}{f} \cdot \ln \frac{F/\rho^2 - G}{ F/\rho_0^2-G}



LaTeX Math Block
anchor1
alignmentleft
 \cos \theta \neq 0



LaTeX Math Block
anchorG0
alignmentleft
L = \frac{1}{2F\, c^* \rho^*} \cdot (\rho_0^2 - \rho^2)
 - \frac{2d}{f} \cdot \ln \frac{\rho_0}{\rho}



LaTeX Math Block
anchor1
alignmentleft
 \cos \theta = 0


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