...
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| LaTeX Math Block |
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| q_0^2 = \frac{2 d A^2 G}{f} \cdot \left[
1 + \frac{ (\rho/\rho_0)^2 -1}{1- (\rho_0/\rho)^{\frac{2}{n-1}} \cdot
\exp \left( \frac{fL/d}{ n-1} \right)}
\right]
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| | LaTeX Math Block |
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|
n = \frac{f \, L^*}{2 \, d}
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| LaTeX Math Block |
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|
L^* = \frac{1}{G \, c^* \, \rho^*}
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| LaTeX Math Block |
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| q_0^2 = \frac{A^2}{c^* \rho^*} \cdot \frac{1 - (\rho/\rho_0)^2}{2 \ln (\rho_0/\rho) + fL/d} |
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where
| LaTeX Math Inline |
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body | --uriencoded--\displaystyle
| LaTeX Math Block |
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| n = \frac{f \, L^*}{2 \, d}
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| LaTeX Math Block |
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| L^* = \frac{1}{G \, c^* \, \rho^*}
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| LaTeX Math Block |
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| anchor | rho_rho0 |
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| alignment | left |
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|
frac%7B1c%5e0%7D%7B1+c%5e* p%7Dwith the following asymptotes:
Low compressible fluids: | LaTeX Math Inline |
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| body | --uriencoded--c%5e* p \ll 1, \, \, c%5e* p_0 \ll 1 |
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| High compressible fluids: | LaTeX Math Inline |
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| body | --uriencoded--c%5e* p \gg 1, \, \, c%5e* p_0 \gg 1 |
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| LaTeX Math Inline |
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| body | --uriencoded--\displaystyle \rho_0/\rho = c%5e* \cdot (p_0-p) |
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| | LaTeX Math Inline |
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| body | \displaystyle \rho_0/\rho = p_0/p |
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Approximations
...
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| LaTeX Math Block |
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| q_0^2 = \frac{A^2}{c^* \rho^*} \cdot \frac{1 - (\rho/\rho_0)^2 \cdot \exp \left( -L/ L^* \right)}{2 \ln (\rho_0/\rho) + fL/d \cdot (1- \exp \left( - L/ L^* \right))/(L/L^*)} |
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| LaTeX Math Block |
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| \dot m^2 = \frac{A^2}{c^* \rho^*} \cdot \frac{\rho_0^2 - \rho^2 \cdot \exp \left( -L/ L^* \right)}{2 \ln (\rho_0/\rho) + fL/d \cdot (1- \exp \left( - L/ L^* \right))/(L/L^*)} |
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| LaTeX Math Block |
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| anchor | static |
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| alignment | left |
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| \rho = \rho_0 \, \exp (c^* \rho^* \, G \, L) \cdot \sqrt{ 1 - \frac{f}{2d} \cdot \frac{j_m^2}{G \, \rho_0^2} \cdot ( 1 - \exp(-2 \, c^* \rho^* \, G \, L))} |
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| LaTeX Math Block |
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| anchor | static |
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| alignment | left |
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| p(L) = \frac{1}{c^*} \cdot \left[
-1 + (1+c^* p_0) \cdot \exp (c^* \rho^* \, G \, L) \cdot \sqrt{ 1 - \frac{f}{2d} \cdot \frac{j_m^2}{G \rho_0^2} \cdot \big(1 - \exp (-2 \, c^* \rho^* \, G \, L) \big) }
\right] |
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| Pressure Profile in GC-proxy static fluid column @model | | LaTeX Math Inline |
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| body | \dot m = 0, \, q_0 = 0 |
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(no flow) |
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| anchor | static |
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| alignment | left |
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| \rho = \rho_0 \, \exp (L/L^*) |
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| LaTeX Math Block |
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| anchor | static |
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| alignment | left |
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| p(L) = \frac{-1 + (1+c^* \, p_0) \cdot \exp(L/L^*)}{c^*} |
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...
