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LaTeX Math Block
anchorNS6JL
alignmentleft
G = g \cdot \cos \theta = \rm const

and rewrite the equation 

LaTeX Math Block Reference
anchorPP
 as:

LaTeX Math Block
anchorPP1
alignmentleft
\frac{\left[\rho(p) -  j_m^2 \cdot c(p)   \right] \, dp}{\rho^2(p) \, G  - \frac{ j_m^2 }{2 d} \cdot  f(p)} = dl
 

The integration of the left side of 

LaTeX Math Block Reference
anchorPP1
 leads to:

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

where

LaTeX Math Inline
body--uriencoded--\displaystyle F = \frac%7B j_m%5e2 %7D%7B2 d%7D \cdot f(p)

See also

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Physics / Fluid Dynamics / Pipe Flow Dynamics / Pipe Flow Simulation / Pressure Profile in Homogeneous Steady-State Pipe Flow @model / Pressure Profile in G-Proxy Pipe Flow @model

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