The base driving equations of a pipe flow are:

(1)	$\begin{array}{l}\displaystyle \frac{d p}{d l} = -\rho \, u \, \frac{d u}{d l} + \rho \, g \, \cos \theta + f_{\rm cnt, \, l}\end{array}$

(2)	$\begin{array}{l}\displaystyle j_m(l) = j_m = \rho(l) \cdot u = \rm const\end{array}$

(3)	$\begin{array}{l}\displaystyle \rho = \rho(p, T)\end{array}$

(4)	$\begin{array}{l}\displaystyle f_{\rm cnt, l} = - f \cdot \frac{ \rho \, u^2 \, }{2 d}\end{array}$

where

$\begin{array}{l}l\end{array}$	distance along the fluid flow streamline
$\begin{array}{l}\theta(l)\end{array}$	inclinational deviation, $\begin{array}{l}\displaystyle \cos \theta = dz/dl\end{array}$
$\begin{array}{l}z(l)\end{array}$	elevation along the 1D flow trajectory
$\begin{array}{l}T(l)\end{array}$	fluid temperature
$\begin{array}{l}p(l)\end{array}$	fluid pressure
$\begin{array}{l}\rho(l)\end{array}$	fluid density
$\begin{array}{l}{\bf u}(l)\end{array}$	fluid velocity vector
$\begin{array}{l}u(l)\end{array}$	superficial velocity of the pipe flow
$\begin{array}{l}{\bf f}_{\rm cnt}(l)\end{array}$	volumetric density of all contact forces exerted on fluid body
$\begin{array}{l}f_{\rm cnt, l}(l) = {\bf e}_u \cdot {\bf f}_{\rm cnt}\end{array}$	projection of $\begin{array}{l}{\bf f}_{\rm cnt}\end{array}$ onto the unit fluid velocity vector: $\begin{array}{l}{\bf e}_u = { \| {\bf u} \|} ^{-1} \, {\bf u}\end{array}$
$\begin{array}{l}j_m = \rho(l) \cdot u(l)\end{array}$	fluid mass flux
$\begin{array}{l}\dot m\end{array}$	mass flowrate
$\begin{array}{l}g\end{array}$	standard gravity constant

Substituting (2) and (4) into (1):

(5)	$\begin{array}{l}\displaystyle \frac{d p}{d l} = -j_m \cdot \frac{d}{d l} \left( \frac{j_m}{\rho} \right) + \rho \, g \, \cos \theta - f \cdot \frac{ \rho \, }{2 d} \cdot \left( \frac{j_m}{\rho} \right)^2\end{array}$

(6)	$\begin{array}{l}\displaystyle \frac{d p}{d l} = j^2_m \cdot \frac{1}{\rho^2} \frac{d \rho}{dl} + \rho \, g \, \cos \theta - \frac{j_m^2}{2 d} \cdot \frac{f}{\rho}\end{array}$

(7)	$\begin{array}{l}\displaystyle \frac{d p}{d l} = j^2_m \cdot \frac{1}{\rho^2} \frac{d \rho}{dp} \cdot \frac{d p}{dl} + \rho \, g \, \cos \theta - \frac{j_m^2}{2 d} \cdot \frac{f}{\rho}\end{array}$

(8)	$\begin{array}{l}\displaystyle \frac{d p}{d l} = j^2_m \cdot \frac{1}{\rho} \cdot c \cdot \frac{d p}{dl} + \rho \, g \, \cos \theta - \frac{j_m^2}{2 d} \cdot \frac{f}{\rho}\end{array}$

and finally

(9)	$\begin{array}{l}\displaystyle \left( 1 - j_m^2 \cdot \frac{c}{\rho} \right ) \frac{dp}{dl} = \rho \, g \, \cos \theta - \frac{j_m^2 }{2 d} \cdot \frac{f}{\rho}\end{array}$

Alternative forms

(10)	$\begin{array}{l}\displaystyle \left[ \rho -j_m^2 \, c \right] \cdot \frac{d p}{dl} = \rho^2 \, g \, \cos \theta - \frac{j_m^2 }{2d} \cdot f(\rho)\end{array}$

(11)	$\begin{array}{l}\displaystyle \left[ \frac{1}{c} - \frac{j_m^2}{\rho} \right] \cdot \frac{d \rho}{dl} = \rho^2 \, g \, \cos \theta - \frac{j_m^2 }{2d} \cdot f(\rho)\end{array}$

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