The base driving equations of a pipe flow is:

(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)	$\begin{array}{l}\displaystyle j_m(l) = j_m = \rm const\end{array}$

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

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: $\begin{array}{l}{\bf e}_u = { \| {\bf u} \|} ^{-1} \, {\bf u}\end{array}$
$\begin{array}{l}j_m\end{array}$	fluid mass flux
$\begin{array}{l}\dot m\end{array}$	mass flowrate
$\begin{array}{l}g\end{array}$	standard gravity constant

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

(5)	$\begin{array}{l}\displaystyle q(l) = \frac{\rho_s \cdot q_s}{\rho}\end{array}$

(6)	$\begin{array}{l}\displaystyle u(l) = \frac{\rho_s \cdot q_s}{\rho(p) \cdot A}\end{array}$

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